diff --git a/-9FRT4oBgHgl3EQfsTcg/content/tmp_files/2301.13623v1.pdf.txt b/-9FRT4oBgHgl3EQfsTcg/content/tmp_files/2301.13623v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..14d223ea33be99a651a55746e908e9675685c252 --- /dev/null +++ b/-9FRT4oBgHgl3EQfsTcg/content/tmp_files/2301.13623v1.pdf.txt @@ -0,0 +1,1097 @@ +arXiv:2301.13623v1 [hep-th] 31 Jan 2023 +Unimodular Gravity in Covariant Formalism +J. Klusoň† and B. Matouš†‡ 1 +† Department of Theoretical Physics and Astrophysics, Faculty of Science, +Masaryk University, Kotlářská 2, 611 37, Brno, Czech Republic +‡ North-Bohemian Observatory and Planetarium in Teplice, +Koperníkova 3062, 415 01, Teplice, Czech Republic +Abstract +In this short note we study unimodular gravity in Weyl-De Donder +formalism. We find corresponding Hamiltonian and study consequence +of the unimodular constraint on the conjugate covariant momenta. We +also find covariant Hamiltonian for Henneaux-Teitelboim unimodular +action and study corresponding equations of motion. +1 +Introduction and Summary +Unimodular gravity was firstly introduced by A. Einstein in his paper [3] +published in 1916. In this work the unimodular constraint √−g = 1 was +used as gauge fixing condition of general diffeomorphism in order to sim- +plify calculations. +Then it was shown in [1, 2] that imposing this condi- +tion before the variation of Einstein-Hilbert action leads to the traceless +equations of motion. +As we review below these equations of motion are +classically equivalent to the general relativity equations of motion with cru- +cial difference that the cosmological constant appears as integration con- +stant rather than true cosmological constant. +This fact brings new hope +how to solve cosmological constant problem which was however questioned +in [4], 2 where it was argued that quantum corrections make the cosmo- +logical constant ultraviolet sensitive in unimodular gravity as well. On the +other hand it is important to stress that no definitive conclusions have been +reached yet regarding this problem and unimodular gravity is still very inten- +sively studied, for some works devoted to unimodular gravity, see for example +[7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 22, 23, 24, 25, 26]. +1Email addresses: J. Klusoň: klu@physics.muni.cz, B. Matouš: bmatous@mail.muni.cz +2For review of unimodular gravity, see for example [5, 6]. +1 + +One of the most interesting aspects of unimodular gravity is the number +of physical degrees of freedom. Naively, unimodular constraint √−g = 1 re- +duces the number of independent components of metric to nine which could +suggest that the number of physical degrees of freedom is less than in general +relativity. On the other hand unimodular gravity is invariant under restricted +diffeomorphism. Taking these two aspects together we find that the num- +ber of local physical degrees of freedom is the same as in ordinary general +relativity. This fact was proved with the help of the Hamiltonian analysis +of unimodular gravity performed in [16, 17, 18, 19, 20, 21]. On the other +hand as was shown in these papers standard analysis of unimodular gravity +based on D + 1 splitting of target space-time is rather non-trivial and shown +complexity of the canonical analysis of systems with constraints. +Then one could ask the question how unimodular gravity could be de- +scribed in covariant canonical formalism that is known as Weyl-De Donder +theory [27, 28]. The key point of this formulation is that we treat all partial +derivatives as equivalent when we define conjugate momenta. For example, +if we have scalar field φ with Lagrangian density in D +1 dimensional space- +time equal to L = −1 +2ηab∂aφ∂bφ − V (φ), we define the conjugate momentum +as 3 +πa = ∂L +∂∂aφ = −ηab∂bφ . +Then covariant canonical Hamiltonian density is defined as +H = πa∂aφ − L = −1 +2πaηabπb + V (φ) . +Clearly such a form of Hamiltonian density preserves diffeomorphism invari- +ance of the theory. This approach is known as multisymplectic field theory, +see for example [29, 30, 31], for review, see [32] and for recent interesting +application of this formalism in string theory, see [33, 34]. +It is clear that such covariant canonical formalism is especially suitable +for manifestly covariant theories as for example general relativity. In fact, +covariant canonical formalism of general relativity was found long time ago +by P. Hořava [35]. This analysis was recently generalized to the case of F(R) +gravity in [37] and further elaborated in [38]. +In this paper we apply this formalism for unimodular theory of gravity in +D + 1 dimensions. This is non-trivial task due to the well known complex- +ity of canonical analysis of unimodular gravity in non-covariant formalism. +3We define ηab = diag(−1, 1, . . ., 1), a, b = 0, 1, . . ., D. +2 + +Further, it is also very interesting to study this system since it contains pri- +mary unimodular constraint and it is non-trivial task how to deal with such +systems in covariant canonical formalism. In more details, we include this +primary constraint to the action with corresponding Lagrange multiplier. +Then we derive corresponding equations of motion. Using these equations +of motion we find that the unimodular constraint implies another constraint +on the canonical conjugate momenta. Then we show that this constraint is +equivalent to the vanishing of the trace of the Christoffel symbols which is +characteristic property of unimodular theory of gravity [10]. This is nice and +non-trivial result. On the other hand the Lagrange multiplier corresponding +to the primary constraint cannot be determined as in non-covariant canon- +ical formalism by imposing condition of the preservation of the secondary +constraint due to the fact that the equations of motion for conjugate mo- +menta are in the form of the divergence of these momenta. For that reason +we determine this constraint in the same way as in the Lagrangian formalism +when we calculate the trace of the equations of motion. As a result we obtain +equations of motion that are traceless and that do not depend on the cos- +mological constant which is in agreement with the Lagrangian formulation +of unimodular gravity. +As the second step in our analysis we find covariant canonical formula- +tion of Henneaux-Teitelboim formulation of unimodular gravity [16]. In this +case we again identify covariant Hamiltonian together with set of primary +constraints. Then we consider canonical form of the action and determine +corresponding equations of motion. Solving these equations of motion we +find that Lagrange multiplier is integration constant. In this case we repro- +duce results well known from Lagrangian analysis. However we mean that +this is nice and interesting application of the covariant canonical analysis to +the constraint systems. +Let us outline our results and suggest possible extension of this work. We +found covariant Hamiltonian formalism for unimodular gravity. First of all +we determined covariant Hamiltonian for general relativity action in D + 1 +dimensions where we again introduced variable f ab = √−ggab. At this place +we would like to stress an importance of this result since it was not apri- +ori known whether f ab is suitable for formulation of gravity in space-time of +dimension different from 4. Then we imposed unimodular constraint using +Lagrange multiplier method and then we studied corresponding equations +of motion. We found that the consistency of the theory demands that the +trace of conjugate momenta is zero. Then we showed that this is character- +3 + +istic property of unimodular gravity when we pass to Lagrangian formalism. +Final we found covariant Hamiltonian for Henneaux-Teltelboim formulation +of unimodular gravity. We identified primary constraints of the theory and +then we studied equations of motion that follow from canonical form of the +action. We showed that they precisely reproduce Lagrangian equations of +motion that is nice consistency check of the covariant canonical formalism. +We mean that the analysis presented in this paper suggests that covariant +Hamiltonian formalism is very close to Lagrangian formalism and in some +situations the covariant Hamiltonian formalism is more suitable than La- +grangian one, as for example study of thermodynamics properties of horizon +[36]. +It is also clear that there are more systems that could be analysed with +the help of covariant canonical formalism. One possibility is to study Weyl +invariant gravity in this formalism. Another possibility would be to perform +analysis of theories of gravity with higher derivatives where the classical +canonical analysis is very complicated, see for example [40]. +We hope to +return to these problems in future. +This paper is organized as follows. +In the next section (2) we review +properties of unimodular gravity.Then in section (3) we proceed to the co- +variant canonical formulation of this theory. Finally in section (4) we perform +covariant canonical formulation of Henneaux-Teltelboim unimodular gravity. +2 +Brief Review of Unimodular Gravity +In this section we review basic facts about unimodular gravity. For recent +very nice and more detailed review, see for example [5, 6]. +Unimodular +gravity is theory with the constraint √−g = 1. Clearly such a condition +has a consequence on allowed differomorphism transformation. In fact, let +us consider general transformation of coordinates +x′a = xa + ξa(x) +(1) +that implies inverse relation +xa = x′a − ξa(x) ≈ x′a − ξa(x′) + O(ξ2) , +(2) +where a, b, c = 0, 1, . . . , D. Under these transformation the metric gab trans- +form as +g′ +ab(x) = gab(x) − ∂cgab(x)ξc(x) − gac(x)∂bξc(x) − ∂aξc(x)gcb(x) +(3) +4 + +that implies following variation of metric +δgab(x) = g′ +ab(x) − gab(x) = −gac∂bxc − ∂aξcgcb − ∂cgabξc +so that the variation of the square root of the determinant of metric is equal +to +δ +� +− det g = −(2∂aξa − ∂cgabgbaξc) +� +− det g . +(4) +In case of unimodular gravity this variation should vanish and hence we +obtain following condition on ξa in the form +∇aξa = ∂aξa + 1 +2gac∂dgcaξd = 0 . +(5) +The most straightforward way how to find an action for unimodular gravity is +to consider standard Einstein-Hilbert action with an unimodular constraint +added +S = +1 +16π +� +dD+1x[√−g(R − 2¯Λ) + Λ(√−g − 1)] + Smatt , +(6) +where Λ is Lagrange multiplier whose variation ensures unimodular condition +and where ¯Λ is constant. +Performing variation of the action (6) with respect to gab we obtain fol- +lowing equations of motion +1 +16π(Rab − 1 +2gab(R − 2¯Λ + Λ)) = Tab , +(7) +where Tab is matter stress energy tensor defined as +Tab = − +1 +√−g +δSmatt +δgab +. +(8) +The crucial point is that Λ is Lagrange multiplier that should be determined +as a consequence of the equations of motion. To do this we perform the trace +of the equation (7) to express Λ as +Λ = (1 − D) +1 + D R − +32π +D + 1T + 2¯Λ , +T ≡ gabTab . +(9) +5 + +Inserting this result into (7) we obtain +Rab − +1 +D + 1gabR = 16π(Tab − +1 +D + 1gabT) . +(10) +These equations of motion are trace-free and also most importantly they do +not contain any information about cosmological constant ¯Λ. +It is important to stress that even equations of motion of general relativ- +ity without unimodular constraint imposed split into 9 trace-free equations +of motion and one additional one. +To see this consider general relativity +equations of motion +Rab − 1 +2gab(R − 2¯Λ) = 16πTab . +(11) +Taking the trace of this equation we can express R as +R = +2 +1 − D(16πT − (D + 1)¯Λ) . +(12) +Note that with the help of this equation we can rewrite (11) into trace-free +form +Rab − +1 +D + 1Rgab = 16π(Tab − +1 +D + 1Tgab) . +(13) +However we should again stress that (12) determines R as function of trace of +matter stress energy tensor and true cosmological constant term in Einstein- +Hilbert action while in case of unimodular gravity we express Λ-which is +Lagrange multiplier and not constant, as function of R, T and ¯Λ, as follows +from equation (9). +In order to check equivalence between unimodular gravity and ordinary +general relativity we should be able to reproduce equation (12) in case of +unimodular gravity as well. We can do this by following procedure. Consider +equations of motion (10) and rewrite them into the form +Rab − 1 +2gabR = 16π(Tab − +1 +D + 1gabT) + +1 − D +2(D + 1)Rgab . +(14) +Now we apply covariant derivative on both sides of the equations above +and using the fact that the covariant derivative of Einstein tensor Gab = +Rab − 1 +2gabR is zero we get +1 +D + 1∇b(16πT − 1 − D +2 +R) = 16π∇aTab . +(15) +6 + +If we consider ordinary form of matter we obtain that divergence of stress +energy tensor is zero as a consequence of matter equations of motion. Then +the right side of the equation above is zero and the left side can be easily +integrated with the result +R = +2 +1 − D(16πT + Ω) , +(16) +where Ω now appears as true integration constant rather than the cosmolog- +ical constant that was imposed in the theory by hand. In other words (16) +is the last equation of motion of unimodular gravity and we fully recovered +equivalence with general relativity however keeping in mind that we should +still have to impose the condition √−g = 1 in the course of calculations. +Having performed basic review of unimodular gravity we proceed in the +next section to its formulation in the covariant Hamiltonian formalism. +3 +Covariant Hamiltonian Formalism For D + 1 +dimensional Unimodular Gravity +In this section we find covariant Hamiltonian formalism for unimodular grav- +ity in D + 1 formalism. +As usual in the covariant formalism we split the Einstein-Hilbert action +into bulk and boundary terms. Since this procedure is well known, see for +example [35, 36] and also recent generalization to the case of F(R) gravity +[37] we write immediately final result +L = Lbulk + Lsurf , +Lbulk = +1 +16π +√−g[Γh +dkΓk +ghggd − Γf +fkΓk +ghggh] + ++ 1 +16π +¯Λ√−g + +1 +16πλ(√−g − 1) ≡ +≡ Lquad + +1 +16π +¯Λ√−g + +1 +16πλ(√−g − 1) , +Lsurf = +1 +16π∂j[√−g(gikΓj +ik − gijΓk +ik)] , +(17) +where Γa +bc are Christoffel symbols +Γa +bc = 1 +2gad(∂bgdc + ∂cgdb − ∂cgab) , +(18) +7 + +and where ¯Λ is cosmological constant. Note that the presence of the term +with Lagrange multiplier allows us to treat all components of metric as in- +dependent. +Now we are ready to proceed to the covariant Hamiltonian formulation +of this theory. The main idea of this formalism is to treat all derivatives +of dynamical variables on the equal footing [27, 29, 35] which is sharp con- +trast with the standard canonical formalism where the time coordinate has +exceptional meaning. This is very attractive idea especially in the context +of generally covariant theories since sometimes it is very difficult to perform +D + 1 splitting of targe-space time and corresponding dynamical fields. In +case of covariant canonical formalism of gravity we define conjugate momenta +Mcmn to gmn in the following way +Mcmn = ∂Lbulk +∂∂cgmn +. +(19) +Note that the momenta are defined by bulk part of the Lagrangian density +only as follows from the fact that equations of motion are derived by variation +of the action when we fix metric and its derivative on the boundary, for careful +discussion see [36]. +Then from (17) we obtain +Mcmn = +1 +32π +√−g[gmkΓc +kdgdn + gnkΓc +kdgdm − +−gmnΓc +ghggh − Γf +fk(gkmgcn + gkngcm) + gmngckΓf +fk] +(20) +using +δΓk +gh +δ∂cgmn += 1 +4(gksδc +g(δm +s δn +h + δn +s δm +h ) + ++gksδc +h(δm +s δn +g + δn +s δm +g ) − gksδc +s(δm +g δn +h + δn +g δm +h )) +(21) +Then we could formulate covariant Hamiltonian formalism using canonical +variales gab and Mcab. However it turns out that the situation is much simpler +when we introduce an alternative set of variables [35, 36] that are defined as +f ab = √−ggab . +(22) +8 + +Then it is easy to see that the conjugate momenta are defined by chain rule +Nc +ab = ∂Lquad +∂∂cf ab = +∂Lquad +∂(∂dgmn) +∂(∂dgmn) +∂(∂cfab) . +(23) +From (22) we see that f ab and gmn are related by point transformations so +that +∂dgmn = ∂gmn +∂f ab ∂df ab . +(24) +Then we have +∂(∂dgmn) +∂(∂cf ab) = ∂gmn +∂f ab δc +d +(25) +and finally +Nc +ab = +∂Lquad +∂(∂cgmn)(−gmkBkl +abgln) , +(26) +where +Bkl +ab = δgkl +δf ab = (−f)− +1 +D−1 +�1 +2(δk +aδl +b + δl +aδk +b ) − +1 +D − 1f klfab +� +, +(27) +where we used the fact that +− det f ≡ −f = (−g) +D+1 +2 (−g)−1 +(28) +and consequently +√−g = (−f) +1 +D−1 , +gab = (−f)− +1 +D−1f ab . +(29) +Then using previous form of Mcmn we obtain +Nc +ab = +∂Lquad +∂(∂cgmn)(−gmkBkl +abgln) = += − 1 +32π[2Γc +ab − Γf +faδc +b − Γf +fbδc +a] . +(30) +9 + +Note that this relation does not depend on the number of space-time di- +mensions. Then in order to find corresponding Hamiltonian we should find +inverse relation between Γa +bc and Na +bc. Let us presume that it has the form +Γc +ab = ANc +ab + B(Nd +daδc +b + Nd +bdδc +a) . +(31) +Inserting (30) into (31) we obtain +Nc +ab = − 1 +32π(2ANc +ab + 2B(Nd +daδc +b + Nd +bdδc +a) − +−(A + B(D + 2))Nf +faδc +b − (A + B(D + 2))Nf +fbδc +a) +(32) +using Γf +fa = (A + B(D + 2))Nf +fa. Comparing left and right side we obtain +that A and B are equal to +A = −16π , +B = −A +D . +(33) +Then it is easy to find kinetic term of covariant Hamiltonian for D + 1 +dimensional unimodular gravity in the form +Hkin = ∂cf abNc +ab − Lquad = 16π +� +Nb +cdf daNc +ab − 1 +DNr +raf abNs +sb +� +, +(34) +where we used the fact that +∂cf ab = ∂c +√−ggab + √−g∂cgab = Γd +dcf ab − Γa +cdf db − Γb +dcf da +(35) +together with the condition ∇cgab = 0 that implies +∂c +√−g = Γd +dc +√−g , +∂cgab = −(Γa +cdgdb + Γb +cdgda) . +(36) +The final form of the covariant Hamiltonian for unimodular gravity con- +tains terms with the unimodular constraint and true cosmological constant +¯Λ. Then the phase-space form of the action has the form +S = +� +dD+1x(Nc +ab∂cfab−Hkin− 1 +16π(−f) +1 +D−1 ¯Λ− 1 +16πλ((−f) +1 +D−1 −1)) , (37) +10 + +where λ is Lagrange multiplier corresponding to unimodular constraint. From +the action above we determine corresponding equations of motion by per- +forming variation with respect to f ab, Nc +ab and λ +δS = +� +dD+1x(δNc +ab∂cfab + Nc +ab∂cδfab − +−δHkin +δNc +ab +δNc +ab − δHkin +δf ab δf ab − +− +1 +16π(D − 1)(λ + ¯Λ)(−f) +1 +D−1δf abfab − δλ((−f) +1 +D−1 − 1)) = 0 +(38) +that implies following equations of motion +∂cf ab = δH +δNc +ab +, +(−f) +1 +D−1 − 1 = 0 , +−∂cNc +ab = δH +δf ab + +λ +16π(D − 1)(−f) +1 +D−1fab + +¯Λ +16π(D − 1)(−f) +1 +D−1fab , +(39) +or explicitly +∂cf ab = 16π[Na +cdf db + Nb +cdf da − 1 +D(f bdNs +sdδa +c + f adNs +sdδb +c)] , +−∂cNc +ab = 16π +2 (Nd +caNc +bd + Nd +cbNc +ad) − +−16π +D Nr +raNs +sb + +λ +16π(D − 1)(−f) +1 +D−1fab + +¯Λ +16π(D − 1)(−f) +1 +D−1fab , +(−f) +1 +D−1 − 1 = 0 . +(40) +Taking the trace of the second equation we can determine λ as +λ = 16π(D − 1) +(D + 1) +(−∂cNc +abf ab − 16πNd +caf abNc +bd + 16π +D Nr +raf abNs +sb) − ¯Λ , +(41) +where we have took into account the equation on the fourth line in (40). +11 + +Then the equations of motion for Nc +ab have the form +−∂cNc +ab = 16π +2 (Nd +caNc +bd + Nd +cbNc +ad) − 16π +D Nr +raNs +sb + ++ +1 +(D + 1)(−∂jNj +ikf ik − 16πNd +cif ikNc +kd + 16π +D Nr +rif ikNs +sk)fab . +(42) +Clearly this equation is traceless and all dependence on the cosmological +constant ¯Λ disappears which is an essence of unimodular gravity. +On the other hand one let us try to calculate the trace of the first equation +that gives +∂cf abfab = 16π[Na +cdf db + Nb +cdf da − 1 +D(f bdNs +sdδa +c + f adNs +sdδb +c)]fba +(43) +that can be simplified into the form +∂cf = 32π[D − 1 +D +]Ns +sc . +Now taking into account unimodular constraint we immediately get the con- +dition +Ns +sc = 0 +(44) +that can be interpreted as secondary constraint. +On the other hand the +condition (44) seems to be too strong so that we should discuss it in more +details. +We begin with the recapitulation that unimodular gravity in the covariant +Hamiltonian formalism is described by canonical conjugate variables f ab, Nc +ab +that are restricted by unimodular condition together with (44). In order to +find proper interpretation of the constraint (44) it is instructive to derive +general relativity variables from f ab, Nc +ab. As the first step let us consider lin- +ear combination of Nc +ab that we denote as Γc +ab and which is given by following +prescription +Γc +ab = −16πNc +ab + 16π +D (Nd +daδc +b + Nd +bdδc +a) . +(45) +This can be always done and we should again stress that Γc +ab is not related +to f ab at all. Clearly Γc +ab = Γc +ba. Then we define covariant derivative where +Γc +ab are coefficients of connection. Let us further define gab and its inverse gab +in the following way +gab = f ab(−f) +1 +1−D , +gab = fab(−f) +1 +D−1 . +(46) +12 + +Let us then define covariant derivative of gab as +∇cgab = ∂cgab + Γa +cdgdb + Γb +cdgda , +(47) +that, using (45), takes the form +∇cgab = (−f) +1 +1−D × +×[∂cf ab − 16πNa +cdf db − 16πNb +cdf da + 16π +D f bdNr +drδa +c + 16π +D Nr +drf daδb +c] = 0 , +(48) +where we used the first equation in (40) that also implies ∂cf mnfmn = +32π D−1 +D Ns +sc. +Now thanks to the equation ∇cgab = 0 we can express Γa +bc +in the form of Christoffel symbols +Γa +bc = 1 +2gad(∂bgdc + ∂cgdb − ∂dgbc) . +(49) +On the other hand let us return to the relation between Γa +bc and Na +bc that +takes the form +Γf +fa = −32π +D Nf +fa +(50) +so that condition that Ns +sa = 0 implies +Γs +sa = 0 . +(51) +On the other hand from (49) we obtain +Γf +fc = 1 +2gfd∂cgdf = ∂c det g = 0 +(52) +so that condition Ns +sc = 0 is equivalent to unimodular condition. It is im- +portant to stress that the fact that unimodular constraint implies Γs +sa = 0 +has not been appreciated too much with exception of recent interesting pa- +per [10] where it was stressed that the equivalence between general relativity +and unimodular gravity is non-trivial. Rather, it was argued there that the +natural geometry for unimodular relativity is equiprojective geometry [39]. +We also see that the condition Ns +sa = 0 emerges naturally in the covariant +canonical formalism of unimodular gravity. +13 + +4 +Covariant Form of Unimodular Gravity +In this section we perform covariant canonical formalism for Henneaux- +Teitelboim formulation of unimodular gravity that has the form +S = +1 +16π +� +dD+1x√−g[R + λ(√−g − ∂aτ a)] , +(53) +where τ a is vector density and λ is Lagrange multiplier. Now the equations +of motion for λ implies +√−g − ∂aτ a = 0 +(54) +while equation of motion for τ a leads to +∂aλ = 0 . +(55) +It is clear that the covariant Hamiltonian formulation of this theory is al- +most the same as in previous case with difference that there is momentum +conjugate to τ a. Writting ∂aτ a = ∂bτ aδb +a we obtain momentum conjugate to +τ a to be equal to +pb +a = +δL +δ∂bτ a = − 1 +16πλδb +a +(56) +however this can be interpreted as primary constraints of the theory +Gb +a ≡ pb +a + +1 +16πλδb +a . +(57) +In fact, the bare Hamiltonian is defined as +HB = pb +a∂bτ a + ∂cf abNc +ab − L = += 16π[Nb +cdf daNc +ab − 1 +DNr +raf abNs +sb] − +1 +16πλ(−f) +1 +D−1 +(58) +and we see that the dependence on momenta pν +µ is missing. For that reason +we should consider Hamiltonian with primary constraints included +HT = 16π[Nb +cdf daNc +ab − 1 +DNr +raf abNs +sb] − +1 +16πλ(−f) +1 +D−1 + Γa +b(pb +a + +1 +16πλδb +a) +(59) +14 + +and consider corresponding equations of motion that arise from the variation +of the canonical form of the action +S = +� +dD+1x(∂cf abNc +ab + pa +b∂aτ b − 16π[Nb +cdf daNc +ab − 1 +DNr +raf abNs +sb] + ++ 1 +16πλ(−f) +1 +D−1 + Γa +b(pb +a + +1 +16πλδb +a)) +(60) +so that the equations of motion have the form +∂cf ab = 16π[Na +cdf db + Nb +cdf da − 1 +D(f bdNs +sdδa +c + f adNs +sdδb +c)] , +−∂cNc +ab = 16π +2 (Nd +caNc +bd + Nd +cbNc +ad) − 16π +D Nr +raNs +sb + +λ +(D − 1)(−f) +1 +D−1fab , +(−f) +1 +D−1 + Γa +a = 0 , +∂bτ a + Γa +b = 0 , +∂apa +b = 0 , +pb +a + +1 +16πλδb +a = 0 . +(61) +If we combine the first and the second equation on the third line we find +(−f) +1 +D−1 = ∂aτ a +(62) +that has exactly the same form as equation (54). We further perform partial +derivative of the fourth equation on the third line and we obtain +∂bpb +a = − 1 +16π∂aλ +(63) +that using the third equation on the same line implies that +∂aλ = 0 . +(64) +This equation also shows that λ is constant and it can be interpreted as +integration constant. Then it can be argued in the same way as in the pre- +vious section that the equations (61) are equivalent to the Lagrangian equa- +tions of Henneaux-Teitelboim gravity. In other words, covariant Hamiltonian +description of Henneaux-Teiltelboim gravity is equivalent to corresponding +Lagrangian description which is nice consistency check. +Acknowledgement: +The work of JK is supported by the grant “Dualitites and higher order +derivatives” (GA23-06498S) from the Czech Science Foundation (GACR). +15 + +References +[1] W. Buchmuller and N. Dragon, “Einstein Gravity From Restricted Coor- +dinate Invariance,” Phys. Lett. B 207 (1988), 292-294 doi:10.1016/0370- +2693(88)90577-1 +[2] W. 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D 89 (2014) no.6, 064043 doi:10.1103/PhysRevD.89.064043 +[arXiv:1311.4141 [hep-th]]. +19 + diff --git a/-9FRT4oBgHgl3EQfsTcg/content/tmp_files/load_file.txt b/-9FRT4oBgHgl3EQfsTcg/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..853e2381c67fd4ec158a35e8e23977d77fe92308 --- /dev/null +++ b/-9FRT4oBgHgl3EQfsTcg/content/tmp_files/load_file.txt @@ -0,0 +1,538 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf,len=537 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='13623v1 [hep-th] 31 Jan 2023 Unimodular Gravity in Covariant Formalism J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Klusoň† and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Matouš†‡ 1 † Department of Theoretical Physics and Astrophysics, Faculty of Science, Masaryk University, Kotlářská 2, 611 37, Brno, Czech Republic ‡ North-Bohemian Observatory and Planetarium in Teplice, Koperníkova 3062, 415 01, Teplice, Czech Republic Abstract In this short note we study unimodular gravity in Weyl-De Donder formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' We find corresponding Hamiltonian and study consequence of the unimodular constraint on the conjugate covariant momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' We also find covariant Hamiltonian for Henneaux-Teitelboim unimodular action and study corresponding equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' 1 Introduction and Summary Unimodular gravity was firstly introduced by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Einstein in his paper [3] published in 1916.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' In this work the unimodular constraint √−g = 1 was used as gauge fixing condition of general diffeomorphism in order to sim- plify calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Then it was shown in [1, 2] that imposing this condi- tion before the variation of Einstein-Hilbert action leads to the traceless equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' As we review below these equations of motion are classically equivalent to the general relativity equations of motion with cru- cial difference that the cosmological constant appears as integration con- stant rather than true cosmological constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' This fact brings new hope how to solve cosmological constant problem which was however questioned in [4], 2 where it was argued that quantum corrections make the cosmo- logical constant ultraviolet sensitive in unimodular gravity as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' On the other hand it is important to stress that no definitive conclusions have been reached yet regarding this problem and unimodular gravity is still very inten- sively studied, for some works devoted to unimodular gravity, see for example [7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 22, 23, 24, 25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' 1Email addresses: J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Klusoň: klu@physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='muni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='cz, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Matouš: bmatous@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='muni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='cz 2For review of unimodular gravity, see for example [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' 1 One of the most interesting aspects of unimodular gravity is the number of physical degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Naively, unimodular constraint √−g = 1 re- duces the number of independent components of metric to nine which could suggest that the number of physical degrees of freedom is less than in general relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' On the other hand unimodular gravity is invariant under restricted diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Taking these two aspects together we find that the num- ber of local physical degrees of freedom is the same as in ordinary general relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' This fact was proved with the help of the Hamiltonian analysis of unimodular gravity performed in [16, 17, 18, 19, 20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' On the other hand as was shown in these papers standard analysis of unimodular gravity based on D + 1 splitting of target space-time is rather non-trivial and shown complexity of the canonical analysis of systems with constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Then one could ask the question how unimodular gravity could be de- scribed in covariant canonical formalism that is known as Weyl-De Donder theory [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' The key point of this formulation is that we treat all partial derivatives as equivalent when we define conjugate momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' For example, if we have scalar field φ with Lagrangian density in D +1 dimensional space- time equal to L = −1 2ηab∂aφ∂bφ − V (φ), we define the conjugate momentum as 3 πa = ∂L ∂∂aφ = −ηab∂bφ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Then covariant canonical Hamiltonian density is defined as H = πa∂aφ − L = −1 2πaηabπb + V (φ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Clearly such a form of Hamiltonian density preserves diffeomorphism invari- ance of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' This approach is known as multisymplectic field theory, see for example [29, 30, 31], for review, see [32] and for recent interesting application of this formalism in string theory, see [33, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' It is clear that such covariant canonical formalism is especially suitable for manifestly covariant theories as for example general relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' In fact, covariant canonical formalism of general relativity was found long time ago by P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Hořava [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' This analysis was recently generalized to the case of F(R) gravity in [37] and further elaborated in [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' In this paper we apply this formalism for unimodular theory of gravity in D + 1 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' This is non-trivial task due to the well known complex- ity of canonical analysis of unimodular gravity in non-covariant formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' 3We define ηab = diag(−1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=', 1), a, b = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=', D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' 2 Further, it is also very interesting to study this system since it contains pri- mary unimodular constraint and it is non-trivial task how to deal with such systems in covariant canonical formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' In more details, we include this primary constraint to the action with corresponding Lagrange multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Then we derive corresponding equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Using these equations of motion we find that the unimodular constraint implies another constraint on the canonical conjugate momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Then we show that this constraint is equivalent to the vanishing of the trace of the Christoffel symbols which is characteristic property of unimodular theory of gravity [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' This is nice and non-trivial result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' On the other hand the Lagrange multiplier corresponding to the primary constraint cannot be determined as in non-covariant canon- ical formalism by imposing condition of the preservation of the secondary constraint due to the fact that the equations of motion for conjugate mo- menta are in the form of the divergence of these momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' For that reason we determine this constraint in the same way as in the Lagrangian formalism when we calculate the trace of the equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' As a result we obtain equations of motion that are traceless and that do not depend on the cos- mological constant which is in agreement with the Lagrangian formulation of unimodular gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' As the second step in our analysis we find covariant canonical formula- tion of Henneaux-Teitelboim formulation of unimodular gravity [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' In this case we again identify covariant Hamiltonian together with set of primary constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Then we consider canonical form of the action and determine corresponding equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Solving these equations of motion we find that Lagrange multiplier is integration constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' In this case we repro- duce results well known from Lagrangian analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' However we mean that this is nice and interesting application of the covariant canonical analysis to the constraint systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Let us outline our results and suggest possible extension of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' We found covariant Hamiltonian formalism for unimodular gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' First of all we determined covariant Hamiltonian for general relativity action in D + 1 dimensions where we again introduced variable f ab = √−ggab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' At this place we would like to stress an importance of this result since it was not apri- ori known whether f ab is suitable for formulation of gravity in space-time of dimension different from 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Then we imposed unimodular constraint using Lagrange multiplier method and then we studied corresponding equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' We found that the consistency of the theory demands that the trace of conjugate momenta is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Then we showed that this is character- 3 istic property of unimodular gravity when we pass to Lagrangian formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Final we found covariant Hamiltonian for Henneaux-Teltelboim formulation of unimodular gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' We identified primary constraints of the theory and then we studied equations of motion that follow from canonical form of the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' We showed that they precisely reproduce Lagrangian equations of motion that is nice consistency check of the covariant canonical formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' We mean that the analysis presented in this paper suggests that covariant Hamiltonian formalism is very close to Lagrangian formalism and in some situations the covariant Hamiltonian formalism is more suitable than La- grangian one, as for example study of thermodynamics properties of horizon [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' It is also clear that there are more systems that could be analysed with the help of covariant canonical formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' One possibility is to study Weyl invariant gravity in this formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Another possibility would be to perform analysis of theories of gravity with higher derivatives where the classical canonical analysis is very complicated, see for example [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' We hope to return to these problems in future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' In the next section (2) we review properties of unimodular gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='Then in section (3) we proceed to the co- variant canonical formulation of this theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Finally in section (4) we perform covariant canonical formulation of Henneaux-Teltelboim unimodular gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' 2 Brief Review of Unimodular Gravity In this section we review basic facts about unimodular gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' For recent very nice and more detailed review, see for example [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Unimodular gravity is theory with the constraint √−g = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Clearly such a condition has a consequence on allowed differomorphism transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' In fact, let us consider general transformation of coordinates x′a = xa + ξa(x) (1) that implies inverse relation xa = x′a − ξa(x) ≈ x′a − ξa(x′) + O(ξ2) , (2) where a, b, c = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' , D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Under these transformation the metric gab trans- form as g′ ab(x) = gab(x) − ∂cgab(x)ξc(x) − gac(x)∂bξc(x) − ∂aξc(x)gcb(x) (3) 4 that implies following variation of metric δgab(x) = g′ ab(x) − gab(x) = −gac∂bxc − ∂aξcgcb − ∂cgabξc so that the variation of the square root of the determinant of metric is equal to δ � − det g = −(2∂aξa − ∂cgabgbaξc) � − det g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (4) In case of unimodular gravity this variation should vanish and hence we obtain following condition on ξa in the form ∇aξa = ∂aξa + 1 2gac∂dgcaξd = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (5) The most straightforward way how to find an action for unimodular gravity is to consider standard Einstein-Hilbert action with an unimodular constraint added S = 1 16π � dD+1x[√−g(R − 2¯Λ) + Λ(√−g − 1)] + Smatt , (6) where Λ is Lagrange multiplier whose variation ensures unimodular condition and where ¯Λ is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Performing variation of the action (6) with respect to gab we obtain fol- lowing equations of motion 1 16π(Rab − 1 2gab(R − 2¯Λ + Λ)) = Tab , (7) where Tab is matter stress energy tensor defined as Tab = − 1 √−g δSmatt δgab .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (8) The crucial point is that Λ is Lagrange multiplier that should be determined as a consequence of the equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' To do this we perform the trace of the equation (7) to express Λ as Λ = (1 − D) 1 + D R − 32π D + 1T + 2¯Λ , T ≡ gabTab .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (9) 5 Inserting this result into (7) we obtain Rab − 1 D + 1gabR = 16π(Tab − 1 D + 1gabT) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (10) These equations of motion are trace-free and also most importantly they do not contain any information about cosmological constant ¯Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' It is important to stress that even equations of motion of general relativ- ity without unimodular constraint imposed split into 9 trace-free equations of motion and one additional one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' To see this consider general relativity equations of motion Rab − 1 2gab(R − 2¯Λ) = 16πTab .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (11) Taking the trace of this equation we can express R as R = 2 1 − D(16πT − (D + 1)¯Λ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (12) Note that with the help of this equation we can rewrite (11) into trace-free form Rab − 1 D + 1Rgab = 16π(Tab − 1 D + 1Tgab) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (13) However we should again stress that (12) determines R as function of trace of matter stress energy tensor and true cosmological constant term in Einstein- Hilbert action while in case of unimodular gravity we express Λ-which is Lagrange multiplier and not constant, as function of R, T and ¯Λ, as follows from equation (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' In order to check equivalence between unimodular gravity and ordinary general relativity we should be able to reproduce equation (12) in case of unimodular gravity as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' We can do this by following procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Consider equations of motion (10) and rewrite them into the form Rab − 1 2gabR = 16π(Tab − 1 D + 1gabT) + 1 − D 2(D + 1)Rgab .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (14) Now we apply covariant derivative on both sides of the equations above and using the fact that the covariant derivative of Einstein tensor Gab = Rab − 1 2gabR is zero we get 1 D + 1∇b(16πT − 1 − D 2 R) = 16π∇aTab .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (15) 6 If we consider ordinary form of matter we obtain that divergence of stress energy tensor is zero as a consequence of matter equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Then the right side of the equation above is zero and the left side can be easily integrated with the result R = 2 1 − D(16πT + Ω) , (16) where Ω now appears as true integration constant rather than the cosmolog- ical constant that was imposed in the theory by hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' In other words (16) is the last equation of motion of unimodular gravity and we fully recovered equivalence with general relativity however keeping in mind that we should still have to impose the condition √−g = 1 in the course of calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Having performed basic review of unimodular gravity we proceed in the next section to its formulation in the covariant Hamiltonian formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' 3 Covariant Hamiltonian Formalism For D + 1 dimensional Unimodular Gravity In this section we find covariant Hamiltonian formalism for unimodular grav- ity in D + 1 formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' As usual in the covariant formalism we split the Einstein-Hilbert action into bulk and boundary terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Since this procedure is well known,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' see for example [35,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' 36] and also recent generalization to the case of F(R) gravity [37] we write immediately final result L = Lbulk + Lsurf ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Lbulk = 1 16π √−g[Γh dkΓk ghggd − Γf fkΓk ghggh] + + 1 16π ¯Λ√−g + 1 16πλ(√−g − 1) ≡ ≡ Lquad + 1 16π ¯Λ√−g + 1 16πλ(√−g − 1) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Lsurf = 1 16π∂j[√−g(gikΓj ik − gijΓk ik)] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (17) where Γa bc are Christoffel symbols Γa bc = 1 2gad(∂bgdc + ∂cgdb − ∂cgab) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (18) 7 and where ¯Λ is cosmological constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Note that the presence of the term with Lagrange multiplier allows us to treat all components of metric as in- dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Now we are ready to proceed to the covariant Hamiltonian formulation of this theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' The main idea of this formalism is to treat all derivatives of dynamical variables on the equal footing [27, 29, 35] which is sharp con- trast with the standard canonical formalism where the time coordinate has exceptional meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' This is very attractive idea especially in the context of generally covariant theories since sometimes it is very difficult to perform D + 1 splitting of targe-space time and corresponding dynamical fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' In case of covariant canonical formalism of gravity we define conjugate momenta Mcmn to gmn in the following way Mcmn = ∂Lbulk ∂∂cgmn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (19) Note that the momenta are defined by bulk part of the Lagrangian density only as follows from the fact that equations of motion are derived by variation of the action when we fix metric and its derivative on the boundary, for careful discussion see [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Then from (17) we obtain Mcmn = 1 32π √−g[gmkΓc kdgdn + gnkΓc kdgdm − −gmnΓc ghggh − Γf fk(gkmgcn + gkngcm) + gmngckΓf fk] (20) using δΓk gh δ∂cgmn = 1 4(gksδc g(δm s δn h + δn s δm h ) + +gksδc h(δm s δn g + δn s δm g ) − gksδc s(δm g δn h + δn g δm h )) (21) Then we could formulate covariant Hamiltonian formalism using canonical variales gab and Mcab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' However it turns out that the situation is much simpler when we introduce an alternative set of variables [35, 36] that are defined as f ab = √−ggab .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (22) 8 Then it is easy to see that the conjugate momenta are defined by chain rule Nc ab = ∂Lquad ∂∂cf ab = ∂Lquad ∂(∂dgmn) ∂(∂dgmn) ∂(∂cfab) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (23) From (22) we see that f ab and gmn are related by point transformations so that ∂dgmn = ∂gmn ∂f ab ∂df ab .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (24) Then we have ∂(∂dgmn) ∂(∂cf ab) = ∂gmn ∂f ab δc d (25) and finally Nc ab = ∂Lquad ∂(∂cgmn)(−gmkBkl abgln) , (26) where Bkl ab = δgkl δf ab = (−f)− 1 D−1 �1 2(δk aδl b + δl aδk b ) − 1 D − 1f klfab � , (27) where we used the fact that − det f ≡ −f = (−g) D+1 2 (−g)−1 (28) and consequently √−g = (−f) 1 D−1 , gab = (−f)− 1 D−1f ab .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (29) Then using previous form of Mcmn we obtain Nc ab = ∂Lquad ∂(∂cgmn)(−gmkBkl abgln) = = − 1 32π[2Γc ab − Γf faδc b − Γf fbδc a] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (30) 9 Note that this relation does not depend on the number of space-time di- mensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Then in order to find corresponding Hamiltonian we should find inverse relation between Γa bc and Na bc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Let us presume that it has the form Γc ab = ANc ab + B(Nd daδc b + Nd bdδc a) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (31) Inserting (30) into (31) we obtain Nc ab = − 1 32π(2ANc ab + 2B(Nd daδc b + Nd bdδc a) − −(A + B(D + 2))Nf faδc b − (A + B(D + 2))Nf fbδc a) (32) using Γf fa = (A + B(D + 2))Nf fa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Comparing left and right side we obtain that A and B are equal to A = −16π , B = −A D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (33) Then it is easy to find kinetic term of covariant Hamiltonian for D + 1 dimensional unimodular gravity in the form Hkin = ∂cf abNc ab − Lquad = 16π � Nb cdf daNc ab − 1 DNr raf abNs sb � , (34) where we used the fact that ∂cf ab = ∂c √−ggab + √−g∂cgab = Γd dcf ab − Γa cdf db − Γb dcf da (35) together with the condition ∇cgab = 0 that implies ∂c √−g = Γd dc √−g , ∂cgab = −(Γa cdgdb + Γb cdgda) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (36) The final form of the covariant Hamiltonian for unimodular gravity con- tains terms with the unimodular constraint and true cosmological constant ¯Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Then the phase-space form of the action has the form S = � dD+1x(Nc ab∂cfab−Hkin− 1 16π(−f) 1 D−1 ¯Λ− 1 16πλ((−f) 1 D−1 −1)) , (37) 10 where λ is Lagrange multiplier corresponding to unimodular constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' From the action above we determine corresponding equations of motion by per- forming variation with respect to f ab,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Nc ab and λ δS = � dD+1x(δNc ab∂cfab + Nc ab∂cδfab − −δHkin δNc ab δNc ab − δHkin δf ab δf ab − − 1 16π(D − 1)(λ + ¯Λ)(−f) 1 D−1δf abfab − δλ((−f) 1 D−1 − 1)) = 0 (38) that implies following equations of motion ∂cf ab = δH δNc ab ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (−f) 1 D−1 − 1 = 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' −∂cNc ab = δH δf ab + λ 16π(D − 1)(−f) 1 D−1fab + ¯Λ 16π(D − 1)(−f) 1 D−1fab ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (39) or explicitly ∂cf ab = 16π[Na cdf db + Nb cdf da − 1 D(f bdNs sdδa c + f adNs sdδb c)] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' −∂cNc ab = 16π 2 (Nd caNc bd + Nd cbNc ad) − −16π D Nr raNs sb + λ 16π(D − 1)(−f) 1 D−1fab + ¯Λ 16π(D − 1)(−f) 1 D−1fab ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (−f) 1 D−1 − 1 = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (40) Taking the trace of the second equation we can determine λ as λ = 16π(D − 1) (D + 1) (−∂cNc abf ab − 16πNd caf abNc bd + 16π D Nr raf abNs sb) − ¯Λ , (41) where we have took into account the equation on the fourth line in (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' 11 Then the equations of motion for Nc ab have the form −∂cNc ab = 16π 2 (Nd caNc bd + Nd cbNc ad) − 16π D Nr raNs sb + + 1 (D + 1)(−∂jNj ikf ik − 16πNd cif ikNc kd + 16π D Nr rif ikNs sk)fab .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (42) Clearly this equation is traceless and all dependence on the cosmological constant ¯Λ disappears which is an essence of unimodular gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' On the other hand one let us try to calculate the trace of the first equation that gives ∂cf abfab = 16π[Na cdf db + Nb cdf da − 1 D(f bdNs sdδa c + f adNs sdδb c)]fba (43) that can be simplified into the form ∂cf = 32π[D − 1 D ]Ns sc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Now taking into account unimodular constraint we immediately get the con- dition Ns sc = 0 (44) that can be interpreted as secondary constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' On the other hand the condition (44) seems to be too strong so that we should discuss it in more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' We begin with the recapitulation that unimodular gravity in the covariant Hamiltonian formalism is described by canonical conjugate variables f ab, Nc ab that are restricted by unimodular condition together with (44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' In order to find proper interpretation of the constraint (44) it is instructive to derive general relativity variables from f ab, Nc ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' As the first step let us consider lin- ear combination of Nc ab that we denote as Γc ab and which is given by following prescription Γc ab = −16πNc ab + 16π D (Nd daδc b + Nd bdδc a) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (45) This can be always done and we should again stress that Γc ab is not related to f ab at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Clearly Γc ab = Γc ba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Then we define covariant derivative where Γc ab are coefficients of connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Let us further define gab and its inverse gab in the following way gab = f ab(−f) 1 1−D , gab = fab(−f) 1 D−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (46) 12 Let us then define covariant derivative of gab as ∇cgab = ∂cgab + Γa cdgdb + Γb cdgda , (47) that, using (45), takes the form ∇cgab = (−f) 1 1−D × ×[∂cf ab − 16πNa cdf db − 16πNb cdf da + 16π D f bdNr drδa c + 16π D Nr drf daδb c] = 0 , (48) where we used the first equation in (40) that also implies ∂cf mnfmn = 32π D−1 D Ns sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Now thanks to the equation ∇cgab = 0 we can express Γa bc in the form of Christoffel symbols Γa bc = 1 2gad(∂bgdc + ∂cgdb − ∂dgbc) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (49) On the other hand let us return to the relation between Γa bc and Na bc that takes the form Γf fa = −32π D Nf fa (50) so that condition that Ns sa = 0 implies Γs sa = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (51) On the other hand from (49) we obtain Γf fc = 1 2gfd∂cgdf = ∂c det g = 0 (52) so that condition Ns sc = 0 is equivalent to unimodular condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' It is im- portant to stress that the fact that unimodular constraint implies Γs sa = 0 has not been appreciated too much with exception of recent interesting pa- per [10] where it was stressed that the equivalence between general relativity and unimodular gravity is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Rather, it was argued there that the natural geometry for unimodular relativity is equiprojective geometry [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' We also see that the condition Ns sa = 0 emerges naturally in the covariant canonical formalism of unimodular gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' 13 4 Covariant Form of Unimodular Gravity In this section we perform covariant canonical formalism for Henneaux- Teitelboim formulation of unimodular gravity that has the form S = 1 16π � dD+1x√−g[R + λ(√−g − ∂aτ a)] , (53) where τ a is vector density and λ is Lagrange multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Now the equations of motion for λ implies √−g − ∂aτ a = 0 (54) while equation of motion for τ a leads to ∂aλ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (55) It is clear that the covariant Hamiltonian formulation of this theory is al- most the same as in previous case with difference that there is momentum conjugate to τ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Writting ∂aτ a = ∂bτ aδb a we obtain momentum conjugate to τ a to be equal to pb a = δL δ∂bτ a = − 1 16πλδb a (56) however this can be interpreted as primary constraints of the theory Gb a ≡ pb a + 1 16πλδb a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (57) In fact, the bare Hamiltonian is defined as HB = pb a∂bτ a + ∂cf abNc ab − L = = 16π[Nb cdf daNc ab − 1 DNr raf abNs sb] − 1 16πλ(−f) 1 D−1 (58) and we see that the dependence on momenta pν µ is missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' For that reason ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='we should consider Hamiltonian with primary constraints included ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='HT = 16π[Nb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='cdf daNc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='ab − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='DNr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='raf abNs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='sb] − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='16πλ(−f) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='D−1 + Γa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='b(pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='a + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='16πλδb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='(59) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='and consider corresponding equations of motion that arise from the variation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='of the canonical form of the action ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='S = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='dD+1x(∂cf abNc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='ab + pa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='b∂aτ b − 16π[Nb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='cdf daNc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='ab − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='DNr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='raf abNs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='sb] + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='16πλ(−f) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='D−1 + Γa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='b(pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='a + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='16πλδb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='a)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='(60) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='so that the equations of motion have the form ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='∂cf ab = 16π[Na ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='cdf db + Nb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='cdf da − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='D(f bdNs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='sdδa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='c + f adNs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='sdδb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content='c)] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' −∂cNc ab = 16π 2 (Nd caNc bd + Nd cbNc ad) − 16π D Nr raNs sb + λ (D − 1)(−f) 1 D−1fab ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (−f) 1 D−1 + Γa a = 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' ∂bτ a + Γa b = 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' ∂apa b = 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' pb a + 1 16πλδb a = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (61) If we combine the first and the second equation on the third line we find (−f) 1 D−1 = ∂aτ a (62) that has exactly the same form as equation (54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' We further perform partial derivative of the fourth equation on the third line and we obtain ∂bpb a = − 1 16π∂aλ (63) that using the third equation on the same line implies that ∂aλ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' (64) This equation also shows that λ is constant and it can be interpreted as integration constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Then it can be argued in the same way as in the pre- vious section that the equations (61) are equivalent to the Lagrangian equa- tions of Henneaux-Teitelboim gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' In other words, covariant Hamiltonian description of Henneaux-Teiltelboim gravity is equivalent to corresponding Lagrangian description which is nice consistency check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FRT4oBgHgl3EQfsTcg/content/2301.13623v1.pdf'} +page_content=' Acknowledgement: The work of JK is supported by 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b/3dE4T4oBgHgl3EQf0Q2d/content/tmp_files/2301.05281v1.pdf.txt @@ -0,0 +1,2214 @@ + DesignCon 2006 +Time Domain Verification of +Differential Transmission Line +Modeling Methods +Jonathan D. Coker, Mayo Clinic +Dr. Erik S. Daniel, Mayo Clinic +Dr. Barry K. Gilbert, Mayo Clinic +gilbert.barry@mayo.edu 507-284-4056 + +Abstract +The advantages and limitations of time-domain pseudo-random binary sequence (PRBS) +excitation methods for system identification of individual modes within a multi- +conductor transmission system are discussed. We develop the modifications necessary +to standard frequency-domain transmission-line models to match time-domain +experimental data from several types of transmission systems. We show a variety of +experimental results showing very good to excellent agreement with our model’s +predictions, up to approximately 10 GHz. + + + + +Author Biographies + +Jon Coker is a Principal Project Engineer of the Special Purpose Processor Development +Group at the Mayo Clinic. Mr. Coker graduated from Wheaton College with a Bachelor +of Arts degree in 1982 and from the University of Minnesota with a Bachelor of Science +degree in electrical engineering in 1984. He is currently pursuing a Ph.D. degree at the +University of Minnesota. + + +Erik Daniel received the B. A. degree in physics and mathematics from Rice University, +Houston, TX, in 1992. He received the Ph.D. degree in solid state physics from the +California Institute of Technology, Pasadena, CA, in 1997, with thesis research focusing +on simulation, fabrication, and characterization of quantum effect semiconductor devices. +He currently is a Staff Scientist in the Department of Physiology and Biomedical +engineering, Mayo Clinic, Rochester, MN, and the Deputy Director of the Special- +Purpose Processor Development Group. + + +Barry Gilbert received the B.S. degree in electrical engineering from Purdue University, +West Lafayette, IN, in 1965 and the Ph.D. degree in physiology and biophysics (with +minors in electrical engineering and applied mathematics) from the University of +Minnesota in 1972. He is currently a Staff Scientist in the Department of Physiology and +Biomedical engineering, Mayo Clinic, Rochester, MN, and the Director of the Special +Purpose Processor Development Group. + + + + + + +Introduction + +High-speed digital back-plane communication channels have long since given up their +“digital design” status and have come to resemble (architecturally) previously-existing, +complex analog communication systems. Equalization, error correction, modulation +codes, or advanced detection schemes are now commonly proposed and implemented for +serializer-deserializer (SERDES) channels (see [10], for example). To achieve the +maximum benefit of these methods, we require increasingly accurate system +identification of the transmission system. At the same time, we find that traditionally +excellent transmission-line models begin to diverge from experiment as bandwidths begin +to extend into the GHz and tens of GHz range [13]. Thus reliable methods for system +identification and model verification for candidate transmission systems supporting such +channel implementations are of increasing and fundamental importance. + +Several methods exist for accurate system identification of linear systems. Analysis +using network analyzers (including vector network analyzers) is a highly-developed +technology using frequency-domain measurements. Advanced time-domain methods +have more recently come on the scene employing TDR and TDT measurements of a step +excitation [7-9]. + +In this paper, we present a technique to predict and experimentally verify transmission +line models purely in the time domain using the method of pseudorandom sequences. +Our technique is an extension of a method developed for system identification commonly +used in magnetic recording data channels and testers [3,6]. We shall focus on the +description of transmission lines which are suitable for a high-speed serial- +communication link. We shall compare expected and experimental results from printed +circuit board (PCB) transmission lines and to results from cables. While we shall not +argue that the proposed method is experimentally superior to established time-domain +techniques for linear systems, we shall discuss the unique nonlinear-system- +identification capabilities of a PRBS waveform. In addition, we shall show a fast +algorithm which is simple enough to consider implementing in modern SERDES +hardware, thus allowing a wide range of inexpensive and highly capable built-in self-test +and board diagnostic capabilities. + +We shall describe our specific model of a transmission line. This section is intended to +outline the assumptions and limitations of the model. We also present our alternative +extensions to the standard telegrapher’s model, which we found necessary to adequately +describe experiment in some cases. + +Transmission Line Modeling +In this section, we briefly review a classical transmission-line model, and discuss the +parameters which we use in the model. +Telegrapher’s Model in the Time Domain + +The basic telegrapher's model for transmission lines is fully worked out in several texts +[1,2]. The standard solution gives the voltage transfer function for a wave traveling in +the positive z direction as + +( +) +( ) +z +H +e γ ω +ω +− +Δ += + +(1.1) +where ω is the radial frequency, z +Δ is the distance down the line, and the complex +propagation constant ( ) +γ ω is: + + +( ) +( +)( +) +( +)( +) +R +j L G +j C +R sL G +sC +γ ω +ω +ω += ++ ++ += ++ ++ + +(1.2) + +If the Fourier transform of an input waveform ( ) +x t is +( ) +X ω , then the Fourier transform of +the waveform at the output of the transmission line, ( ) +y t , is + + +( ) +( ) +( ) +Y +H +X +ω +ω +ω += + +(1.3) + +and the time-domain version of the output waveform is + + +1 +( ) +{ +( ) +( )} +y t +H +X +ω +ω +− += F + +(1.4) + +Cm +L +C +C +R +G +C +G +C +L + L +m +R +L + L +m +Cm +G +L +L +to +Infinity +to +Infinity +G +L +R +R +L +L +R +R +Gm +C +C +Cm +G +G +Gm +C +C +Cm +G +G +Gm +Lm +Lm +L +C +C +Lm +(x1 (t) + x +2(t)) +GENERAL EQUIVALENT CIRCUIT FOR +SYMMETRIC SIGNAL CONDUCTORS +EVEN-MODE EQUIVALENT CIRCUIT +( Lines Toggling in Same Direction ) +R +G + 2G +m +G + 2G +m +C + 2C +m +C + 2C +m +L - L +m +R +L - L +m +to +Infinity +(x1 (t) - x +2(t)) +x1 (t) +x2 (t) +ODD-MODE EQUIVALENT CIRCUIT +( Lines Toggling in Opposite Directions ) +1 +2 + +Figure 1: Simplified Odd Mode and Even Mode Equivalent Circuits for a Symmetric Two-Signal- +Conductor Generalized Transmission Line (18500). + +In Figure 1, we show the standard, generalized equivalent circuit of a symmetric two- +signal-conductor transmission line, and reduced-complexity, single-ended equivalent +circuits for each transmission mode. Note that the simplified equivalent circuits allow + +direct application of the RLGC model, using the correctly-transformed versions of the +RLGC parameters appropriate for the transmission mode. + +Variation of RLGC Parameters with Frequency + +Having set the stage for the time-domain application of a generic RLGC model, we now +focus on the specific forms of each component used our RLGC model. The following +section specifies the formulas used in our basic RLGC model (which are fairly standard) +and identifies modifications we thought necessary to adequately explain observed +laboratory behavior (which are not always standard). + +Series Impedance Variation with Frequency +In the case of simple, homogeneous signal conductors, we use the classical result for the +series impedance based on surface impedance concepts, which we repeat here: + + +( +) +AC +R +sL +R +s +L s ++ += ++ ∞ +(1.5) + +where L∞ may be interpreted as the inductance of the system when all currents flow +uniformly on the surface of the signal conductors (that is, at moderately high frequency) +and +AC +R + is a constant which we approximate as + +2 +AC +R +S +η +μ +σ += + +(1.6) + +where + and +μ +σ are the permeability and conductivity of the signal conductor, and S is the +length of the effective perimeter of the signal conductor through which the surface +currents flow. The geometry-dependent constant η represents a factor determining the +increase in resistive losses due to currents in the return paths. In a thin stripline +configuration, one might expect the value of η to be in the neighborhood of +2 +η = +because the widths of the expected return paths are about the same as the +circumference of the signal conductor. In a coaxial cable, one might expect η to be less +than 2, because the return paths in the outer shield are significantly wider than the +circumference of the signal conductor. In practice, any of the parameters of equation +(1.6), including +AC +R + itself, may be varied in equation (1.5) to match laboratory data. + +However, there exists a common transmission system which does not fit equation (1.5) +well. The Gore EyeOpener ™ cable, for example, uses signal conductors constructed +from a heterogeneous combination of metal layers to achieve self-equalizing properties. +We now derive an approximation to the surface impedance for a thick bulk material, +covered by a relatively thin layer of another conductor, to address this case. General +field solutions to this type of problem were generated by Wait [5], which we here apply +to transmission lines. + + +We presume a planar, infinitely-thick conductor of bulk conductivity +2 +σ underneath a +thin layer of thickness +1τ and conductivity +1 +σ . As in the classical case, the electric field +will decay throughout the finite thickness +1τ to the value + +1 +1 +1 +1 +0 +( ) | +z +y +j +y +E +E e +τ +τ +δ += ++ +− += + +(1.7) + +where +1δ is the skin depth in the outer conductor. At the interface between the two +different conductors, the electric field will have a new behavior given by the boundary +condition requirements of Maxwell’s equations. The appropriate constraint is that of +continuous tangential electric field across the interface. Therefore, at the interface, the +electric field begins a new exponential behavior in the bulk material with new depth +constant +2 +δ : + + +1 +1 +1 +2 +1 +1 +( +) +0 +( ) +j +j y +z +E +y +E e +e +τ +τ +δ +δ ++ ++ +− +− +− += + +(1.8) + +Working out the integral for the total current under a width S, the resulting surface +impedance is: + + +1 +1 +1 +1 +1 +1 +1 +2 +1 1 +1 +2 +1 +2 +1 +1 +{ +}{ +( +1) +1} +{ +}{ +( +1) +1} +j +z +s +AC +j +Z +e +S +R +s +e +τ +δ +μ +σ +η +σ +σ δ +σ +σ +σ ++ +− +− +− +− ++ += +− ++ += +− ++ + +(1.9) + +Equation (1.9) adds a new factor to the classical surface impedance. The factor is a +function of the outer-conductor thickness +1τ and the ratio of the conductivities of the two +materials. In general, we see that the resistive and reactive portions of the surface +impedance are no longer equal when the conductor is composite. In our physical +approximation, we take the conductor width S to be the effective length of the perimeter +of the signal conductor. The overall series impedance is then: + + +1 +1 +1 +1 +2 +1 +2 +1 +{ +}{ +( +1) +1} +s +AC +R +sL +R +s +e +sL +μ +ητ +σ +σ +σ +− +− +∞ ++ += +− ++ ++ + +(1.10) + +Equation (1.9) is valid when the permeability of the all conductors is that of free space. +When the bulk conductor is magnetic, the conductivity of the bulk conductor +2 +σ can be +replaced by an effective conductivity + + +2 +2' +R +σ +σ +μ += + +(1.11) + + +where +R +μ is the relative permeability of the bulk conductor. + + +Shunt Conductance Variation with Frequency +Traditionally, the dielectric losses are characterized by a shunt conductance G per unit +length: + +1 +tan +tan +G +C +sC +j +ω +δ +δ += += + +(1.12) +The “loss tangent”, tanδ , is commonly specified by dielectric manufacturers to be in the +0.01 to 0.001 range. Typically, users are left to presume that the loss tangent is +independent of frequency. In such a case, the total shunt admittance can be written as + + +tan +(1 +tan ) +G +Cs +G +Cs +j +Cs +ω +δ +δ ++ += ++ += +− + +(1.13) + +We shall see that this form can give good agreement with laboratory data at frequencies +in the few-GHz range (and when dielectric losses are relatively small compared to the +series resistance). At higher frequencies (perhaps in the 10 GHz range) the model’s +main deficiency becomes apparent: the form of equation (1.13) cannot be physically +reasonable. It is well known that this form of dielectric loss is not a physically consistent +possibility [14]. In the present case, it is relatively straightforward to see why this is so. +If we take the series impedance to be lossless, that is, +0 +AC +R += +, then using equations +(1.13), (1.5), (1.2), and (1.1), the transform of the impulse response of the transmission +line can be written as: + + +(1 +tan +) +( ( )) +j +LC +j +z +h t +e +e +ω +δ +γ +− +− +− Δ += += +F + +(1.14) + +Equation (1.14) has a closed-form inverse transform, which is: + + +2 +2 +( ) +[( +) +] +h t +t +α +π +τ +α += +− ++ + +(1.15) + +where the parameters are given as + +Re{ 1 +tan } +Im{ 1 +tan } +z LC +j +z LC +j +τ +δ +α +δ += Δ +− += Δ +− + +(1.16) +Therefore, the impulse response of a transmission line with dielectric possessing constant +loss tangent is a delayed Lorentzian pulse. The Lorentzian form gives experimentally +plausible insights, such as: the amplitude of the pulse is inversely proportional to the +length of the transmission line, with proportionality constant simply related to the loss +tangent. However, we can observe that the impulse response extends back infinitely in +time even though its impulse excitation occurred at the time origin. The model predicts +non-causal behavior and is therefore not plausible as a physical model. We have found +that this feature of the constant-loss-tangent model is often the root cause of the failure to +match our experimental data at high frequency. + + +Other forms for the variation of the loss tangent with frequency must be applied in these +cases. In Figure 2 we highlight the difference in expected pulse shape between the +constant-loss-tangent model and that of a loss tangent varying linearly with frequency. +The response in the linear-loss-tangent case is derived numerically using equation (1.4) +because the problem does not have a closed-form solution. We will show later that the +linear-variation version can exhibit good fit to experimental data at frequencies in excess +of 10 GHz (which is our only justification for using it). + +0.8 +0.9 +1.0 +1.1 +1.2 +1.3 +-0.1 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 + Constant Loss Tangent + Linear Loss Tangent +Modeled Pulse Response, Volts +Time, ns +Constant Loss Tangent +Response is Lorentzian +1 +1 + Dt2 +Linear Loss Tangent Response Identically Zero Outside These Limits + +Figure 2: Comparison of Modeled Time Responses of a Transmission Line with No Series Loss but +Finite Dielectric Loss Due to Two Loss Models, Showing Causality Failure of Constant Loss Tangent +Assumptions. (20573) + +Time-Domain Laboratory Techniques + +Transmission lines (and other linear circuits) are often characterized by s-parameter +analysis using a vector network analyzer (VNA). The methodology for calibration, de- +embedding, and interpretation of VNA results is a highly-developed specialty, which +(when properly applied) will fully characterize the transmission line over a wide +bandwidth. + +For this work, we have elected to use a time-domain method, for the following reasons. +First, most VNAs have two ports and do not simply support differential-mode excitation. + +Second, in many applications, the primary information needed is the classical transfer- +function of the transmission line system (i.e., the information present in +21 +s in the +absence of reflections). The complexities in testing, calibration, de-embedding, and +interpretation for the other s-parameters may not be strictly necessary in some +applications. Third, a full-fledged experimental characterization of systems utilizing +transmission lines is often not limited to pure system-identification techniques, but also +may include direct measurements of higher-level system performance quantities (such as +error rate or eye diagrams). In some applications it is beneficial to integrate as many +measurement schemes as feasible into an experimental setup that is as simple (and +inexpensive) as possible, so long as the resulting measurements are “good enough” for +the application. + +Finally, there may exist nonlinearities in the transmission system which will introduce +interpretation errors in the s-parameter analysis without warning. These nonlinearities +are not likely to be in the transmission lines and interconnect (which are linear to an +excellent approximation) but may exist, for example, in the driver circuitry of a buffer +amplifier. Most (if not all) nonlinearities cannot be characterized by the magnitude and +phase of single-sinusoid reception; however, there exist waveforms which do broaden the +scope of complete nonlinear characterization. We shall see that a PRBS is one such +waveform. + +In the following section, we describe a method which addresses all four of these points. +Naturally, the described method will have its own disadvantages, which are generally +related to experimental approximations which do not exist (or are unnecessary) in a +VNA-type of analysis. + +Pseudorandom Sequence Excitation Method + +We describe a time-domain system identification method in this section. We follow the +scheme developed in [3,6]. A performance analyzer (‘bit error rate tester’) is used to +generate a 127-bit pseudo-random binary sequence (PRBS). The analyzer has true and +complement outputs, which are used to drive the differential transmission line. A digital +oscilloscope is used to sample the differential signal at the end of the transmission line. +The scope is triggered by the performance-analyzer synch-out pulse, which fires once +every PRBS period. This pulse provides a consistent time datum which is independent of +the transmission-line length. A typical experimental setup (shown for a cable) is as +shown in Figure 3. + +The upper panel in Figure 4 shows an example of the raw waveform as sampled at the +end of a transmission line. Because the raw waveform is too complicated to interpret +easily by eye, the waveform can be processed to generate a discrete-time pulse response. + +15 Meter Cable Assembly Eye Diagram Test +Board With Cable Assembly Under Test + +Figure 3: Test Setup for Measuring Error Rate, Eye Diagrams, and System Transfer Functions in the +Time Domain. (18045) +0 +10 +20 +30 +40 +50 +0 +.5 +1 +Extracted Pulse Response +Time from Signal Source Trigger Datum, nsec +-1 +0 +1 +Measured Voltage + +Figure 4: Example of a Measured PRBS Waveform and Its Extracted Pulse Response. (18322) + +The response (which we shall call extracted pulse response in keeping with the literature +[3,6]), is numerically generated using a two-step process. First, the raw waveform is re- +sampled to take care of experimental frequency errors between the PRBS generator and +the oscilloscope, and to provide an appropriate sampling rate for subsequent analysis. +Second, the re-sampled waveform is mathematically deconvolved with a theoretically +perfect version of the same PRBS. We shall return to describe a very efficient method +for this deconvolution. + +The extracted pulse response is analogous to the impulse response in continuous time +systems. In this case, the extracted response is the expected response of the transmission +line (and measurement system and signal drivers) if the PRBS driver had output a simple +base-to-peak digital pulse instead of the PRBS. + +The lower panel in Figure 4 shows the extracted response for the given raw waveform. +While both waveforms contain exactly the same information, it is clear that the extracted +pulse is a short primary response, with some small echoes of interesting shape and +location on the time axis. In Figure 5, the extracted pulse response for two different, but +short, PCB transmission lines is shown. All connectors, test cables, and test conditions +are nominally identical, except for the length of the transmission line. +5 +0.0 +0.2 +0.4 +0.6 +0.8 +6 +7 +8 +10 +11 +12 +13 +14 +15 +5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +6 +7 +8 +9 +10 +Time from Signal Source Trigger Datum, nsec +Extracted Pulse Response +( 6 Inch Line ), Volts +Extracted Pulse Response +( 1.5 Inch Line ), Volts +11 +12 +13 +14 +15 +Nonlinear +Echo +Main Pulse +First +Reflection +Second +Reflection + +Figure 5: Extracted Pulse Responses from Two Lengths of Otherwise Identical Lines (1.5” and 6”) +(18323) + + +In the upper panel of Figure 5, we zoom in on the interesting sections of the extracted +pulse response, for a 1.5 inch PCB line. As might be expected from the significant +impedance mismatches, the signal appears to reflect back and forth between the +impedance discontinuities before settling down. At 2.5 Gbits/second excitation, the +pulse response is nominally 400 picoseconds wide, and the first forward-moving +reflection should begin to arrive about a half nanosecond after the onset of the main +pulse; the second forward-moving reflection arrives a half nanosecond after that, and so +forth. Thus we should not expect to see separation between the main pulse and its +subsequent, exponentially decaying reflections. + +In the lower panel of Figure 5, we see the effects of quadrupling the transmission-line +length. The now well-resolved reflections occur at approximately 2 nanosecond +intervals (as expected). The main pulse is resolved, and is proportional in magnitude to +what the pulse would have been down 6 inches of this transmission line without the +reflections due to impedance mismatches, but including any pulse modification due +directly to impedance mismatches. (The interface of two perfect transmission lines of +real, but different, impedance at all frequencies will modify the main pulse in amplitude, +only). + +The main pulse and reflections may also be resolved by increasing the excitation +frequency of the PRBS on a given line (instead of changing line lengths, as above). +This technique may be limited by the rise time of the drivers, the time span of the impulse +response of the transmission lines under test, and the analog bandwidth of the sampling +oscilloscope. + +These observations suggest a type of de-embedding technique which is very simple, +though approximate. We assume that the combination of the transmission line length, +the resolution of the channel response, and the frequency of the PRBS is such that the +main pulse is resolvable from reflections or other imperfections. We then claim, to the +extent these assumptions are true, that the resolvable main pulse may be written as + + + + +1 +1 +0 +1 +1 +0 +1 +( ) +{ +( ) +( ) +( ) +( )... +( , +)} +{ +( ) +( ) +( , +)} +driver +launches +scope +tline +T +tline +x t +F +X +H +H +H +H +z +F +X +H +H +z +ω +ω +ω +ω +ω +ω +ω +ω +− +− +≅ +Δ +≅ +Δ + (1.17) + +where +0( ) +X ω is the Fourier transform of an ideal PRBS sequence, +1z +Δ +is the length of +transmission line, and the total transfer function +( ) +T +H +ω is the composite linear response +of all non-ideal elements in a practical test. Note that the interpretation of +1( ) +x t is not +exactly that of the time response related to the s-parameter +21 +s ; it is only equivalent to s- +parameter analysis when there are no reflections. In principle, and with sufficient +cleverness, we could also resolve each reflection in the measured waveforms to +independently and fully analyze the entire characteristic described by +21 +s . However, we + +shall not develop those techniques in this report; instead, we shall focus only on resolving +the response that would have occurred without reflections. + +If we now consider a transmission line of length +2 +1 +z +z +Δ +> Δ +, then its response can be +written as + +1 +2 +0 +2 +1 +0 +1 +2 +1 +( ) +{ +( ) +( ) +( , +)} +{ +( ) +( ) +( , +)} +T +tline +tline +x t +X +H +H +z +X +X +H +z +z +ω +ω +ω +ω +ω +ω +− +− += +Δ += +Δ +− Δ +F +F + +(1.18) + +where the second line follows from the separability of the transmission line transfer +function (equation (1.1)) into component factors, that is, + + +2 +1 +2 +1 +2 +1 +2 +1 +( +) +( +)( +) +( +)( +) +( +)( +) +2 +1 +2 +1 +( , +) +( , +) +( , +) +z +z +z +z +z +z +z +z +tline +tline +tline +H +z +e +e +e +e +H +z H +z +z +γ +γ ω +γ ω +γ ω +ω +ω +ω +− +Δ +− +Δ +Δ +−Δ +− +Δ +−Δ +− +Δ +−Δ +Δ += += += += +Δ +Δ +− Δ + +(1.19) + +Despite its mathematical look, the suggested modeling technique is exceedingly simple: + +1. Measure +1( ) +x t of a relatively short line and find its extracted pulse response. +2. Calculate the expected response of a longer transmission line of length +2z +Δ + by +applying equation (1.4) for a transmission line of length +2 +1 +z +z +Δ +− Δ . +3. Compare the output of step 2 to an actual measurement of the extracted pulse +response of a transmission line of length +2z +Δ +. Focus on the (by assumption, +resolvable) main pulse only; ignoring all other interesting blips. + +We have argued that this technique will give an expected waveform, and an actual +waveform, to measure the goodness-of-fit of the RLGC model; furthermore this +comparison de-embeds all systematic linear effects in the test, to the extent that the main +pulse is resolvable in time from all reflections or other phenomenon. + +The “resolvability” criterion gives this technique its simplicity, but is its major source of +interpretation uncertainty when compared to more accurate experimental methods (such +as, a VNA). A typical “eyeball” comparison is good to perhaps 30-40 dB (a few percent +in voltage). In many applications (such as, the identification of eye diagrams), the +“eyeball” criterion is, nearly by definition, good enough. However, the engineer must +judge what kind of accuracy is required for a particular application. + +We have argued that the proposed technique effectively de-embeds effects which can be +described by a linear transfer function. We now briefly digress to discuss the effects of +stochastic jitter in the trigger waveform and oscilloscope timings. These effects are not +present in frequency-domain measurement techniques, and therefore deserve further +discussion. + +In general the analysis of the all jitter effects on the waveform captured by the +oscilloscope is very difficult, but the following simplifications allow some insight into + +the problem. If the jitter is small enough that the waveform is well approximated by its +first-order Taylor’s series, that is: + + +( +) +( ) +'( ) +x t +x t +x t +τ +τ ++ Δ +≅ ++ Δ + +(1.20) + +and all jitter phenomenon are lumped into a single effective jitter at the scope sampler +(relative to the signal ( ) +x t ), with jitter variance +2 +σ , then the signal can be thought of in +two components. There exists a stochastic portion (whose spectrum is continuous even +if ( ) +x t is periodic) whose power is linearly related to +2 +σ and the average square of the +derivative of ( ) +x t . This power comes at the expense of the high frequencies in the +deterministic portion of the identified waveform (that is, its averaged sequence value as +sampled on the scope). It can be shown that the jitter produces a stochastic filter whose +average transfer characteristic is + + +2 +2 +( ) +jitt +H +e σ ω +ω +− += +% + +(1.21) + +if the jitter is Gaussian. Therefore, if +2 / +f +π σ +<< + , the jitter effects should be negligible +in the sampled waveform ( ) +x t . If the standard deviation of the jitter is in the few pS +range, then frequencies below about 10 GHz are affected by less than 1 percent. + +In a sense, +( ) +jitt +H +ω +% + can be thought of as one of components of the total deterministic +transfer function +( ) +T +H +ω , but only if the various realizations of +( ) +jitt +H +ω +% + are averaged +sufficiently to converge to the average transfer function. This convergence might +happen, for instance, when the waveform reported from the scope is the average of a +large number of sample sequences. In such a case, this portion of the transfer function +cancels out in our method (as do the deterministic transfer functions). However, this is +dangerous ground, and the experimenter should take care to ensure that the stochastic +conditions of the tests are identical when exploiting this effect. For example, if trigger +jitter is significant, the realization of +( ) +jitt +H +ω +% + for a waveform derived by “averaging” +only one sample waveform is in general quite different from the realization of +( ) +jitt +H +ω +% + +when many averages are taken. (The frequency response will look better, in general, +with only a few averages than it does with long averages). + +Numerical Methods + +In the preceding section, we developed the concept of the extracted pulse response, +which is the circular deconvolution of the measured PRBS sequence with an ideal version +of the same sequence. A good, standard method for deconvolution is to take the Fourier +transform of both waveforms, perform a complex division, and then take the inverse +Fourier transform. + + +In the present case, the number of points in the transform is +2 +1 +N +n = +− , which is very +often a prime number, and is never highly composite. In such cases, the best fast +algorithms for discrete Fourier transforms (DFTs) are relatively useless. In most such +cases, the deconvolution will require about +2 +n complex multiplication operations and +about n complex divisions. + +We now show a special deconvolution method which takes optimal advantage of the fact +that measured waveform is a filtered version of a PRBS. This method requires about +2 +n +real additions, zero multiplications, zero divisions, and zero complex operations of any +sort. + +We develop the algorithm in the following way. If the sampled digital-pulse response of +the transmission line system is denoted by [ ] +( +) +h n +h nT += +, with n an integer and T the bit +period of the sequence, then the expected sampled measured output is giving by the +circular convolution of the PRBS and the pulse response: + + +[ ] +[ +] [ ] +k +y n +h k +n p k += +− +∑ + +(1.22) +where [ ] +{ 1,1} +p n ∈ − + represents the PRBS and k ranges over the length L of the PRBS, +which is always of the form +2 +1 +N +L = +− . We shall also call [ ] +h n the extracted pulse +response, after we have back-calculated it from measured data. Because this is a circular +convolution, the sequences y, p, and h are assumed to be periodic in L; therefore the +indices can always be taken modulo L. + +We now assume that the mathematical convolution was actually completed in the +physical world by playing the ideal PRBS through a linear system, and we are able to +measure the analog waveform ( ) +y t directly. We then resample this waveform to obtain +the measured version of [ ] +y n . By inspiration, we can generate a new function from [ ] +y n +and note how else it can be expressed: + + +1 +[ ] +[ ] [ +] +1 +1 +[ +] [ ] [ +] +1 +[ ] +l +l +k +x +Z n +y n p n +l +L +h k +n p k p n +l +L +h n +μ += +− ++ += +− +− ++ += ++ +∑ +∑∑ + +(1.23) + +where +x +μ is the mean of the pulse response [ ] +x n . The last line of equation (1.23) is +general only when the excitation is a PRBS, and is derived using the properties of a +PRBS [4]. Therefore we can recover, within a small DC shift, the extracted pulse +response [ ] +h n from the measured sequence [ ] +y n by a simple transformation using +2 +( +1) +L L +L +− +≅ + additions and L trivial multiplications by 1 +± . Depending on the +application, the division by L+1 may or may not be necessary; if it is necessary, +implementers should note that the divisor is always exactly a power of two. With a little + +more complexity in the algorithm it is possible to reconstruct exactly the DC conditions +to get a new function +'[ ] +[ ] +Z n +h n +≡ +, but in our case we have simply ignored the small DC +offset as it makes little practical difference in our analysis. + +We have developed this algorithm in a general way when we are interested in analyzing +or plotting waveforms with more than one point per PRBS bit. In such a case, we can +decimate a waveform representing M points per bit into M waveforms representing +[ ] +m +y +n , that is, the sampled version of (( +/ +) ) +y n +m M T ++ +. We can determine the extracted +pulse response for each sub-channel, +[ ] +m +h +n , using equation (1.23) and +[ ] +m +y +n as input. +The resultant response +[ ] +m +h n is the sampled version of the analog waveform +(( +/ +) ) +h n +m M T ++ +. Therefore, we can reconstruct the M-point-per-bit version of the +response by re-interleaving the results of each +[ ] +m +h +n . Note that the calculations of +[ ] +m +h n +are only dependent on values of y in its own interleave; whereas in general this property +is not true of DFT-type deconvolutions. This property is another advantage of this type +of specialized deconvolution. + +Therefore, we have shown a very efficient method for deconvolution when PRBS +excitation is used. Clearly, for offline analysis using relatively small L, there is little +practical difference between a few microseconds and a several milliseconds of +computation time. Even so, these differences can become quite significant for large +enough L even for offline analysis because L grows exponentially with each PRBS order +N. Perhaps more significantly, the algorithm in equation (1.23) is simple enough to +realistically be implemented at-speed in hardware. For example, the algorithm +implementation is simply a single accumulator if we are willing to calculate one +( ) +Z n at a +time in an operation. + +Nonlinear Analysis with PRBS Excitation + +Up to this point, we have not described anything of particular value in using a PRBS as +the excitation waveform over other types of waveforms (other than its alternative utility +as a convenient bit-error-rate test pattern). Indeed, a TDT-type step waveform gives the +same information as does an extracted pulse response for a linear system. We now +digress to explain an interesting and useful feature of this type of PRBS-deconvolution +analysis, which allows native nonlinear Volterra analysis of the waveforms. + +In each of the extracted pulse waveforms discussed thus far (shown in Figure 4 and +Figure 6), a strange blip appears to arrive at the oscilloscope 2.5 nanoseconds before the +main pulse. This blip always shows up in exactly the same place (relative to the main +pulse), independent of the transmission-line length or type, or even whether there is a +transmission line under test at all. If the bit excitation frequency is changed, then the +time spacing between the blip and the main pulse also changes. However, if the time +spacing is normalized to bit time units, now this number now remains constant, +independent of excitation frequency. Therefore, we have discovered an apparently +physical delay effect which seems to know something about the bit times. No signal + +phenomenon of any kind shows up at these locations when a simple TDR pulse is used as +excitation. + +The reason for this type of behavior was resolved in the 1980s [3]. The phenomenon +came to be understood in the following way. If a system is linear, then the +deconvolution operation, described above, always gives the same results for the extracted +pulse response no matter what the underlying data pattern is. If a system is somewhat +nonlinear, then in the general case, the deconvolution operation will contain garbage +which will appear as a “noise” everywhere on the extracted pulse response. However, +if the data pattern is a PRBS, then for many types of nonlinearities, the disturbances due +to such nonlinearities destructively interfere almost everywhere after the deconvolution +operation. The nonlinearities tend to constructively interfere at specific locations on the +extracted pulse response, and with specific shapes and magnitude, all of which vary +depending on the nature and severity of the nonlinearity. This phenomenon is now used +universally in magnetic recording to identify various impairments of the magnetic +system. + +In our case, the blip phenomenon is due to circuit-driver nonlinearity somewhere in the +test system. Analysis shows that this imperfection (if so it may be called) is due to a +non-linearity in the driver circuit in the performance analyzer. Close examination of the +output differential pulses show an asymmetry in the rise times (as compared to the fall +times) of the output driver. As a result, a positive-going pulse looks noticeably different +than the opposite of a negative-going pulse. This difference is by definition a nonlinear +effect. + +It has been shown that the PRBS phenomenon is related to the Volterra-series nonlinear- +system identification technique for discrete-time systems [4]. This property is another +example of the apparently endless set of practical and useful features of a PRBS. + +To an excellent approximation, the transmissions lines are natively linear, and (in +principle) the choice of excitation waveform is immaterial. However, circuit drivers, +receivers, digital-to-analog converters, equalizers, analog-to-digital converters, and other +elements of a complex analog communication system, are not natively linear. In such a +case, analysis using linear assumptions, under conditions of PRBS excitation yields (for +no additional work), nonlinear system analysis capability. + +Comparison to Theory +We now present experimental PRBS results gathered from several different types of +transmission lines, and compare the results to expectations from our RLGC model. We +found that a constant loss tangent assumption was adequate at lower frequencies (~2.5 +Gbits/second excitation) but completely inadequate at 40 Gbits/sec excitation. We +concluded that the series losses were modeled adequately by the standard form at all +frequencies tested, except for any frequency using a bilayer-conductor transmission +system. + +2.5 Gbits/second Excitation Results + + +We applied the preceding techniques to a PCB transmission line. We designed and +procured a passive printed circuit board (PCB) using GETEK ™ dielectric materials. +This board was fabricated to verify the accuracy of the RLGC model described above. + +Manual RLGC-model fitting procedures produced model results which compared quite +favorably to the experimentally-measured results at 2.5 Gbits/second. These results are +compared, for four different lengths of the transmission line, in Figure 6. All model +parameters are frozen at the given values (except, of course, for the transmission-line +length) for all model results on this graph. We used a 6-inch version of the lines as the +de-embedding reference. The important model parameters for the GETEK materials are: + + +378 nH/m, +117 pF/m, tan +0.011, +5.45 7, +/ +5 mils. +L +C +E +S +δ +σ +η +∞ = += += += += + +(1.24) + +We tested several lengths of 100 Ohm Skewclear Infiniband 12X cable from +Amphenol/SpectraStrip ™. We merely used the advertised (nominal) odd-mode +impedance and velocity numbers to calculate the odd-mode reactive parameters. + +The model parameters for the SKEWCLEAR cable are: + + +377 nH/m, +37.7 pF/m, tan +0.0001, +6.0 7, +/ +17mils. +L +C +E +S +δ +σ +η +∞ = += += += += + (1.25) + + + +10 +12 +14 +0.0 +0.2 +0.4 +0.6 +0.8 +12 +14 +16 +0.0 +0.2 +0.4 +0.6 +0.8 +14 +16 +18 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +18 +20 +22 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Time from Signal Source Trigger Datum, nsec +Time from Signal Source Trigger Datum, nsec +Amplitude of Pulse Response, +Volts +Amplitude of Pulse Response, +Volts + Measured 12 inch Trace + Modeled 12 inch Trace + Measured 36 inch Trace + Modeled 36 inch Trace + Measured 24 inch Trace + Modeled 24 inch Trace + Measured 60 inch Trace + Modeled 60 inch Trace + +Figure 6: Experimental Comparison of GETEK PCB to RLGC Model in Odd Mode with Parameters +378 nH/m, +117 pF/m, tan +0.011, +5.45 7, +/ +5 mils. +L +C +E +S +δ +σ +η +∞ = += += += += + (18334) + +Figure 7 shows the modeled and actual pulse responses for several lengths of cable. In +this case, the cable connectors were clearly a significant factor in the test system response +as is seen by the ringing in the response tail. However, the de-embedding method The +reference measurement (shown in the first panel of Figure 7, used as input to the +numerical model, was taken from a relatively short, 1 meter cable. + + +12 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +14 +16 +18 +0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +2 +4 +Time from Signal Source Trigger Datum, nsec +Time from Signal Source Trigger Datum, nsec +Amplitude of Pulse Response, +Volts +Amplitude of Pulse Response, +Volts +6 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +22 +24 +26 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +30 +32 +34 +Measured 10 meter Cable +Modeled 10 meter Cable +Measured 15 meter Cable +Modeled 15 meter Cable +Measured 5 meter Cable +Modeled 5 meter Cable +Measured-Data +Reference Waveform +From 1 meter Cable + +Connectors + +Figure 7: Experimental Comparison of SkewClear Cable to RLGC Model using Parameters +377 nH/m, +37.7 pF/m, tan +0.0001, +6.0 7, +/ +17mils. +L +C +E +S +δ +σ +η +∞ = += += += += + (18324) +We also analyzed a Gore EyeOpener Plus cable (26 AWG), which was optimized by +Gore for operation at 2.5 Gbits/second. The experimental data was not fitted well with +the traditional model for surface impedance. We assumed that the cable was built with +stratified signal conductors and were able to fit the data much better using equation (1.10) +for the surface impedance. The fitted parameters for EyeOpener cable are: + + +1 +2 +1 +387 nH/m, +37.7 pF/m, tan +0.0004, +6 7, +1 7, +0.115 mil, +/ +21 mils. +L +C +E +E +S +δ +σ +σ +τ +η +∞ = += += += += += += + +(1.26) + +The effective conductivity of the core conductor is so low, for a metal, that the material is +likely to be magnetic. The effective conductivity of the bulk material is “reduced” by a +factor equal to the relative permeability of the material in our model. + +Figure 8 shows the modeled and actual pulse responses for two lengths of the Gore +EyeOpener Plus cable. The reference measurement, used as numerical model input, was +taken from a relatively short, 1 meter cable. Because little or no geometry or +composition information was available for the cables, we do not claim that the fit +physical parameters truly reflect the physical cable characteristics. We do claim that this + +combination of parameters adequately describes the differential behavior of the cables +over a significant range of cable lengths. + +0 +1 +2 +3 +4 +5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Time from Signal Source Trigger Datum, nsec +32 +33 +34 +35 +36 +37 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Amplitude of Pulse Response, Volts + Measured 5 meter cable + Modeled 5 meter cable + Measured 10 meter cable + Modeled 10 meter cable + +Figure 8: Experimental Comparison of EyeOpener Cable to RLGC Model using Parameters +1 +2 +1 +387 nH/m, +37.7 pF/m, tan +0.0004, +6 7, +1 6, +0.115 mil, +/ +21 mils. +L +C +E +E +S +δ +σ +σ +τ +η +∞ = += += += += += += + +(18446) + +We conclude that the RLGC model, with appropriate extensions when necessary, +accurately describes a variety of cable transmission lines and PCB lines at 2.5 +Gbits/second excitation. + +40 Gbits/second Excitation + +We shall now present the time-domain analysis with data taken at 40 Gbits/second +excitation. While (in principle) an infinitely-sharp rise time at 2.5 Gbits/second will also +excite very high frequencies which will then be identified with by our method even at +low excitation frequency, it’s also true that equipment made to produce 40 Gbits/second +data will have significantly smaller rise times and that of equipment made to generate a +PRBS at lower frequencies. Using such equipment is therefore experimentally better if +high frequencies are to be accurately modeled by this method. + +In Figure 9, we show comparisons for model versus actual pulse responses for various +lengths of a PCB trace made using Nelco 1300 ™ material. In this case we found much + +improved experimental fit using a linear loss tangent model. The graphs show quite a bit +of attenuation for this type of transmission line at these frequencies (which is not at all +surprising as this material is not intended for use up at these frequencies over any +reasonable length of trace). We find the fit of these curves to be fairly good, but clearly +there exist differences between expected and actual traces which do not exist in the 2.5 +Gbit/second data. The frequency content of these waveforms is only significant up to +approximately 10 GHz of bandwidth, so we do not claim to have a model which matches +data above this bandwidth. + +Model parameters for the 40 Gbits/second Nelco 1300 PCB traces are: + + +327 nH/m, +125 pF/m, / +5.75 mils, tan =2E-13 , =5E7 S/m +L +C +S η +δ +ω σ +∞ = += += +(1.27) + +Note that this model uses a loss tangent which is proportional to frequency. + +Measured Reference (3" Trace) +20" Trace +34" Trace +42" Trace +Measured +Modeled +Measured +Modeled +Measured +Modeled +2.4 +0.0 +0.1 +0.2 +0.3 +2.5 +2.6 +2.7 +Time, ns +Pulse Response, Volts +2.8 +2.9 +3.0 +1.1 +0.0 +0.1 +0.2 +0.3 +1.2 +1.3 +1.4 +Time, ns +Pulse Response, Volts +1.5 +1.6 +1.7 +2.0 +0.0 +0.1 +0.2 +0.3 +2.1 +2.2 +2.3 +Time, ns +Pulse Response, Volts +2.4 +2.5 +2.6 +2.4 +0.0 +0.1 +0.2 +0.3 +2.5 +2.6 +2.7 +Time, ns +Pulse Response, Volts +2.8 +2.9 +3.0 + +Figure 9: Experimental Comparison of Nelco 1300 PCB to RLGC Model at 40 Gbits/s using +Parameters +327 nH/m, +125 pF/m, / +5.75 mils, tan =2E-13 , =5E7 S/m +L +C +S η +δ +ω σ +∞ = += += +(20568) +Finally, in Figure 10 we show the results from data taken at 40 Gbits/second from an +advanced version of the SkewClear cable. This version is designed for higher +frequencies and includes a modified ground conductor. + + + +The model parameters for the upgraded SkewClear cable are: + + +451 nH/m, +40 pF/m, / +20 mils, tan =1E-13 , +=5E7 S/m +L +C +S η +δ +ω σ +∞ = += += + (1.28) + +Note that again the loss tangent model is linear in frequency. +0.9 +1.0 +1.1 +1.2 +1.3 +1.4 +1.5 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +Pulse Response, Volts +Time, ns +2.5 +2.6 +2.7 +2.8 +2.9 +3.0 +3.1 +0.0 +0.1 +0.2 +0.3 +0.4 +Pulse Response, Volts +Time, ns + Measured + Modeled +1 Meter Reference, +Differentially Excited and Received +1 Meter Reference, +Singly Excited, Differentially Received +5 Meter Cable, +Differentially Excited and Received +5 Meter Cable, +Singly Excited, Differentially Received +0.9 +1.0 +1.1 +1.2 +1.3 +1.4 +1.5 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +Pulse Response, Volts +Time, ns +2.5 +2.6 +2.7 +2.8 +2.9 +3.0 +3.1 +0.0 +0.1 +0.2 +0.3 +0.4 +Pulse Response, Volts +Time, ns + Measured + Modeled +Mismatch Indicates +Even Mode Leakage + +Figure 10: Experimental Comparison of Upgraded SkewClear Cable to RLGC model at 40Gb/s +under Single Ended and Differential Excitation Conditions using Model Parameters +451 nH/m, +40 pF/m, / +20 mils, tan =1E-13 , +=5E7 S/m +L +C +S η +δ +ω σ +∞ = += += + (20571) +In this case, the fit is nearly as excellent as for the 2.5 GHz case, even though the +significant frequencies in this waveform extend up to approximately 20 GHz. The usual +model comparison (in the upper right graph) shows excellent fit except for the area +around 100 pS before the main pulse, which shows a relatively small blip of apparently +unknown origin. + +We shall take this opportunity to show the types of conclusions one may draw from +investigations using time domain techniques. We have shown similar data on the lower +portion of the figure except in this case the excitation is provided only on one side of the +differential cable, with the other input terminated. Theoretically, or ideally, this +condition should produce exactly half the differential signal at the output terminals (and +indeed this is very well approximated for the 1 meter reference signals). However, the +singly-excited model comparison is rather weak, though (evidently) the errors apparently +nearly cancel when the system is both differentially excited and differentially received +(the upper right case). The simplest explanation for this behavior is a mode leakage +between even and odd modes down the length of the line; and indeed, we have shown by + +similar methods that the even mode signal has a slightly higher velocity than does the odd +mode, such that the even mode shows up about 100 pS before the odd mode at a distance +of 5 meters. + +Conclusions +We have developed a time-domain method for experimental verification of an RLGC +model, and showed adequate experimental fit over a wide variety of transmission line +types up to approximately 10 GHz in bandwidth. We found that, depending on the +frequency and transmission line type, specific extensions were required to adequately +explain laboratory behavior. We found that modifications to both the dielectric (shunt) +loss model and the resistive (series) loss models were necessary to adequately cover all +cases. + +We have shown that, despite a fairly substantial list of assumptions, a PRBS time-domain +method can be used in a de-embedding fashion to remove many experimental +impediments. Indeed, the PRBS technique can be used to easily identify and quantify +nonlinear effects. + +We proposed an extremely efficient method for implementing the PRBS identification +process, which can either speed up offline computation when the waveform is large, or, +can be used in hardware to directly implement the method in real-time so long as an +appropriately complex ADC is available. + +We note that the proposed technique is similar to (in fact, it is a simplification of) a +generalized TDT method, possessing a similar set of advantages and disadvantages when +compared to frequency-domain methods. For the price of software processing and a real +time scope, this method can be used to turn a bit-error-rate test system into a reasonable +system identification station. + +While we have shown laboratory data indicating the utility of this measurement method +using high-quality lab equipment, we believe that this type of analysis will become most +attractive in self-test applications in product hardware. As the analog-to-digital +converters (ADCs) in a SERDES migrate from traditional 1-bit comparators to those +required to support advanced signaling and detection schemes, we expect that methods +such as those described in this paper will be used to advance the operational and +diagnostic capabilities of the overall system, as they have in other systems [12]. +Acknowledgements + +The authors would like to thank Eric Hanlon and Devon Post for providing test boards +and cables for this activity; Jason Prairie for his data-taking expertise; Patrick Zabinski +for many instructive and seminal conversations; and Steve Richardson, Elaine Doherty, +and Deanna Jensen for generating the graphics for this report. + +References + + +[1] Richard E. Matick. Transmission Lines for Digital and Communication Networks. +IEEE Press, 1999. + +[2] Simon Rano, John R. Whinnery, and Theodore Van Duzer. Fields and Waves in +Communication Electronics. John Wiley and Sons, second edition, 1984. + +[3] Dean Palmer, Jon Coker, Michael Meyer, and Pablo Ziperovich. Overwrite in thin +media measured by the method of pseudorandom sequences. IEEE Transactions on +Magnetics, 24(6), November 1988. + +[4] Reto Hermann. Volterra Modeling of digital magnetic saturation recording channels. +IEEE Transactions on Magnetics, September 1990. + +[5] James R. Wait. Electromagnetic Waves in Stratified Media. IEEE Press, 1995. + +[6] Dean Palmer, Pablo Ziperovich, Roger Wood, and Thomas Howell. Identification of +nonlinear write effects using pseudorandom sequences. IEEE Transactions on +Magnetics, volume 24, November 1988. + +[7] D.A Smolyansky and S.D Corey. Computing self and mutual capacitance and +inductance using even and odd TDR measurements. Electrical Performance of Electronic +Packaging, 2002. + +[8] Cherry Wakayama and Jeff Loyer. Correlation between VNA and TDR/TDT +extracted s-parameters up to 20 GHz. TDA Systems customer note. + +[9] James R. Andrews. Time domain spectrum analysis and s-parameter vector network +analysis. Picosecond Labs app note AN-16A, November 2004. + +[10] Martin Schmatz et al. A 22 Gbit/s PAM-4 receiver in 90 nm CMOS-SOI +technology. Symposium on VLSI Circuits, 2005. + +[11] Fidanboylu, K.M.; Riad, S.M. and A. Elshabini-Riad. An enhanced time-domain +approach for dielectric characterization using stripeline geometry. IEEE Transactions on +Instrumentation and Measurement, Volume 41, Issue 1, Feb. 1992 + +[12] United States Patent 5392295. Error measurement circuit. February 21, 1995. + +[13] Engin, A.E.; Mathis, W.; John, W.; Sommer, G.; and Reichl, H. Time-domain +modeling of lossy substrates with constant loss tangent. Proceeding of the 8th IEEE +Workshop on Signal Propagation on Interconnects, May 2004. + +[14] Coperich Branch, K.M et al. Physically consistent transmission line models for high- +speed interconnects in lossy dielectrics. Advanced Packaging, IEEE Transactions on +Volume 25, Issue 2, May 2002 Page(s):129 - 135 + diff --git a/3dE4T4oBgHgl3EQf0Q2d/content/tmp_files/load_file.txt b/3dE4T4oBgHgl3EQf0Q2d/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4cf288c9d7f9eee0d2b37f5d316f7143ae68c3d4 --- /dev/null +++ b/3dE4T4oBgHgl3EQf0Q2d/content/tmp_files/load_file.txt @@ -0,0 +1,646 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf,len=645 +page_content='DesignCon 2006 Time Domain Verification of Differential Transmission Line Modeling Methods Jonathan D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Coker, Mayo Clinic Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Erik S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Daniel, Mayo Clinic Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Barry K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Gilbert, Mayo Clinic gilbert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='barry@mayo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='edu 507-284-4056 Abstract The advantages and limitations of time-domain pseudo-random binary sequence (PRBS) excitation methods for system identification of individual modes within a multi- conductor transmission system are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' We develop the modifications necessary to standard frequency-domain transmission-line models to match time-domain experimental data from several types of transmission systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' We show a variety of experimental results showing very good to excellent agreement with our model’s predictions, up to approximately 10 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Author Biographies Jon Coker is a Principal Project Engineer of the Special Purpose Processor Development Group at the Mayo Clinic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Coker graduated from Wheaton College with a Bachelor of Arts degree in 1982 and from the University of Minnesota with a Bachelor of Science degree in electrical engineering in 1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' He is currently pursuing a Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' degree at the University of Minnesota.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Erik Daniel received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' degree in physics and mathematics from Rice University, Houston, TX, in 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' He received the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' degree in solid state physics from the California Institute of Technology, Pasadena, CA, in 1997, with thesis research focusing on simulation, fabrication, and characterization of quantum effect semiconductor devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' He currently is a Staff Scientist in the Department of Physiology and Biomedical engineering, Mayo Clinic, Rochester, MN, and the Deputy Director of the Special- Purpose Processor Development Group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Barry Gilbert received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' degree in electrical engineering from Purdue University, West Lafayette, IN, in 1965 and the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' degree in physiology and biophysics (with minors in electrical engineering and applied mathematics) from the University of Minnesota in 1972.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' He is currently a Staff Scientist in the Department of Physiology and Biomedical engineering, Mayo Clinic, Rochester, MN, and the Director of the Special Purpose Processor Development Group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Introduction High-speed digital back-plane communication channels have long since given up their “digital design” status and have come to resemble (architecturally) previously-existing, complex analog communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Equalization, error correction, modulation codes, or advanced detection schemes are now commonly proposed and implemented for serializer-deserializer (SERDES) channels (see [10], for example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' To achieve the maximum benefit of these methods, we require increasingly accurate system identification of the transmission system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' At the same time, we find that traditionally excellent transmission-line models begin to diverge from experiment as bandwidths begin to extend into the GHz and tens of GHz range [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Thus reliable methods for system identification and model verification for candidate transmission systems supporting such channel implementations are of increasing and fundamental importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Several methods exist for accurate system identification of linear systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Analysis using network analyzers (including vector network analyzers) is a highly-developed technology using frequency-domain measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Advanced time-domain methods have more recently come on the scene employing TDR and TDT measurements of a step excitation [7-9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' In this paper, we present a technique to predict and experimentally verify transmission line models purely in the time domain using the method of pseudorandom sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Our technique is an extension of a method developed for system identification commonly used in magnetic recording data channels and testers [3,6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' We shall focus on the description of transmission lines which are suitable for a high-speed serial- communication link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' We shall compare expected and experimental results from printed circuit board (PCB) transmission lines and to results from cables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' While we shall not argue that the proposed method is experimentally superior to established time-domain techniques for linear systems, we shall discuss the unique nonlinear-system- identification capabilities of a PRBS waveform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' In addition, we shall show a fast algorithm which is simple enough to consider implementing in modern SERDES hardware, thus allowing a wide range of inexpensive and highly capable built-in self-test and board diagnostic capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' We shall describe our specific model of a transmission line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' This section is intended to outline the assumptions and limitations of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' We also present our alternative extensions to the standard telegrapher’s model, which we found necessary to adequately describe experiment in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Transmission Line Modeling In this section, we briefly review a classical transmission-line model, and discuss the parameters which we use in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=" Telegrapher’s Model in the Time Domain The basic telegrapher's model for transmission lines is fully worked out in several texts [1,2]." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' The standard solution gives the voltage transfer function for a wave traveling in the positive z direction as ( ) ( ) z H e γ ω ω − Δ = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='1) where ω is the radial frequency, z Δ is the distance down the line, and the complex propagation constant ( ) γ ω is: ( ) ( )( ) ( )( ) R j L G j C R sL G sC γ ω ω ω = + + = + + (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='2) If the Fourier transform of an input waveform ( ) x t is ( ) X ω , then the Fourier transform of the waveform at the output of the transmission line, ( ) y t , is ( ) ( ) ( ) Y H X ω ω ω = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='3) and the time-domain version of the output waveform is 1 ( ) { ( ) ( )} y t H X ω ω − = F (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='Cm L C C R G C G C L + L m R L + L m Cm G L L to Infinity to Infinity G L R R L L R R Gm C C Cm G G Gm C C Cm G G Gm Lm Lm L C C Lm (x1 (t) + x 2(t)) GENERAL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='EQUIVALENT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='CIRCUIT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='FOR SYMMETRIC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='SIGNAL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='CONDUCTORS EVEN-MODE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='EQUIVALENT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='CIRCUIT ( Lines ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='Toggling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='Same ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='Direction ) R G + 2G m G + 2G m C + 2C m C + 2C m L - L m R L - L m to Infinity (x1 (t) - x 2(t)) x1 (t) x2 (t) ODD-MODE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='EQUIVALENT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='CIRCUIT ( Lines ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='Toggling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='in Opposite ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='Directions ) 1 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='Figure 1: Simplified Odd Mode and Even Mode Equivalent Circuits for a Symmetric Two-Signal- Conductor Generalized Transmission Line (18500).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' In Figure 1, we show the standard, generalized equivalent circuit of a symmetric two- signal-conductor transmission line, and reduced-complexity, single-ended equivalent circuits for each transmission mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Note that the simplified equivalent circuits allow direct application of the RLGC model, using the correctly-transformed versions of the RLGC parameters appropriate for the transmission mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Variation of RLGC Parameters with Frequency Having set the stage for the time-domain application of a generic RLGC model, we now focus on the specific forms of each component used our RLGC model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' The following section specifies the formulas used in our basic RLGC model (which are fairly standard) and identifies modifications we thought necessary to adequately explain observed laboratory behavior (which are not always standard).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Series Impedance Variation with Frequency In the case of simple, homogeneous signal conductors, we use the classical result for the series impedance based on surface impedance concepts, which we repeat here: ( ) AC R sL R s L s + = + ∞ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='5) where L∞ may be interpreted as the inductance of the system when all currents flow uniformly on the surface of the signal conductors (that is, at moderately high frequency) and AC R is a constant which we approximate as 2 AC R S η μ σ = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='6) where and μ σ are the permeability and conductivity of the signal conductor, and S is the length of the effective perimeter of the signal conductor through which the surface currents flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' The geometry-dependent constant η represents a factor determining the increase in resistive losses due to currents in the return paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' In a thin stripline configuration, one might expect the value of η to be in the neighborhood of 2 η = because the widths of the expected return paths are about the same as the circumference of the signal conductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' In a coaxial cable, one might expect η to be less than 2, because the return paths in the outer shield are significantly wider than the circumference of the signal conductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' In practice, any of the parameters of equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='6), including AC R itself, may be varied in equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='5) to match laboratory data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' However, there exists a common transmission system which does not fit equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='5) well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' The Gore EyeOpener ™ cable, for example, uses signal conductors constructed from a heterogeneous combination of metal layers to achieve self-equalizing properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' We now derive an approximation to the surface impedance for a thick bulk material, covered by a relatively thin layer of another conductor, to address this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' General field solutions to this type of problem were generated by Wait [5], which we here apply to transmission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' We presume a planar, infinitely-thick conductor of bulk conductivity 2 σ underneath a thin layer of thickness 1τ and conductivity 1 σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' As in the classical case, the electric field will decay throughout the finite thickness 1τ to the value 1 1 1 1 0 ( ) | z y j y E E e τ τ δ = + − = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='7) where 1δ is the skin depth in the outer conductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' At the interface between the two different conductors, the electric field will have a new behavior given by the boundary condition requirements of Maxwell’s equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' The appropriate constraint is that of continuous tangential electric field across the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Therefore, at the interface, the electric field begins a new exponential behavior in the bulk material with new depth constant 2 δ : 1 1 1 2 1 1 ( ) 0 ( ) j j y z E y E e e τ τ δ δ + + − − − = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='8) Working out the integral for the total current under a width S, the resulting surface impedance is: 1 1 1 1 1 1 1 2 1 1 1 2 1 2 1 1 { }{ ( 1) 1} { }{ ( 1) 1} j z s AC j Z e S R s e τ δ μ σ η σ σ δ σ σ σ + − − − − + = − + = − + (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='9) Equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='9) adds a new factor to the classical surface impedance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' The factor is a function of the outer-conductor thickness 1τ and the ratio of the conductivities of the two materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' In general, we see that the resistive and reactive portions of the surface impedance are no longer equal when the conductor is composite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' In our physical approximation, we take the conductor width S to be the effective length of the perimeter of the signal conductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' The overall series impedance is then: 1 1 1 1 2 1 2 1 { }{ ( 1) 1} s AC R sL R s e sL μ ητ σ σ σ − − ∞ + = − + + (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='10) Equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='9) is valid when the permeability of the all conductors is that of free space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=" When the bulk conductor is magnetic, the conductivity of the bulk conductor 2 σ can be replaced by an effective conductivity 2 2' R σ σ μ = (1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='11) where R μ is the relative permeability of the bulk conductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Shunt Conductance Variation with Frequency Traditionally, the dielectric losses are characterized by a shunt conductance G per unit length: 1 tan tan G C sC j ω δ δ = = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='12) The “loss tangent”, tanδ , is commonly specified by dielectric manufacturers to be in the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='01 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='001 range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Typically, users are left to presume that the loss tangent is independent of frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' In such a case, the total shunt admittance can be written as tan (1 tan ) G Cs G Cs j Cs ω δ δ + = + = − (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='13) We shall see that this form can give good agreement with laboratory data at frequencies in the few-GHz range (and when dielectric losses are relatively small compared to the series resistance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' At higher frequencies (perhaps in the 10 GHz range) the model’s main deficiency becomes apparent: the form of equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='13) cannot be physically reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' It is well known that this form of dielectric loss is not a physically consistent possibility [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' In the present case, it is relatively straightforward to see why this is so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' If we take the series impedance to be lossless, that is, 0 AC R = , then using equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='13), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='5), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='2), and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='1), the transform of the impulse response of the transmission line can be written as: (1 tan ) ( ( )) j LC j z h t e e ω δ γ − − − Δ = = F (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='14) Equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='14) has a closed-form inverse transform, which is: 2 2 ( ) [( ) ] h t t α π τ α = − + (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='15) where the parameters are given as Re{ 1 tan } Im{ 1 tan } z LC j z LC j τ δ α δ = Δ − = Δ − (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='16) Therefore, the impulse response of a transmission line with dielectric possessing constant loss tangent is a delayed Lorentzian pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' The Lorentzian form gives experimentally plausible insights, such as: the amplitude of the pulse is inversely proportional to the length of the transmission line, with proportionality constant simply related to the loss tangent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' However, we can observe that the impulse response extends back infinitely in time even though its impulse excitation occurred at the time origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' The model predicts non-causal behavior and is therefore not plausible as a physical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' We have found that this feature of the constant-loss-tangent model is often the root cause of the failure to match our experimental data at high frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Other forms for the variation of the loss tangent with frequency must be applied in these cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' In Figure 2 we highlight the difference in expected pulse shape between the constant-loss-tangent model and that of a loss tangent varying linearly with frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' The response in the linear-loss-tangent case is derived numerically using equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='4) because the problem does not have a closed-form solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' We will show later that the linear-variation version can exhibit good fit to experimental data at frequencies in excess of 10 GHz (which is our only justification for using it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='3 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='6 Constant Loss Tangent Linear Loss Tangent Modeled Pulse Response, Volts Time, ns Constant Loss Tangent Response is Lorentzian 1 1 + Dt2 Linear Loss Tangent Response Identically Zero Outside These Limits Figure 2: Comparison of Modeled Time Responses of a Transmission Line with No Series Loss but Finite Dielectric Loss Due to Two Loss Models, Showing Causality Failure of Constant Loss Tangent Assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' (20573) Time Domain Laboratory Techniques Transmission lines (and other linear circuits) are often characterized by s-parameter analysis using a vector network analyzer (VNA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' The methodology for calibration, de- embedding, and interpretation of VNA results is a highly-developed specialty, which (when properly applied) will fully characterize the transmission line over a wide bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' For this work, we have elected to use a time-domain method, for the following reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' First, most VNAs have two ports and do not simply support differential-mode excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Second, in many applications, the primary information needed is the classical transfer- function of the transmission line system (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=', the information present in 21 s in the absence of reflections).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' The complexities in testing, calibration, de-embedding, and interpretation for the other s-parameters may not be strictly necessary in some applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Third, a full-fledged experimental characterization of systems utilizing transmission lines is often not limited to pure system-identification techniques, but also may include direct measurements of higher-level system performance quantities (such as error rate or eye diagrams).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' In some applications it is beneficial to integrate as many measurement schemes as feasible into an experimental setup that is as simple (and inexpensive) as possible, so long as the resulting measurements are “good enough” for the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Finally, there may exist nonlinearities in the transmission system which will introduce interpretation errors in the s-parameter analysis without warning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' These nonlinearities are not likely to be in the transmission lines and interconnect (which are linear to an excellent approximation) but may exist, for example, in the driver circuitry of a buffer amplifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Most (if not all) nonlinearities cannot be characterized by the magnitude and phase of single-sinusoid reception;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' however, there exist waveforms which do broaden the scope of complete nonlinear characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' We shall see that a PRBS is one such waveform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' In the following section, we describe a method which addresses all four of these points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Naturally, the described method will have its own disadvantages, which are generally related to experimental approximations which do not exist (or are unnecessary) in a VNA-type of analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Pseudorandom Sequence Excitation Method We describe a time-domain system identification method in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' We follow the scheme developed in [3,6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' A performance analyzer (‘bit error rate tester’) is used to generate a 127-bit pseudo-random binary sequence (PRBS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' The analyzer has true and complement outputs, which are used to drive the differential transmission line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' A digital oscilloscope is used to sample the differential signal at the end of the transmission line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' The scope is triggered by the performance-analyzer synch-out pulse, which fires once every PRBS period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' This pulse provides a consistent time datum which is independent of the transmission-line length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' A typical experimental setup (shown for a cable) is as shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' The upper panel in Figure 4 shows an example of the raw waveform as sampled at the end of a transmission line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Because the raw waveform is too complicated to interpret easily by eye, the waveform can be processed to generate a discrete-time pulse response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' 15 Meter Cable Assembly Eye Diagram Test Board With Cable Assembly Under Test Figure 3: Test Setup for Measuring Error Rate, Eye Diagrams, and System Transfer Functions in the Time Domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' (18045) 0 10 20 30 40 50 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='5 1 Extracted Pulse Response Time from Signal Source Trigger Datum, nsec -1 0 1 Measured Voltage Figure 4: Example of a Measured PRBS Waveform and Its Extracted Pulse Response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' (18322) The response (which we shall call extracted pulse response in keeping with the literature [3,6]), is numerically generated using a two-step process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' First, the raw waveform is re- sampled to take care of experimental frequency errors between the PRBS generator and the oscilloscope, and to provide an appropriate sampling rate for subsequent analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Second, the re-sampled waveform is mathematically deconvolved with a theoretically perfect version of the same PRBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' We shall return to describe a very efficient method for this deconvolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' The extracted pulse response is analogous to the impulse response in continuous time systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' In this case, the extracted response is the expected response of the transmission line (and measurement system and signal drivers) if the PRBS driver had output a simple base-to-peak digital pulse instead of the PRBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' The lower panel in Figure 4 shows the extracted response for the given raw waveform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' While both waveforms contain exactly the same information, it is clear that the extracted pulse is a short primary response, with some small echoes of interesting shape and location on the time axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' In Figure 5, the extracted pulse response for two different, but short, PCB transmission lines is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' All connectors, test cables, and test conditions are nominally identical, except for the length of the transmission line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='8 6 7 8 10 11 12 13 14 15 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='0 6 7 8 9 10 Time from Signal Source Trigger Datum, nsec Extracted Pulse Response ( 6 Inch Line ), Volts Extracted Pulse Response ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='5 Inch Line ), Volts 11 12 13 14 15 Nonlinear Echo Main Pulse First Reflection Second Reflection Figure 5: Extracted Pulse Responses from Two Lengths of Otherwise Identical Lines (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='5” and 6”) (18323) In the upper panel of Figure 5, we zoom in on the interesting sections of the extracted pulse response, for a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='5 inch PCB line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' As might be expected from the significant impedance mismatches, the signal appears to reflect back and forth between the impedance discontinuities before settling down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' At 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='5 Gbits/second excitation, the pulse response is nominally 400 picoseconds wide, and the first forward-moving reflection should begin to arrive about a half nanosecond after the onset of the main pulse;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' the second forward-moving reflection arrives a half nanosecond after that, and so forth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Thus we should not expect to see separation between the main pulse and its subsequent, exponentially decaying reflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' In the lower panel of Figure 5, we see the effects of quadrupling the transmission-line length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' The now well-resolved reflections occur at approximately 2 nanosecond intervals (as expected).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' The main pulse is resolved, and is proportional in magnitude to what the pulse would have been down 6 inches of this transmission line without the reflections due to impedance mismatches, but including any pulse modification due directly to impedance mismatches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' (The interface of two perfect transmission lines of real, but different, impedance at all frequencies will modify the main pulse in amplitude, only).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' The main pulse and reflections may also be resolved by increasing the excitation frequency of the PRBS on a given line (instead of changing line lengths, as above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' This technique may be limited by the rise time of the drivers, the time span of the impulse response of the transmission lines under test, and the analog bandwidth of the sampling oscilloscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' These observations suggest a type of de-embedding technique which is very simple, though approximate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' We assume that the combination of the transmission line length, the resolution of the channel response, and the frequency of the PRBS is such that the main pulse is resolvable from reflections or other imperfections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' We then claim, to the extent these assumptions are true, that the resolvable main pulse may be written as 1 1 0 1 1 0 1 ( ) { ( ) ( ) ( ) ( ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' ( , )} { ( ) ( ) ( , )} driver launches scope tline T tline x t F X H H H H z F X H H z ω ω ω ω ω ω ω ω − − ≅ Δ ≅ Δ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='17) where 0( ) X ω is the Fourier transform of an ideal PRBS sequence, 1z Δ is the length of transmission line, and the total transfer function ( ) T H ω is the composite linear response of all non-ideal elements in a practical test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Note that the interpretation of 1( ) x t is not exactly that of the time response related to the s-parameter 21 s ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' it is only equivalent to s- parameter analysis when there are no reflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' In principle, and with sufficient cleverness, we could also resolve each reflection in the measured waveforms to independently and fully analyze the entire characteristic described by 21 s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' However, we shall not develop those techniques in this report;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' instead, we shall focus only on resolving the response that would have occurred without reflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' If we now consider a transmission line of length 2 1 z z Δ > Δ , then its response can be written as 1 2 0 2 1 0 1 2 1 ( ) { ( ) ( ) ( , )} { ( ) ( ) ( , )} T tline tline x t X H H z X X H z z ω ω ω ω ω ω − − = Δ = Δ − Δ F F (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='18) where the second line follows from the separability of the transmission line transfer function (equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='1)) into component factors, that is, 2 1 2 1 2 1 2 1 ( ) ( )( ) ( )( ) ( )( ) 2 1 2 1 ( , ) ( , ) ( , ) z z z z z z z z tline tline tline H z e e e e H z H z z γ γ ω γ ω γ ω ω ω ω − Δ − Δ +Δ −Δ − Δ −Δ − Δ −Δ Δ = = = = Δ Δ − Δ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='19) Despite its mathematical look, the suggested modeling technique is exceedingly simple: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Measure 1( ) x t of a relatively short line and find its extracted pulse response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Calculate the expected response of a longer transmission line of length 2z Δ by applying equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='4) for a transmission line of length 2 1 z z Δ − Δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Compare the output of step 2 to an actual measurement of the extracted pulse response of a transmission line of length 2z Δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Focus on the (by assumption, resolvable) main pulse only;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' ignoring all other interesting blips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' We have argued that this technique will give an expected waveform, and an actual waveform, to measure the goodness-of-fit of the RLGC model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' furthermore this comparison de-embeds all systematic linear effects in the test, to the extent that the main pulse is resolvable in time from all reflections or other phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' The “resolvability” criterion gives this technique its simplicity, but is its major source of interpretation uncertainty when compared to more accurate experimental methods (such as, a VNA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' A typical “eyeball” comparison is good to perhaps 30-40 dB (a few percent in voltage).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' In many applications (such as, the identification of eye diagrams), the “eyeball” criterion is, nearly by definition, good enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' However, the engineer must judge what kind of accuracy is required for a particular application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' We have argued that the proposed technique effectively de-embeds effects which can be described by a linear transfer function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' We now briefly digress to discuss the effects of stochastic jitter in the trigger waveform and oscilloscope timings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' These effects are not present in frequency-domain measurement techniques, and therefore deserve further discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' In general the analysis of the all jitter effects on the waveform captured by the oscilloscope is very difficult, but the following simplifications allow some insight into the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=" If the jitter is small enough that the waveform is well approximated by its first-order Taylor’s series, that is: ( ) ( ) '( ) x t x t x t τ τ + Δ ≅ + Δ (1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='20) and all jitter phenomenon are lumped into a single effective jitter at the scope sampler (relative to the signal ( ) x t ), with jitter variance 2 σ , then the signal can be thought of in two components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' There exists a stochastic portion (whose spectrum is continuous even if ( ) x t is periodic) whose power is linearly related to 2 σ and the average square of the derivative of ( ) x t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' This power comes at the expense of the high frequencies in the deterministic portion of the identified waveform (that is, its averaged sequence value as sampled on the scope).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' It can be shown that the jitter produces a stochastic filter whose average transfer characteristic is 2 2 ( ) jitt H e σ ω ω − = % (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='21) if the jitter is Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Therefore, if 2 / f π σ << , the jitter effects should be negligible in the sampled waveform ( ) x t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' If the standard deviation of the jitter is in the few pS range, then frequencies below about 10 GHz are affected by less than 1 percent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' In a sense, ( ) jitt H ω % can be thought of as one of components of the total deterministic transfer function ( ) T H ω , but only if the various realizations of ( ) jitt H ω % are averaged sufficiently to converge to the average transfer function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' This convergence might happen, for instance, when the waveform reported from the scope is the average of a large number of sample sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' In such a case, this portion of the transfer function cancels out in our method (as do the deterministic transfer functions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' However, this is dangerous ground, and the experimenter should take care to ensure that the stochastic conditions of the tests are identical when exploiting this effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' For example, if trigger jitter is significant, the realization of ( ) jitt H ω % for a waveform derived by “averaging” only one sample waveform is in general quite different from the realization of ( ) jitt H ω % when many averages are taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' (The frequency response will look better, in general, with only a few averages than it does with long averages).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Numerical Methods In the preceding section, we developed the concept of the extracted pulse response, which is the circular deconvolution of the measured PRBS sequence with an ideal version of the same sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' A good, standard method for deconvolution is to take the Fourier transform of both waveforms, perform a complex division, and then take the inverse Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' In the present case, the number of points in the transform is 2 1 N n = − , which is very often a prime number, and is never highly composite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' In such cases, the best fast algorithms for discrete Fourier transforms (DFTs) are relatively useless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' In most such cases, the deconvolution will require about 2 n complex multiplication operations and about n complex divisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' We now show a special deconvolution method which takes optimal advantage of the fact that measured waveform is a filtered version of a PRBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' This method requires about 2 n real additions, zero multiplications, zero divisions, and zero complex operations of any sort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' We develop the algorithm in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' If the sampled digital-pulse response of the transmission line system is denoted by [ ] ( ) h n h nT = , with n an integer and T the bit period of the sequence, then the expected sampled measured output is giving by the circular convolution of the PRBS and the pulse response: [ ] [ ] [ ] k y n h k n p k = − ∑ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='22) where [ ] { 1,1} p n ∈ − represents the PRBS and k ranges over the length L of the PRBS, which is always of the form 2 1 N L = − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' We shall also call [ ] h n the extracted pulse response, after we have back-calculated it from measured data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Because this is a circular convolution, the sequences y, p, and h are assumed to be periodic in L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' therefore the indices can always be taken modulo L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' We now assume that the mathematical convolution was actually completed in the physical world by playing the ideal PRBS through a linear system, and we are able to measure the analog waveform ( ) y t directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' We then resample this waveform to obtain the measured version of [ ] y n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' By inspiration, we can generate a new function from [ ] y n and note how else it can be expressed: 1 [ ] [ ] [ ] 1 1 [ ] [ ] [ ] 1 [ ] l l k x Z n y n p n l L h k n p k p n l L h n μ = − + = − − + = + ∑ ∑∑ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='23) where x μ is the mean of the pulse response [ ] x n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' The last line of equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='23) is general only when the excitation is a PRBS, and is derived using the properties of a PRBS [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Therefore we can recover, within a small DC shift, the extracted pulse response [ ] h n from the measured sequence [ ] y n by a simple transformation using 2 ( 1) L L L − ≅ additions and L trivial multiplications by 1 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Depending on the application, the division by L+1 may or may not be necessary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' if it is necessary, implementers should note that the divisor is always exactly a power of two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=" With a little more complexity in the algorithm it is possible to reconstruct exactly the DC conditions to get a new function '[ ] [ ] Z n h n ≡ , but in our case we have simply ignored the small DC offset as it makes little practical difference in our analysis." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' We have developed this algorithm in a general way when we are interested in analyzing or plotting waveforms with more than one point per PRBS bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' In such a case, we can decimate a waveform representing M points per bit into M waveforms representing [ ] m y n , that is, the sampled version of (( / ) ) y n m M T + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' We can determine the extracted pulse response for each sub-channel, [ ] m h n , using equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='23) and [ ] m y n as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' The resultant response [ ] m h n is the sampled version of the analog waveform (( / ) ) h n m M T + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Therefore, we can reconstruct the M-point-per-bit version of the response by re-interleaving the results of each [ ] m h n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Note that the calculations of [ ] m h n are only dependent on values of y in its own interleave;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' whereas in general this property is not true of DFT-type deconvolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' This property is another advantage of this type of specialized deconvolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Therefore, we have shown a very efficient method for deconvolution when PRBS excitation is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Clearly, for offline analysis using relatively small L, there is little practical difference between a few microseconds and a several milliseconds of computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Even so, these differences can become quite significant for large enough L even for offline analysis because L grows exponentially with each PRBS order N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Perhaps more significantly, the algorithm in equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='23) is simple enough to realistically be implemented at-speed in hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' For example, the algorithm implementation is simply a single accumulator if we are willing to calculate one ( ) Z n at a time in an operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Nonlinear Analysis with PRBS Excitation Up to this point, we have not described anything of particular value in using a PRBS as the excitation waveform over other types of waveforms (other than its alternative utility as a convenient bit-error-rate test pattern).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Indeed, a TDT-type step waveform gives the same information as does an extracted pulse response for a linear system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' We now digress to explain an interesting and useful feature of this type of PRBS-deconvolution analysis, which allows native nonlinear Volterra analysis of the waveforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' In each of the extracted pulse waveforms discussed thus far (shown in Figure 4 and Figure 6), a strange blip appears to arrive at the oscilloscope 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='5 nanoseconds before the main pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' This blip always shows up in exactly the same place (relative to the main pulse), independent of the transmission-line length or type, or even whether there is a transmission line under test at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' If the bit excitation frequency is changed, then the time spacing between the blip and the main pulse also changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' However, if the time spacing is normalized to bit time units, now this number now remains constant, independent of excitation frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Therefore, we have discovered an apparently physical delay effect which seems to know something about the bit times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' No signal phenomenon of any kind shows up at these locations when a simple TDR pulse is used as excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' The reason for this type of behavior was resolved in the 1980s [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' The phenomenon came to be understood in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' If a system is linear, then the deconvolution operation, described above, always gives the same results for the extracted pulse response no matter what the underlying data pattern is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' If a system is somewhat nonlinear, then in the general case, the deconvolution operation will contain garbage which will appear as a “noise” everywhere on the extracted pulse response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' However, if the data pattern is a PRBS, then for many types of nonlinearities, the disturbances due to such nonlinearities destructively interfere almost everywhere after the deconvolution operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' The nonlinearities tend to constructively interfere at specific locations on the extracted pulse response, and with specific shapes and magnitude, all of which vary depending on the nature and severity of the nonlinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' This phenomenon is now used universally in magnetic recording to identify various impairments of the magnetic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' In our case, the blip phenomenon is due to circuit-driver nonlinearity somewhere in the test system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Analysis shows that this imperfection (if so it may be called) is due to a non-linearity in the driver circuit in the performance analyzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Close examination of the output differential pulses show an asymmetry in the rise times (as compared to the fall times) of the output driver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' As a result, a positive-going pulse looks noticeably different than the opposite of a negative-going pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' This difference is by definition a nonlinear effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' It has been shown that the PRBS phenomenon is related to the Volterra-series nonlinear- system identification technique for discrete-time systems [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' This property is another example of the apparently endless set of practical and useful features of a PRBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' To an excellent approximation, the transmissions lines are natively linear, and (in principle) the choice of excitation waveform is immaterial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' However, circuit drivers, receivers, digital-to-analog converters, equalizers, analog-to-digital converters, and other elements of a complex analog communication system, are not natively linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' In such a case, analysis using linear assumptions, under conditions of PRBS excitation yields (for no additional work), nonlinear system analysis capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Comparison to Theory We now present experimental PRBS results gathered from several different types of transmission lines, and compare the results to expectations from our RLGC model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' We found that a constant loss tangent assumption was adequate at lower frequencies (~2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='5 Gbits/second excitation) but completely inadequate at 40 Gbits/sec excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' We concluded that the series losses were modeled adequately by the standard form at all frequencies tested, except for any frequency using a bilayer-conductor transmission system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='5 Gbits/second Excitation Results We applied the preceding techniques to a PCB transmission line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' We designed and procured a passive printed circuit board (PCB) using GETEK ™ dielectric materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' This board was fabricated to verify the accuracy of the RLGC model described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Manual RLGC-model fitting procedures produced model results which compared quite favorably to the experimentally-measured results at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='5 Gbits/second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' These results are compared, for four different lengths of the transmission line, in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' All model parameters are frozen at the given values (except, of course, for the transmission-line length) for all model results on this graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' We used a 6-inch version of the lines as the de-embedding reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' The important model parameters for the GETEK materials are: 378 nH/m, 117 pF/m, tan 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='011, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='45 7, / 5 mils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' L C E S δ σ η ∞ = = = = = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='24) We tested several lengths of 100 Ohm Skewclear Infiniband 12X cable from Amphenol/SpectraStrip ™.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' We merely used the advertised (nominal) odd-mode impedance and velocity numbers to calculate the odd-mode reactive parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' The model parameters for the SKEWCLEAR cable are: 377 nH/m, 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='7 pF/m, tan 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='0001, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='0 7, / 17mils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' L C E S δ σ η ∞ = = = = = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='25) 10 12 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='8 12 14 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='8 14 16 18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='0 18 20 22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='0 Time from Signal Source Trigger Datum, nsec Time from Signal Source Trigger Datum, nsec Amplitude of Pulse Response, Volts Amplitude of Pulse Response, Volts Measured 12 inch Trace Modeled 12 inch Trace Measured 36 inch Trace Modeled 36 inch Trace Measured 24 inch Trace Modeled 24 inch Trace Measured 60 inch Trace Modeled 60 inch Trace Figure 6: Experimental Comparison of GETEK PCB to RLGC Model in Odd Mode with Parameters 378 nH/m, 117 pF/m, tan 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='011, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='45 7, / 5 mils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' L C E S δ σ η ∞ = = = = = (18334) Figure 7 shows the modeled and actual pulse responses for several lengths of cable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' In this case, the cable connectors were clearly a significant factor in the test system response as is seen by the ringing in the response tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' However, the de-embedding method The reference measurement (shown in the first panel of Figure 7, used as input to the numerical model, was taken from a relatively short, 1 meter cable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='0 14 16 18 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='0 2 4 Time from Signal Source Trigger Datum, nsec Time from Signal Source Trigger Datum, nsec Amplitude of Pulse Response, Volts Amplitude of Pulse Response, Volts 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='0 22 24 26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='0 30 32 34 Measured 10 meter Cable Modeled 10 meter Cable Measured 15 meter Cable Modeled 15 meter Cable Measured 5 meter Cable Modeled 5 meter Cable Measured-Data Reference Waveform From 1 meter Cable + Connectors Figure 7: Experimental Comparison of SkewClear Cable to RLGC Model using Parameters 377 nH/m, 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='7 pF/m, tan 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='0001, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='0 7, / 17mils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' L C E S δ σ η ∞ = = = = = (18324) We also analyzed a Gore EyeOpener Plus cable (26 AWG), which was optimized by Gore for operation at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='5 Gbits/second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' The experimental data was not fitted well with the traditional model for surface impedance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' We assumed that the cable was built with stratified signal conductors and were able to fit the data much better using equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='10) for the surface impedance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' The fitted parameters for EyeOpener cable are: 1 2 1 387 nH/m, 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='7 pF/m, tan 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='0004, 6 7, 1 7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='115 mil, / 21 mils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' L C E E S δ σ σ τ η ∞ = = = = = = = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='26) The effective conductivity of the core conductor is so low, for a metal, that the material is likely to be magnetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' The effective conductivity of the bulk material is “reduced” by a factor equal to the relative permeability of the material in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Figure 8 shows the modeled and actual pulse responses for two lengths of the Gore EyeOpener Plus cable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' The reference measurement, used as numerical model input, was taken from a relatively short, 1 meter cable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Because little or no geometry or composition information was available for the cables, we do not claim that the fit physical parameters truly reflect the physical cable characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' We do claim that this combination of parameters adequately describes the differential behavior of the cables over a significant range of cable lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' 0 1 2 3 4 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='0 Time from Signal Source Trigger Datum, nsec 32 33 34 35 36 37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='0 Amplitude of Pulse Response, Volts Measured 5 meter cable Modeled 5 meter cable Measured 10 meter cable Modeled 10 meter cable Figure 8: Experimental Comparison of EyeOpener Cable to RLGC Model using Parameters 1 2 1 387 nH/m, 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='7 pF/m, tan 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='0004, 6 7, 1 6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='115 mil, / 21 mils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' L C E E S δ σ σ τ η ∞ = = = = = = = (18446) We conclude that the RLGC model, with appropriate extensions when necessary, accurately describes a variety of cable transmission lines and PCB lines at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='5 Gbits/second excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' 40 Gbits/second Excitation We shall now present the time-domain analysis with data taken at 40 Gbits/second excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' While (in principle) an infinitely-sharp rise time at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='5 Gbits/second will also excite very high frequencies which will then be identified with by our method even at low excitation frequency, it’s also true that equipment made to produce 40 Gbits/second data will have significantly smaller rise times and that of equipment made to generate a PRBS at lower frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Using such equipment is therefore experimentally better if high frequencies are to be accurately modeled by this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' In Figure 9, we show comparisons for model versus actual pulse responses for various lengths of a PCB trace made using Nelco 1300 ™ material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' In this case we found much improved experimental fit using a linear loss tangent model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' The graphs show quite a bit of attenuation for this type of transmission line at these frequencies (which is not at all surprising as this material is not intended for use up at these frequencies over any reasonable length of trace).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' We find the fit of these curves to be fairly good, but clearly there exist differences between expected and actual traces which do not exist in the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='5 Gbit/second data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' The frequency content of these waveforms is only significant up to approximately 10 GHz of bandwidth, so we do not claim to have a model which matches data above this bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Model parameters for the 40 Gbits/second Nelco 1300 PCB traces are: 327 nH/m, 125 pF/m, / 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='75 mils, tan =2E 13 , =5E7 S/m L C S η δ ω σ ∞ = = = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='27) Note that this model uses a loss tangent which is proportional to frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Measured Reference (3" Trace) 20" Trace 34" Trace 42" Trace Measured Modeled Measured Modeled Measured Modeled 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='7 Time, ns Pulse Response, Volts 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='4 Time, ns Pulse Response, Volts 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='3 Time, ns Pulse Response, Volts 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='7 Time, ns Pulse Response, Volts 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='0 Figure 9: Experimental Comparison of Nelco 1300 PCB to RLGC Model at 40 Gbits/s using Parameters 327 nH/m, 125 pF/m, / 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='75 mils, tan =2E-13 , =5E7 S/m L C S η δ ω σ ∞ = = = (20568) Finally, in Figure 10 we show the results from data taken at 40 Gbits/second from an advanced version of the SkewClear cable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' This version is designed for higher frequencies and includes a modified ground conductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' The model parameters for the upgraded SkewClear cable are: 451 nH/m, 40 pF/m, / 20 mils, tan =1E 13 , =5E7 S/m L C S η δ ω σ ∞ = = = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='28) Note that again the loss tangent model is linear in frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='7 Pulse Response, Volts Time, ns 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='4 Pulse Response, Volts Time, ns Measured Modeled 1 Meter Reference, Differentially Excited and Received 1 Meter Reference, Singly Excited, Differentially Received 5 Meter Cable, Differentially Excited and Received 5 Meter Cable, Singly Excited, Differentially Received 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='2 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='7 Pulse Response, Volts Time, ns 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='4 Pulse Response, Volts Time, ns Measured Modeled Mismatch Indicates Even Mode Leakage Figure 10: Experimental Comparison of Upgraded SkewClear Cable to RLGC model at 40Gb/s under Single Ended and Differential Excitation Conditions using Model Parameters 451 nH/m, 40 pF/m, / 20 mils, tan =1E-13 , =5E7 S/m L C S η δ ω σ ∞ = = = (20571) In this case, the fit is nearly as excellent as for the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content='5 GHz case, even though the significant frequencies in this waveform extend up to approximately 20 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' The usual model comparison (in the upper right graph) shows excellent fit except for the area around 100 pS before the main pulse, which shows a relatively small blip of apparently unknown origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' We shall take this opportunity to show the types of conclusions one may draw from investigations using time domain techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' We have shown similar data on the lower portion of the figure except in this case the excitation is provided only on one side of the differential cable, with the other input terminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Theoretically, or ideally, this condition should produce exactly half the differential signal at the output terminals (and indeed this is very well approximated for the 1 meter reference signals).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' However, the singly-excited model comparison is rather weak, though (evidently) the errors apparently nearly cancel when the system is both differentially excited and differentially received (the upper right case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' The simplest explanation for this behavior is a mode leakage between even and odd modes down the length of the line;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' and indeed, we have shown by similar methods that the even mode signal has a slightly higher velocity than does the odd mode, such that the even mode shows up about 100 pS before the odd mode at a distance of 5 meters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Conclusions We have developed a time-domain method for experimental verification of an RLGC model, and showed adequate experimental fit over a wide variety of transmission line types up to approximately 10 GHz in bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' We found that, depending on the frequency and transmission line type, specific extensions were required to adequately explain laboratory behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' We found that modifications to both the dielectric (shunt) loss model and the resistive (series) loss models were necessary to adequately cover all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' We have shown that, despite a fairly substantial list of assumptions, a PRBS time-domain method can be used in a de-embedding fashion to remove many experimental impediments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Indeed, the PRBS technique can be used to easily identify and quantify nonlinear effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' We proposed an extremely efficient method for implementing the PRBS identification process, which can either speed up offline computation when the waveform is large, or, can be used in hardware to directly implement the method in real-time so long as an appropriately complex ADC is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' We note that the proposed technique is similar to (in fact, it is a simplification of) a generalized TDT method, possessing a similar set of advantages and disadvantages when compared to frequency-domain methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' For the price of software processing and a real time scope, this method can be used to turn a bit-error-rate test system into a reasonable system identification station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' While we have shown laboratory data indicating the utility of this measurement method using high-quality lab equipment, we believe that this type of analysis will become most attractive in self-test applications in product hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' As the analog-to-digital converters (ADCs) in a SERDES migrate from traditional 1-bit comparators to those required to support advanced signaling and detection schemes, we expect that methods such as those described in this paper will be used to advance the operational and diagnostic capabilities of the overall system, as they have in other systems [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Acknowledgements The authors would like to thank Eric Hanlon and Devon Post for providing test boards and cables for this activity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Jason Prairie for his data-taking expertise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Patrick Zabinski for many instructive and seminal conversations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' and Steve Richardson, Elaine Doherty, and Deanna Jensen for generating the graphics for this report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' References [1] Richard E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Matick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Transmission Lines for Digital and Communication Networks.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Physically consistent transmission line models for high- speed interconnects in lossy dielectrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} +page_content=' Advanced Packaging, IEEE Transactions on Volume 25, Issue 2, May 2002 Page(s):129 - 135' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE4T4oBgHgl3EQf0Q2d/content/2301.05281v1.pdf'} diff --git a/4NE1T4oBgHgl3EQfmASv/content/tmp_files/2301.03293v1.pdf.txt b/4NE1T4oBgHgl3EQfmASv/content/tmp_files/2301.03293v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..76b7c3eeefcfed32e02688468f901e43c3830c7e --- /dev/null +++ b/4NE1T4oBgHgl3EQfmASv/content/tmp_files/2301.03293v1.pdf.txt @@ -0,0 +1,957 @@ +Distributed Multirobot Control for +Non-Cooperative Herding +Nishant Mohanty∗, Jaskaran Grover∗, Changliu Liu, Katia Sycara +The Robotics Institute, Carnegie Mellon University, Pittsburgh, USA +{nishantm,jaskarag,cliu6,sycara}@andrew.cmu.edu +Abstract. In this paper, we consider the problem of protecting a high- +value area from being breached by sheep agents by crafting motions for +dog robots. We use control barrier functions to pose constraints on the +dogs’ velocities that induce repulsions in the sheep relative to the high- +value area. This paper extends the results developed in our prior work +on the same topic in three ways. Firstly, we implement and validate our +previously developed centralized herding algorithm on many robots. We +show herding of up to five sheep agents using three dog robots. Secondly, +as an extension to the centralized approach, we develop two distributed +herding algorithms, one favoring feasibility while the other favoring opti- +mality. In the first algorithm, we allocate a unique sheep to a unique dog, +making that dog responsible for herding its allocated sheep away from the +protected zone. We provide feasibility proof for this approach, along with +numerical simulations. In the second algorithm, we develop an iterative +distributed reformulation of the centralized algorithm, which inherits the +optimality (i.e. budget efficiency) from the centralized approach. Lastly, +we conduct real-world experiments of these distributed algorithms and +demonstrate herding of up to five sheep agents using five dog robots. +Videos of these results are available at https://bit.ly/3bZq0dB. +Keywords: Herding, Barrier Functions, Quadratic Programming +1 +Introduction +Recent developments in robotics and sensing have created significant interest +among researchers to deploy multiple robots to operate cooperatively towards +achieving a common goal. Many works have developed techniques to tackle real- +world problems using multi-robot systems (MRS), like conducting surveys or +automating warehouses [1,2], [3]. The major developments in MRS for enabling +multiple robots to behave cooperatively have been based on interactions within +a single team, i.e., a robot interacts with other robots in its group to achieve a +given objective [4,5]. The main features of these types of algorithms are a) local +∗ These authors contributed equally to this work. +This research is supported by AFOSR FA9550-18-1-0097 and AFRL/AFOSR +FA9550-18-1-0251 +arXiv:2301.03293v1 [cs.RO] 9 Jan 2023 + +2 +Nishant Mohanty∗, Jaskaran Grover∗, Changliu Liu, Katia Sycara +interaction, b) collision-free motion within the group, and c) achieving collective +behavior using local interaction [6]. +In literature, there are studies on MRS that involve interaction between mul- +tiple groups of agents. Here, along with the local interaction with group members, +the individuals also interact with an external agent from another group. An ex- +ample of this is a scenario where a group of adversarial robots has a goal of their +own that might damage a given high-value unit. Here, a group of defenders must +interact with the adversarial robots to ensure the safety of the unit. [7,8]. In this +paper, we propose a provably correct controller for the group of defenders (“dog +robots”) to prevent an adversarial group (the “sheep robots”) from breaching a +protected zone. This is challenging because dog robots do not control the sheep +robots directly; rather have to rely on the interaction dynamics between the dogs +and sheep to influence the sheep’s behavior. +In our prior work [9], we developed a centralized algorithm to solve this +problem using control barrier functions. In this work, a) we provide more ex- +perimental validation of the centralized algorithm, b) propose two distributed +algorithms, and c) provide simulations and experiments to validate these algo- +rithms. Our formulation computes the velocity of each dog locally to prevent +sheep from breaching the protected zone(s). In the first distributed algorithm, +we allocate each sheep to a unique dog and pose a constraint on that dog’s +velocity to herd its allocated sheep away from the protected zone. We provide +proof of feasibility of this approach, thus showing that whenever the number +of sheep and dogs are equal, the herding problem is well-posed. Our previously +proposed centralized algorithm lacked this feasibility guarantee. However, it did +not necessitate equal numbers of dogs and sheep; in fact, in many experiments, +fewer dogs than sheep were sufficient to herd all the sheep away. This obser- +vation led us to develop the second algorithm. In this algorithm, we construct +an iterative distributed approach that asymptotically attains the same veloci- +ties as computed by the centralized approach, thereby attaining the same total +optimality (measured in terms of the total movement the dogs exhibit) as the +centralized approach and obviating the need to have equal numbers of dogs and +sheep. We build on the dual-decomposition algorithms proposed in [10,11] for de- +veloping this distributed algorithm. Both of our proposed distributed algorithms +are compositional in nature i.e., we can protect multiple zones by including more +constraints, as shown in figure 1(c). To highlight the performance of our formu- +lation, we provide results from numerical simulations showing the success of our +approach for multiple dogs against multiple sheep. Finally, we demonstrate our +algorithm on real robots and show multiple dog robots successfully preventing +the breaching of protected zones against multiple sheep robots. +The outline of this paper is as follows: in section 2, we give a brief review of +the prior work in this area. In section 3, we provide a mathematical formulation +of the problem statement. In section 4, we show how to use control barrier +functions to pose constraints on dog velocities. Section 5 provides simulations +accepted in IEEE Conference on Decision and Control 2022 + +Distributed Herding +3 +and experimental results to demonstrate the proposed approach. Finally, we +summarize our work in section 6 along with our directions for future work. +2 +Prior Work +The framework of multi-group interaction within MRS has many applications +beyond the adversarial problem statements. The shepherding problem is an ex- +ample of such a category. In [12,13], the authors have proposed methods to +enable multiple shepherd agents to influence a flock of sheep by modeling the +interaction as repulsion forces. The Robot Sheepdog Project [14,15] conducted +a real-world demonstration of a shepherding algorithm where a group of mobile +ground robots cooperatively herded a flock of ducks to a given goal location. +In the literature, there are several works on non-cooperative shepherding as +an example of a multi-group interaction type problem. The works like [13], [16], +[17], [18], [19], [20]. deal with a problem where the sheep robots do not exhibit +adversarial behavior. They do not have any goals of their own. However, they +experience a repulsive force from the dog robots, which is exploited to produce +the desired behavior in the sheep robots. For example, collecting all the sheep +at some location and then driving them to a target goal. +Differently from prior work, our sheep may or may not be adversarial. We call +them adversarial if their goal lies inside the protected zone and non-adversarial +otherwise. Our safe control synthesis approach remains the same regardless. The +dog robots observe and generate their control commands considering the cohesion +between the sheep robots, the attraction to their goal location, and the repulsion +experienced by them from the dog robots. And as we use control barrier functions +to generate the constraints on the velocity of the dog robots, it only requires the +dynamics of the sheep to be represented as a symbolic function. Thus allowing +for the sheep to experience any kind of attractive or repulsive forces. +3 +Problem Formulation +Consider a scenario with n sheep agents flocking towards a common goal loca- +tion. One commonly assumed model for flocking is the Reynolds-Boids dynamics +[21] that considers inter-sheep cohesive forces, inter-sheep repulsive forces, and +attraction to a common goal. In the presence of dog agents, each sheep’s dy- +namics would include repulsive forces from each dog robot. While en route to +their goal, the sheep, having no knowledge about high-value regions in workspace +(protected zones), pose a risk of breaching them. Thus, our problem is to orches- +trate the motions of dog robots by capitalizing on the repulsions that the sheep +experience from the dogs to prevent this breaching. Next, we pose this problem +in formal terms. +Consider the protected zone P ⊂ R2 as a disc centered at xP with radius +Rp, i.e., P := {x ∈ R2| ∥x − xP ∥ ≤ Rp}. We denote the flock of sheep as S and +the position of the ith sheep as xSi ∈ R2. The collective positions of all sheep +is denoted as xall +S := (xS1, xS2, ..., xSm). Similarly, we denote the set of all dogs + +4 +Nishant Mohanty∗, Jaskaran Grover∗, Changliu Liu, Katia Sycara +using D. The position of the kth dog is xDk ∈ R2 and the positions of all dogs +collectively is xall +D := (xD1, xD2, ..., xDn). Each sheep follows single integrator +dynamics ˙xSi := f i(xS1, ..., xSn, xD1, ..., xDn), given by +˙xSi = uSi = kS +� +j∈S\i +� +1 − +R3 +S +∥xSj − xSi∥3 +� +(xSj − xSi) +� +�� +� +inter-sheep cohesion and repulsion ++ +kG (xG − xSi) +� +�� +� +attraction to goal ++ kD +� +l∈D +xSi − xDl +∥xSi − xDl∥3 +� +�� +� +repulsion from dogs +(1) +Here, RS is a safety margin that each sheep tends to maintain with every other +sheep, xG is the sheep’s desired goal and kS, kG and kD are proportional gains +corresponding to the attractive and repulsive forces. We model each dog as a +velocity controlled robot with the following dynamics: +˙xDk = uDk ∀k ∈ {1, 2, · · · , n} +(2) +Before posing the problem, we state some assumptions on the dogs’ knowledge: +Assumption 1. The dog robots have knowledge about the sheep’s dynamics i.e. +(1) and can measure the sheep’s positions accurately. +Assumption 2. Each dog robot can measure the velocities of other dog robots +(by using numerical differentiation, for example). +Problem 1. Assuming that the initial positions of the sheep xSi(0) /∈ P ∀i ∈ +S, the dog robots’ problem is to synthesize controls {uD1, · · · , uDn} such that +xSi(t) /∈ P ∀t ≥ 0 ∀i ∈ S. +4 +Controller Design +In this section, we show two approaches to solve Problem 1, building on our +previously proposed centralized algorithm [9]. Define a safety index h(·) : R2 −→ +R that quantifies the distance of Si from P: +h(xSi) = ∥xSi − xP ∥2 − (r + Rp)2 +(3) +Here r is a safety buffer distance. Thus, we require h(xSi(t)) ≥ 0 ∀t ≥ 0. We +define x = (xall +S , xall +D ) as the aggregated state of all sheep and all dogs. To +ensure, h(xSi(t)) ≥ 0 ∀t ≥ 0, we treat h(·) as a control barrier function require +its derivative to satisfy +˙h(x) + p1h(xSi) ≥ 0. +(4) + +Distributed Herding +5 +Here p1 is a design parameter and is chosen based to satisfy +p1 > 0 +and +p1 > − +˙h(x(0)) +h(xSi(0)). +(5) +The first condition on p1 requires that the pole is real and negative. The second +depends on the initial positions x(0) of all the sheep and dogs relative to the +protected zone. Note that the constraint in (4) does not contain any dog velocity +terms, which is what we require to control each dog. Therefore, we define the +LHS of (4) as another control barrier function v(x) : R4n −→ R: +v = ˙h + p1h, +(6) +and require its derivative to satisfy the constraint: ˙v(x) + p2v(x) ≥ 0. Here p2 +is another design parameter which must satisfy +p2 > 0 +and +p2 > − +¨h(x(0)) + p1 ˙h(x(0)) +˙h(x(0)) + p1h(xSi(0)) +(7) +Using (3), (6) and the constraint on the derivative, we get +¨h(x) + α˙h(x) + βh(xSi) ≥ 0 +(8) +where α := p1 + p2 and β := p1p2. The derivatives of h(·) are: +˙h(x) = 2(xSi − xP )T ˙xSi = 2(xSi − xP )T f i(x) +(9) +¨h(x) = 2f T +i f i + 2(xSi − xP )T +� � +j∈S +JS +jif i + +� +l∈D +JD +li uDl +� +(10) +where the jacobians are defined as JS +ji := ∇xSj f i(x) and JD +li := ∇xDl f i(x) +Note that (10) contains the velocity terms of all dogs. In [9], we leveraged this +observation to obtain a linear constraint on the velocity of all dogs collectively +for preventing sheep Si from breaching P: +AH +i uall +D ≤ bH +i , +where +(11) +AH +i := (xP − xSi)T � +JD +1i, JD +2i, · · · , JD +ni +� +bH +i := f T +i f i + (xSi − xP )T � +j∈S +JS +jif j + α(xSi − xP )T f i + β h +2 +A centralized algorithm was developed that collectively computes the velocities +of all dogs using the following QP +uall +D = arg min +uall +D +∥uall +D ∥2 +subject to +AH +i uall +D ≤ bH +i ∀i ∈ S. +(12) +Building on this centralized approach, in this paper, we develop two distributed +approaches wherein we allow each dog to compute its velocity locally such that +the computed velocities will make the dog herd the sheep away from P. + +6 +Nishant Mohanty∗, Jaskaran Grover∗, Changliu Liu, Katia Sycara +4.1 +Approach 1: One dog to one sheep allocation based approach +In this approach, we assume that we have an equal number of dogs and sheep. By +exploiting this equality, we assign a unique sheep Si for i ∈ {1, · · · , n} to a unique +dog Dk for k ∈ {1, · · · , n} and make Dk responsible for herding Si away from +P. In other words, Dk computes a velocity uDk that repels Si from P thereby +ensuring that xSi(t) /∈ P ∀t ≥ 0. The premise is that owing to the equality, +each sheep will end up being herded by a unique dog, therefore, no sheep will +breach the protected zone . Now while this strategy necessitates having an equal +number of dogs and sheep, the benefit of this approach stems from the feasibility +guarantee (that we prove shortly), which the centralized approach lacks. Simple +algebraic manipulation of constraint (11) yields a constraint on the velocity of +Dk as follows +AH +i uDk ≤ bH +i , +where +(13) +AH +i := (xP − xSi)T JD +ki +bH +i := f T +i f i + (xSi − xP )T �� +j∈S +JS +jif j + αf i + β h +2 + +� +l∈D\k +JD +li uDl +� +Here AH +i ∈ R1×2 and bH +i ∈ R. The term uDl in the expression of bH +i is computed +by using numerical differentiation of the positions xDl. We pose a QP to obtain +the min-norm velocity for Dk as follows +u∗ +Dk = arg min +uDk +∥uDk∥2 +subject to +AH +i uDk ≤ bH +i +(14) +The obtained velocity u∗ +Dk guarantees that the protected zone P will not be +breached by sheep Si by ensuring that h(xSi(t)) ≥ 0 ∀t ≥ 0. Since each dog +in D is in-charge of herding exactly one sheep in S, feasibility of (13) ∀k ∈ D +would ensure no sheep breaches P. Next, we show the conditions under which +(14) remains feasible but first state some assumptions. +Assumption 3. We make the following assumptions on the distances between +pairs of agents: +1. There exists a lower bound and upper bound on the distance between any pair +of sheep, i.e, LS ⩽ +��xSi − xSj +�� ⩽ MS, ∀i, j ∈ S and i ̸= j. +2. There exists a lower bound on the distance between every sheep and dog, i.e., +∥xSi − xDk∥ ≥ LD ∀i ∈ S and k ∈ D. +3. There exists a upper bound on the distance between each sheep and its goal +i.e., ∥xSi − xG∥ ⩽ MG and between the sheep and the center of the protected +zone i.e., ∥xSi − xP ∥ ⩽ MP . +Note that although Si is assigned to Dk, the position of the remaining dogs +{1, · · · , n}\k and the remaining sheep {1, · · · , n}\i do influence Dk’s constraint pa- +rameters (AH +i , bH +i ), and in turn, its computed velocity u∗ +Dk. + +Distributed Herding +7 +Theorem 1. In a scenario with ‘n’ dogs and ‘n’ sheep, with each dog assigned +a unique sheep, the herding constraint (13) for a given dog is always feasible, +provided assumptions 3 are met. +Proof. See appendix (section 7). +4.2 +Approach 2: Iterative distributed reformulation of (12) +The distributed formulation proposed in (14) comes with a feasibility guarantee +ensuring that all sheep will be herded away from P. While vital, this comes at +the cost of requiring as many dog robots as the number of sheep agents. This +is because, in a way, this equality ensures that controlling the sheep from the +perspective of dog robots is not an underactuated problem. Be that as it may, +in our simulations and experiments involving the centralized approach with an +equal number of dogs and sheep, we frequently observed that not all dog robots +needed to move to repel the sheep away from P i.e., equality may have been an +overkill. Thus, in terms of budget efficiency, at least empirically, the centralized +approach outweighs the distributed approach. +This raises the question, can we convert the centralized algorithm of (12) into +a distributed version that inherits the budget efficiency (optimality) promised by +(12)? Indeed, we found out that [10,11] propose algorithms to convert constrained- +coupled convex optimization problems (such as (12)) into distributed counter- +parts. They combine techniques called dual decomposition and proximal min- +imization and develop iterative distributed schemes which consist of local op- +timization problems. The solutions to these optimization problems asymptoti- +cally converge to the solution of centralized optimization under mild convexity +assumptions and connectivity properties of the communication network. In our +case, this network refers to the communication between dog robots. Below, we +present the distributed dual sub-gradient method of [10,11] adapted to the costs +and constraints of (12). This algorithm calculates an estimate of dog Dk’s ve- +locity ˆuDk which, given large enough iterations Kmax, matches with the kth +velocity component in the optimal velocities u∗all +D +returned by (12). Ak ∈ RnS×2 +refers to those columns of AH that correspond to uDk in uall +D . +Algorithm 1 Distributed Dual Subgradient for (12) (based on sec. 3.4.2 in [11]) +Initialize Lagrange Multiplier: µ0 +k = 0 ∈ RnS +Evolution: t = 1, 2, · · · , Kmax +Gather Multipliers µt +r from Dr ∀r ∈ {1, · · · , nD}\k +Average Multipliers: vt+1 +k += +1 +nD +� +r∈{1,··· ,nD}\k µt +r +Local Solution: ut+1 +Dk = arg min +u +∥u∥2 + (vt+1 +k +)T (Aku − +1 +nD bH) = − 1 +2AT +k vt+1 +k +Update Multiplier: µt+1 +k += +� +vt+1 +k ++ γt +� +Akut+1 +Dk − +1 +nD b +�� ++ +Return Average: ˆuDk = (1/Kmax) �Kmax +t=1 +ut +Dk + +8 +Nishant Mohanty∗, Jaskaran Grover∗, Changliu Liu, Katia Sycara +5 +Results +In this section, we provide simulation and real-world experimental results demon- +strating our proposed distributed algorithms. +5.1 +Simulation Results +We first validate the first distributed algorithm and the feasibility proof given +in 4.1. For this, we model the sheep with the Reynolds-Boids dynamics (1) with +gains kS = 0.5, kG = 1 and kD = 0.1. The dogs use (14) to compute their +velocities, where hyperparameters α and β are computed following (5) and (7). +We chose a circular protected zone of radius Rp = 0.6m and center xP at origin. +The sheep are initialized outside of the protected zone, and their goal location xG +is chosen such that their nominal trajectory would make them breach the zone, +thus necessitating intervention from dogs. The positions of dogs are initialized +randomly within a certain range of the protected zone. In figures 1(a) and 1(b), +we show two examples involving a) two dog robots vs. two sheep robots and b) +three dog robots vs. three sheep robots. To demonstrate the compositionality of +our approach, we consider two protected zones in figure 1(c) where we have four +dogs defending both zones from four sheep. In all these simulations, none of the +sheep breach any zone, thus demonstrating the correctness of our approach. In +the interest of space, we skip the simulation results for the algorithm in 4.2 but +do provide experimental results. +(a) Two dogs v. two sheep. (b) Three dogs v. three sheep (c) Four dogs v. four sheep. +Fig. 1: Preventing the breaching of the protected zone using our proposed dis- +tributed algorithm in section 4.1. Here dogs are shown in blue and sheep in red. +The green disc represents the protected zone. The nominal task of the sheep is +to go straight towards goal xG. However, since this would result in infiltration +of the protected zone, the dog intervenes using the control algorithm presented +in (14). In Fig. 1(c), we defend two protected zones from four sheep. + +1.5 +1 +0.5 +0 +-0.5 +-1 +-1.5 +1 +01.5 +1 +0.5 +0 +-0.5 +1 +-1.5 +1 +01.5 +1 +0.5 +0 +-0.5 +1 +-1.5 +0 +-1Distributed Herding +9 +5.2 +Robot Experiments +In this section, we show the results obtained by performing robot experiments +by implementing the distributed algorithms of section 4.1 and section 4.2. Ad- +ditionally, we also present more experimental results for our prior centralized +algorithm from [9] (because at the time, we did not have as many robots). We +conduct these experiments in our lab’s multi-robot arena, which consists of a +14ft × 7ft platform with multiple Khepera IV robots and eight Vicon cameras +for motion tracking. Although Khepera robots have unicycle dynamics, [9] con- +sists of a technique to convert the single-integrator dynamics (assumed for dogs +and sheep) to linear and angular velocity commands for the robots. +First of all, to build upon our previous work, we show additional experiments +using centralized velocity computation of the dog robots (12). Figure 2 shows a +case with 2 dog and 4 sheep robots. The dog robots have a green tail, and the +sheep robots have an orange tail. The tails are pointing in the opposite direction +of the robot’s heading angle. The protected zone is the green-colored circular re- +gion. This figure shows the performance in the case of an underactuated system, +i.e, there are more sheep against less number of dogs. Another example is shown +in figure 3 where 3 dogs successfully prevent breaching against 5 sheep robots. +Following that, multiple experiments were conducted using the distributed +algorithm presented in section 4.1, which requires equal numbers of dogs and +sheep. Figure 4 shows 4 dog robots against 4 sheep robots scenario. Here we +take two protected zones and show that the dogs can protect both of them. This +highlights the compositional nature of our proposed algorithm. We conducted +experiments with 5 dog robots and 5 sheep robots, as shown in Figure 5. Here we +can see some dog robots did not require to move as the assigned sheep were being +prevented from entering the protected zone due to the configuration of the flock +itself. Finally, we test our distributed algorithm presented in section 4.2. Figure 6 +shows a case where 2 dogs prevent the breaching of protected zone against three +dogs. This highlights that our distributed approach can handle under-actuated +scenarios. Figure 7 and figure 2 can be compared to see both centralized and +distributed algorithm handling a similar scenario of 2 dogs against 4 sheep. +6 +Conclusions +In this paper, we developed a novel optimization-based distributed control tech- +niques to enable multiple dog robots to prevent the breaching of protected zones +by sheep agents. We provided proof of feasibility of the controller when n dog +robots face an equal number of sheep robots. Additionally, we developed another +distributed algorithm that iteratively computes a solution that agrees with the +solution returned by the centralized problem without requiring equal number of +dogs and sheep. We experimentally validated both distributed algorithms in ad- +dition to validating our previously developed centralized control. We show that +multiple dog robots can prevent breaching of protected zone in both simulation +and real-world experiments. In future work, we aim for the dog robots to learn +the dynamics of the sheep robots online while preventing them from breaching. + +10 +Nishant Mohanty∗, Jaskaran Grover∗, Changliu Liu, Katia Sycara +(a) t = 0s +(b) t = 5s +(c) t = 12s +(d) t = 30s +Fig. 2: Experiments for Centralized Control: Two dogs defending the pro- +tected zone from four sheep using centralized control algorithm (12) from our +prior work [9]. Video at https://bit.ly/3OTAnOu. +(a) t = 0s +(b) t = 5s +(c) t = 30s +(d) t = 50s +Fig. 3: Experiment for Centralized Control: Three dogs (green-tailed +robots) defending a protected zone from five sheep (orange-tailed robots) us- +ing centralized control (12) from our prior work [9]. Video at https://youtu.be/ +2 Xuxnd9jZw. + +leepGoaleepGoalleepGoaleepGoaleepGoaleepooaeepGoaieep GoalDistributed Herding +11 +(a) t = 0s +(b) t = 6s +(c) t = 12s +(d) t = 20s +Fig. 4: Experiment for the distributed algorithm in section 4.1 : Four +dogs (green-tailed robots) defending two protected zone from four sheep (orange- +tailed robots). The goal position xG (red disc) is in extreme left that would +encourage sheep to breach both zones. However, our proposed algorithm moves +the dogs so that none of the zones get breached. Video at https://bit.ly/3yo9ziC. +(a) t = 0s +(b) t = 12s +(c) t = 25s +(d) t = 40s +Fig. 5: Experiment for the distributed algorithm in section 4.1) : Five +dogs (green-tailed robots) defending the protected zone from five sheep (orange- +tailed robots). The sheep’s goal (red disc) is in the center of the protected zone. +Eventually, in this scenario a deadlock occurs where all sheep come to a stop +outside the protected zone. Video at https://bit.ly/3o51Cu1. + +leenGnaeereepGoal12 +Nishant Mohanty∗, Jaskaran Grover∗, Changliu Liu, Katia Sycara +(a) t = 0s +(b) t = 4s +(c) t = 15s +(d) t = 30s +Fig. 6: Experiment for distributed algorithm in section 4.2) : Two dogs +(green-tailed robots) defending the protected zone from three sheep (orange- +tailed robots). The goal position xG (red disc) is at the center of the zone. Video +at https://youtu.be/IbCjkR1ye0c. +(a) t = 0s +(b) t = 4s +(c) t = 15s +(d) t = 30s +Fig. 7: Experiment for distributed algorithm in section 4.2) : Two dogs +(green-tailed robots) defending the protected zone from four sheep (orange-tailed +robots). This case is similar to the one shown in fig. 2. Video at https://youtu. +be/51FoHZWFYC4. + +leepGoalheepGoalheepGoalheepGoalheepGoalheepGoalheepGoalDistributed Herding +13 +7 +Appendix: Proof of feasibility for Approach 1 +Theorem 1. In a scenario with ‘n’ dogs and ‘n’ sheep, with each dog assigned +a unique sheep, the herding constraint (13) for a given dog is always feasible, +provided assumptions 3 are met. +Proof. Our strategy to guarantee feasibility of constraint (13) relies on ruling +out situations in which it is infeasible. (13) can become infeasible +– either when AH +i = 0 and bH +i < 0 (possibility 1) +– or when bH +i = −∞ (possibility 2). +To determine the conditions in which possibility 1 occurs, we calculate the de- +terminant of JD +ki as +det(JD +ki) = +−2k2 +D +∥xDk − xSi∥3 +The determinant det(JD +ki) is non-zero as long as the distance between dog Dk and +sheep Si is finite. Therefore, JD +ki will have no null space, implying that AH +i ̸= 0 +∀xSi ∈ R2, xDk ∈ R2. This rules out possibility 1 for infeasibility. To rule out +possibility 2, we need to check for condition when bH +i −→ −∞. Given bH +i in (13), +we find its worst case lower bound. Here f T +i f i ≥ 0 and as we assume that at +the current time step, the sheep is outside P, this ensures β h +2 ≥ 0. By removing +these terms, the lower bound of bH +i +can be given as +bH +i ≥ +� +j∈S\i +(xSi − xP )T JS +jif j + (xSi − xP )T JS +iif i + +� +l∈D\k +(xSi − xP )T JD +li uDl ++ α(xSi − xP )T f i +(1) +Using the triangle inequality on the RHS and Cauchy-Schwarz inequality on +individual terms, we get +bH +i ≥ +� +j∈S\i +� +−σmax +� +JS +ji +� +∥xSi − xP ∥ ∥f j∥ +� +− σmax +� +JS +ii +� +∥xSi − xP ∥ ∥f i∥ +(2) ++ +� +l∈D\k +� +−σmax +� +JD +li +� +∥xSi − xP ∥ ∥uDl∥ +� +− α∥xSi − xP ∥∥f i∥ +where σmax is the largest singular value of a matrix. Further, using the fact that +the largest singular value of a matrix (σmax) is upper bounded by its Frobenius +norm (σF ), we obtain +bH +i ≥ +� +j∈S\i +� +−σF +� +JS +ji +� +∥xSi − xP ∥ ∥f j∥ +� +− σF +� +JS +ii +� +∥xSi − xP ∥ ∥f i∥ +(3) +� +l∈D\k +� +−σF +� +JD +ki +� +∥xSi − xP ∥ ∥uDl∥ +� +− α∥xSi − xP ∥∥f i∥ +Now to compute this lower bound we make use of assumption 3. We use the +dynamics in (1) to compute JS +ii and obtain the upper bound on σF +� +JS +ii +� +and use +the bounds on distances from assumption 3 to get following upper bound: + +14 +Nishant Mohanty∗, Jaskaran Grover∗, Changliu Liu, Katia Sycara +σF +� +JS +ii +� +⩽ +� +j∈S\i +kS +� +√ +2 + (3 + +√ +2)R3 +∥xSi − xSj∥3 +� ++ +√ +2kG + +� +l∈D\k +� +3 + +√ +2 +� +kD +∥xSi − xDl∥3 +⩽ (n − 1) +� +√ +2kS + (3 + +√ +2)kSR3 +L3 +S +� ++ +√ +2kG + n +�� +3 + +√ +2 +� +kD +L3 +D +� +:= λM +We omit the proof of this computation in the interest of space. Similarly, using +the dynamics in (1), we compute an expression for JS +ji and obtain an upper +bound on σF +� +JS +ji +� +as follows: +σF +� +JS +ji +� +⩽ +√ +2kS + (3 + +√ +2)kSR3 +∥xS1 − xSj∥3 ⩽ +√ +2kS + (3 + +√ +2)kSR3 +L3 +S +:= λS +Likewise, an upper bound of σF +� +JS +li +� +, is given by +σF +� +JS +li +� +⩽ (3 + +√ +2)kSR3 +∥xS1 − xDl∥3 ⩽ (3 + +√ +2)kSR3 +L3 +D +:= λD +Lastly, we use obtain an upper bound on the dynamics of each sheep f i as: +∥f i∥ ⩽ +� +j∈S\i +kS +� +∥xSi − xSj∥ + +R3 +∥xSi − xSj∥2 +� ++ kG∥xG − xSi∥ ++ +� +l∈D +kD +∥xSi − xDl∥ +∥xSi − xDl∥3 +(4) +Now we need to compute the maximum possible value of the RHS to get the +upper bound of the sheep dynamics. The first term has a local minima at ∥xSi − +xSj∥ = (2)1/3R. Therefore the maximum value can occur at either the lower +bound or upper bound of ∥xSi − xSj∥. Thus the maximum value of the first +term can be given as Fmax := max(kSLS + kS R3 +L2 +S , kSMS + kS R3 +M 2 +S ). Second term +is maximum when ∥xG − xSi∥ = MG. The last term is maximum when distance +of the sheep to the dogs are minimum, ∥xSi −xDk∥ = LD. Using these the upper +bound on the sheep dynamics is computed as: +∥f i∥ ⩽ (n − 1)Fmax + kGMG + nkD +� 1 +L2 +D +� +Assuming that the velocity of the dog robots have an upper bound, and by +taking the upper bound on the dynamics of all the sheep to be equal, the lower +bound on bH +i +from 3 is (taking γ = −(α + λM + (n − 1)λS)Mp) +bH +i ⩾ γ +� +(n − 1)Fmax + kGMG + nkD +L2 +D +� +− (n − 1)λDMP ∥uD∥max +This shows that bH +i has a finite lower bound, thus ruling out possibility 2. Thus, +the herding constraint (13) for a one dog to repel one sheep from the protected +zone is always feasible. Since each sheep in S is allocated to one unique dog in +D, extension of this feasibility result to all sheep ensures that none of them will +breach the protected zone. + +Distributed Herding +15 +References +1. R. D’Andrea, “Guest editorial: A revolution in the warehouse: A retrospective on +kiva systems and the grand challenges ahead,” IEEE Transactions on Automation +Science and Engineering, vol. 9, no. 4, pp. 638–639, 2012. +2. R. D’Andrea and G. E. Dullerud, “Distributed control design for spatially inter- +connected systems,” IEEE Transactions on automatic control, vol. 48, no. 9, pp. +1478–1495, 2003. +3. W. Kazmi, M. Bisgaard, F. Garcia-Ruiz, K. D. Hansen, and A. la Cour-Harbo, +“Adaptive surveying and early treatment of crops with a team of autonomous +vehicles,” in Proceedings of the 5th European Conference on Mobile Robots ECMR +2011, 2011, pp. 253–258. +4. M. Ji and M. Egerstedt, “Distributed coordination control of multiagent systems +while preserving connectedness,” IEEE Transactions on Robotics, vol. 23, no. 4, +pp. 693–703, 2007. +5. J. Lin, A. S. Morse, and B. D. Anderson, “The multi-agent rendezvous problem- +the asynchronous case,” in 2004 43rd IEEE Conference on Decision and Control +(CDC)(IEEE Cat. No. 04CH37601), vol. 2. +IEEE, 2004, pp. 1926–1931. +6. C. W. Reynolds, “Flocks, herds and schools: A distributed behavioral model,” in +Proceedings of the 14th annual conference on Computer graphics and interactive +techniques, 1987, pp. 25–34. +7. C. Walton, I. Kaminer, Q. Gong, A. Clark, T. Tsatsanifos et al., “Defense against +adversarial swarms with parameter uncertainty,” arXiv preprint arXiv:2108.04205, +2021. +8. T. Tsatsanifos, A. H. Clark, C. Walton, I. Kaminer, and Q. Gong, “Model- +ing and control of large-scale adversarial swarm engagements,” arXiv preprint +arXiv:2108.02311, 2021. +9. J. Grover, N. Mohanty, W. Luo, C. Liu, and K. Sycara, “Noncooperative herd- +ing with control barrier functions: Theory and experiments,” arXiv preprint +arXiv:2204.10945, 2022. +10. A. Falsone, K. Margellos, S. Garatti, and M. Prandini, “Dual decomposition +for multi-agent distributed optimization with coupling constraints,” Automatica, +vol. 84, pp. 149–158, 2017. +11. G. Notarstefano, I. Notarnicola, A. Camisa et al., “Distributed optimization for +smart cyber-physical networks,” Foundations and Trends® in Systems and Con- +trol, vol. 7, no. 3, pp. 253–383, 2019. +12. J.-M. Lien, O. B. Bayazit, R. T. Sowell, S. Rodriguez, and N. M. Amato, “Shep- +herding behaviors,” in IEEE International Conference on Robotics and Automa- +tion, 2004. Proceedings. ICRA’04. 2004, vol. 4. +IEEE, 2004, pp. 4159–4164. +13. A. Pierson and M. Schwager, “Controlling noncooperative herds with robotic +herders,” IEEE Transactions on Robotics, vol. 34, no. 2, pp. 517–525, 2017. +14. R. Vaughan, N. Sumpter, J. Henderson, A. Frost, and S. Cameron, “Robot con- +trol of animal flocks,” in Proceedings of the 1998 IEEE International Symposium +on Intelligent Control (ISIC) held jointly with IEEE International Symposium on +Computational Intelligence in Robotics and Automation (CIRA) Intell. +IEEE, +1998, pp. 277–282. +15. ——, “Experiments in automatic flock control,” Robotics and autonomous systems, +vol. 31, no. 1-2, pp. 109–117, 2000. +16. A. Pierson and M. Schwager, “Bio-inspired non-cooperative multi-robot herding,” +in 2015 IEEE International Conference on Robotics and Automation (ICRA). +IEEE, 2015, pp. 1843–1849. + +16 +Nishant Mohanty∗, Jaskaran Grover∗, Changliu Liu, Katia Sycara +17. R. A. Licitra, Z. I. Bell, E. A. Doucette, and W. E. Dixon, “Single agent indirect +herding of multiple targets: A switched adaptive control approach,” IEEE Control +Systems Letters, vol. 2, no. 1, pp. 127–132, 2017. +18. R. A. Licitra, Z. D. Hutcheson, E. A. Doucette, and W. E. Dixon, “Single agent +herding of n-agents: A switched systems approach,” IFAC-PapersOnLine, vol. 50, +no. 1, pp. 14 374–14 379, 2017. +19. E. Sebasti´an and E. Montijano, “Multi-robot implicit control of herds,” in 2021 +IEEE International Conference on Robotics and Automation (ICRA). IEEE, 2021, +pp. 1601–1607. +20. M. Bacon and N. Olgac, “Swarm herding using a region holding sliding mode +controller,” Journal of Vibration and Control, vol. 18, no. 7, pp. 1056–1066, 2012. +21. C. W. Reynolds, “Flocks, herds and schools: A distributed behavioral model,” ser. +SIGGRAPH ’87. +New York, NY, USA: Association for Computing Machinery, +1987, p. 25–34. + diff --git a/4NE1T4oBgHgl3EQfmASv/content/tmp_files/load_file.txt b/4NE1T4oBgHgl3EQfmASv/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..262fac8d696c5d62be5f370e20ee22f4855fde9d --- /dev/null +++ b/4NE1T4oBgHgl3EQfmASv/content/tmp_files/load_file.txt @@ -0,0 +1,507 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf,len=506 +page_content='Distributed Multirobot Control for Non-Cooperative Herding Nishant Mohanty∗, Jaskaran Grover∗, Changliu Liu, Katia Sycara The Robotics Institute, Carnegie Mellon University, Pittsburgh, USA {nishantm,jaskarag,cliu6,sycara}@andrew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='cmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='edu Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' In this paper, we consider the problem of protecting a high- value area from being breached by sheep agents by crafting motions for dog robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' We use control barrier functions to pose constraints on the dogs’ velocities that induce repulsions in the sheep relative to the high- value area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' This paper extends the results developed in our prior work on the same topic in three ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Firstly, we implement and validate our previously developed centralized herding algorithm on many robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' We show herding of up to five sheep agents using three dog robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Secondly, as an extension to the centralized approach, we develop two distributed herding algorithms, one favoring feasibility while the other favoring opti- mality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' In the first algorithm, we allocate a unique sheep to a unique dog, making that dog responsible for herding its allocated sheep away from the protected zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' We provide feasibility proof for this approach, along with numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' In the second algorithm, we develop an iterative distributed reformulation of the centralized algorithm, which inherits the optimality (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' budget efficiency) from the centralized approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Lastly, we conduct real-world experiments of these distributed algorithms and demonstrate herding of up to five sheep agents using five dog robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Videos of these results are available at https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='ly/3bZq0dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Keywords: Herding, Barrier Functions, Quadratic Programming 1 Introduction Recent developments in robotics and sensing have created significant interest among researchers to deploy multiple robots to operate cooperatively towards achieving a common goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Many works have developed techniques to tackle real- world problems using multi-robot systems (MRS), like conducting surveys or automating warehouses [1,2], [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' The major developments in MRS for enabling multiple robots to behave cooperatively have been based on interactions within a single team, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=', a robot interacts with other robots in its group to achieve a given objective [4,5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' The main features of these types of algorithms are a) local ∗ These authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' This research is supported by AFOSR FA9550-18-1-0097 and AFRL/AFOSR FA9550-18-1-0251 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='03293v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='RO] 9 Jan 2023 2 Nishant Mohanty∗, Jaskaran Grover∗, Changliu Liu, Katia Sycara interaction, b) collision-free motion within the group, and c) achieving collective behavior using local interaction [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' In literature, there are studies on MRS that involve interaction between mul- tiple groups of agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Here, along with the local interaction with group members, the individuals also interact with an external agent from another group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' An ex- ample of this is a scenario where a group of adversarial robots has a goal of their own that might damage a given high-value unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Here, a group of defenders must interact with the adversarial robots to ensure the safety of the unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' [7,8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' In this paper, we propose a provably correct controller for the group of defenders (“dog robots”) to prevent an adversarial group (the “sheep robots”) from breaching a protected zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' This is challenging because dog robots do not control the sheep robots directly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' rather have to rely on the interaction dynamics between the dogs and sheep to influence the sheep’s behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' In our prior work [9], we developed a centralized algorithm to solve this problem using control barrier functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' In this work, a) we provide more ex- perimental validation of the centralized algorithm, b) propose two distributed algorithms, and c) provide simulations and experiments to validate these algo- rithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Our formulation computes the velocity of each dog locally to prevent sheep from breaching the protected zone(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' In the first distributed algorithm, we allocate each sheep to a unique dog and pose a constraint on that dog’s velocity to herd its allocated sheep away from the protected zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' We provide proof of feasibility of this approach, thus showing that whenever the number of sheep and dogs are equal, the herding problem is well-posed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Our previously proposed centralized algorithm lacked this feasibility guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' However, it did not necessitate equal numbers of dogs and sheep;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' in fact, in many experiments, fewer dogs than sheep were sufficient to herd all the sheep away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' This obser- vation led us to develop the second algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' In this algorithm, we construct an iterative distributed approach that asymptotically attains the same veloci- ties as computed by the centralized approach, thereby attaining the same total optimality (measured in terms of the total movement the dogs exhibit) as the centralized approach and obviating the need to have equal numbers of dogs and sheep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' We build on the dual-decomposition algorithms proposed in [10,11] for de- veloping this distributed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Both of our proposed distributed algorithms are compositional in nature i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=', we can protect multiple zones by including more constraints, as shown in figure 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' To highlight the performance of our formu- lation, we provide results from numerical simulations showing the success of our approach for multiple dogs against multiple sheep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Finally, we demonstrate our algorithm on real robots and show multiple dog robots successfully preventing the breaching of protected zones against multiple sheep robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' The outline of this paper is as follows: in section 2, we give a brief review of the prior work in this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' In section 3, we provide a mathematical formulation of the problem statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' In section 4, we show how to use control barrier functions to pose constraints on dog velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Section 5 provides simulations accepted in IEEE Conference on Decision and Control 2022 Distributed Herding 3 and experimental results to demonstrate the proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Finally, we summarize our work in section 6 along with our directions for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' 2 Prior Work The framework of multi-group interaction within MRS has many applications beyond the adversarial problem statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' The shepherding problem is an ex- ample of such a category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' In [12,13], the authors have proposed methods to enable multiple shepherd agents to influence a flock of sheep by modeling the interaction as repulsion forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' The Robot Sheepdog Project [14,15] conducted a real-world demonstration of a shepherding algorithm where a group of mobile ground robots cooperatively herded a flock of ducks to a given goal location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' In the literature, there are several works on non-cooperative shepherding as an example of a multi-group interaction type problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' The works like [13], [16], [17], [18], [19], [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' deal with a problem where the sheep robots do not exhibit adversarial behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' They do not have any goals of their own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' However, they experience a repulsive force from the dog robots, which is exploited to produce the desired behavior in the sheep robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' For example, collecting all the sheep at some location and then driving them to a target goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Differently from prior work, our sheep may or may not be adversarial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' We call them adversarial if their goal lies inside the protected zone and non-adversarial otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Our safe control synthesis approach remains the same regardless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' The dog robots observe and generate their control commands considering the cohesion between the sheep robots, the attraction to their goal location, and the repulsion experienced by them from the dog robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' And as we use control barrier functions to generate the constraints on the velocity of the dog robots, it only requires the dynamics of the sheep to be represented as a symbolic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Thus allowing for the sheep to experience any kind of attractive or repulsive forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' 3 Problem Formulation Consider a scenario with n sheep agents flocking towards a common goal loca- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' One commonly assumed model for flocking is the Reynolds-Boids dynamics [21] that considers inter-sheep cohesive forces, inter-sheep repulsive forces, and attraction to a common goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' In the presence of dog agents, each sheep’s dy- namics would include repulsive forces from each dog robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' While en route to their goal, the sheep, having no knowledge about high-value regions in workspace (protected zones), pose a risk of breaching them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Thus, our problem is to orches- trate the motions of dog robots by capitalizing on the repulsions that the sheep experience from the dogs to prevent this breaching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Next, we pose this problem in formal terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Consider the protected zone P ⊂ R2 as a disc centered at xP with radius Rp, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=', P := {x ∈ R2| ∥x − xP ∥ ≤ Rp}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' We denote the flock of sheep as S and the position of the ith sheep as xSi ∈ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' The collective positions of all sheep is denoted as xall S := (xS1, xS2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=', xSm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Similarly, we denote the set of all dogs 4 Nishant Mohanty∗, Jaskaran Grover∗, Changliu Liu, Katia Sycara using D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' The position of the kth dog is xDk ∈ R2 and the positions of all dogs collectively is xall D := (xD1, xD2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=', xDn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Each sheep follows single integrator dynamics ˙xSi := f i(xS1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=', xSn, xD1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=', xDn), given by ˙xSi = uSi = kS � j∈S\\i � 1 − R3 S ∥xSj − xSi∥3 � (xSj − xSi) � �� � inter-sheep cohesion and repulsion + kG (xG − xSi) � �� � attraction to goal + kD � l∈D xSi − xDl ∥xSi − xDl∥3 � �� � repulsion from dogs (1) Here, RS is a safety margin that each sheep tends to maintain with every other sheep, xG is the sheep’s desired goal and kS, kG and kD are proportional gains corresponding to the attractive and repulsive forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' We model each dog as a velocity controlled robot with the following dynamics: ˙xDk = uDk ∀k ∈ {1, 2, · · · , n} (2) Before posing the problem, we state some assumptions on the dogs’ knowledge: Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' The dog robots have knowledge about the sheep’s dynamics i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' (1) and can measure the sheep’s positions accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Each dog robot can measure the velocities of other dog robots (by using numerical differentiation, for example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Assuming that the initial positions of the sheep xSi(0) /∈ P ∀i ∈ S, the dog robots’ problem is to synthesize controls {uD1, · · · , uDn} such that xSi(t) /∈ P ∀t ≥ 0 ∀i ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' 4 Controller Design In this section, we show two approaches to solve Problem 1, building on our previously proposed centralized algorithm [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Define a safety index h(·) : R2 −→ R that quantifies the distance of Si from P: h(xSi) = ∥xSi − xP ∥2 − (r + Rp)2 (3) Here r is a safety buffer distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Thus, we require h(xSi(t)) ≥ 0 ∀t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' We define x = (xall S , xall D ) as the aggregated state of all sheep and all dogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' To ensure, h(xSi(t)) ≥ 0 ∀t ≥ 0, we treat h(·) as a control barrier function require its derivative to satisfy ˙h(x) + p1h(xSi) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' (4) Distributed Herding 5 Here p1 is a design parameter and is chosen based to satisfy p1 > 0 and p1 > − ˙h(x(0)) h(xSi(0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' (5) The first condition on p1 requires that the pole is real and negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' The second depends on the initial positions x(0) of all the sheep and dogs relative to the protected zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Note that the constraint in (4) does not contain any dog velocity terms, which is what we require to control each dog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Therefore, we define the LHS of (4) as another control barrier function v(x) : R4n −→ R: v = ˙h + p1h, (6) and require its derivative to satisfy the constraint: ˙v(x) + p2v(x) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Here p2 is another design parameter which must satisfy p2 > 0 and p2 > − ¨h(x(0)) + p1 ˙h(x(0)) ˙h(x(0)) + p1h(xSi(0)) (7) Using (3), (6) and the constraint on the derivative, we get ¨h(x) + α˙h(x) + βh(xSi) ≥ 0 (8) where α := p1 + p2 and β := p1p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' The derivatives of h(·) are: ˙h(x) = 2(xSi − xP )T ˙xSi = 2(xSi − xP )T f i(x) (9) ¨h(x) = 2f T i f i + 2(xSi − xP )T � � j∈S JS jif i + � l∈D JD li uDl � (10) where the jacobians are defined as JS ji := ∇xSj f i(x) and JD li := ∇xDl f i(x) Note that (10) contains the velocity terms of all dogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' In [9],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' we leveraged this observation to obtain a linear constraint on the velocity of all dogs collectively for preventing sheep Si from breaching P: AH i uall D ≤ bH i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' where (11) AH i := (xP − xSi)T � JD 1i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' JD 2i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' JD ni � bH i := f T i f i + (xSi − xP )T � j∈S JS jif j + α(xSi − xP )T f i + β h 2 A centralized algorithm was developed that collectively computes the velocities of all dogs using the following QP uall D = arg min uall D ∥uall D ∥2 subject to AH i uall D ≤ bH i ∀i ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' (12) Building on this centralized approach, in this paper, we develop two distributed approaches wherein we allow each dog to compute its velocity locally such that the computed velocities will make the dog herd the sheep away from P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' 6 Nishant Mohanty∗, Jaskaran Grover∗, Changliu Liu, Katia Sycara 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='1 Approach 1: One dog to one sheep allocation based approach In this approach, we assume that we have an equal number of dogs and sheep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' By exploiting this equality, we assign a unique sheep Si for i ∈ {1, · · · , n} to a unique dog Dk for k ∈ {1, · · · , n} and make Dk responsible for herding Si away from P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' In other words, Dk computes a velocity uDk that repels Si from P thereby ensuring that xSi(t) /∈ P ∀t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' The premise is that owing to the equality, each sheep will end up being herded by a unique dog, therefore, no sheep will breach the protected zone .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Now while this strategy necessitates having an equal number of dogs and sheep, the benefit of this approach stems from the feasibility guarantee (that we prove shortly), which the centralized approach lacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Simple algebraic manipulation of constraint (11) yields a constraint on the velocity of Dk as follows AH i uDk ≤ bH i , where (13) AH i := (xP − xSi)T JD ki bH i := f T i f i + (xSi − xP )T �� j∈S JS jif j + αf i + β h 2 + � l∈D\\k JD li uDl � Here AH i ∈ R1×2 and bH i ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' The term uDl in the expression of bH i is computed by using numerical differentiation of the positions xDl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' We pose a QP to obtain the min-norm velocity for Dk as follows u∗ Dk = arg min uDk ∥uDk∥2 subject to AH i uDk ≤ bH i (14) The obtained velocity u∗ Dk guarantees that the protected zone P will not be breached by sheep Si by ensuring that h(xSi(t)) ≥ 0 ∀t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Since each dog in D is in-charge of herding exactly one sheep in S, feasibility of (13) ∀k ∈ D would ensure no sheep breaches P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Next, we show the conditions under which (14) remains feasible but first state some assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' We make the following assumptions on the distances between pairs of agents: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' There exists a lower bound and upper bound on the distance between any pair of sheep, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='e, LS ⩽ ��xSi − xSj �� ⩽ MS, ∀i, j ∈ S and i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' There exists a lower bound on the distance between every sheep and dog, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=', ∥xSi − xDk∥ ≥ LD ∀i ∈ S and k ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' There exists a upper bound on the distance between each sheep and its goal i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=', ∥xSi − xG∥ ⩽ MG and between the sheep and the center of the protected zone i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=', ∥xSi − xP ∥ ⩽ MP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Note that although Si is assigned to Dk, the position of the remaining dogs {1, · · · , n}\\k and the remaining sheep {1, · · · , n}\\i do influence Dk’s constraint pa- rameters (AH i , bH i ), and in turn, its computed velocity u∗ Dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Distributed Herding 7 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' In a scenario with ‘n’ dogs and ‘n’ sheep, with each dog assigned a unique sheep, the herding constraint (13) for a given dog is always feasible, provided assumptions 3 are met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' See appendix (section 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='2 Approach 2: Iterative distributed reformulation of (12) The distributed formulation proposed in (14) comes with a feasibility guarantee ensuring that all sheep will be herded away from P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' While vital, this comes at the cost of requiring as many dog robots as the number of sheep agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' This is because, in a way, this equality ensures that controlling the sheep from the perspective of dog robots is not an underactuated problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Be that as it may, in our simulations and experiments involving the centralized approach with an equal number of dogs and sheep, we frequently observed that not all dog robots needed to move to repel the sheep away from P i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=', equality may have been an overkill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Thus, in terms of budget efficiency, at least empirically, the centralized approach outweighs the distributed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' This raises the question, can we convert the centralized algorithm of (12) into a distributed version that inherits the budget efficiency (optimality) promised by (12)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Indeed, we found out that [10,11] propose algorithms to convert constrained- coupled convex optimization problems (such as (12)) into distributed counter- parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' They combine techniques called dual decomposition and proximal min- imization and develop iterative distributed schemes which consist of local op- timization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' The solutions to these optimization problems asymptoti- cally converge to the solution of centralized optimization under mild convexity assumptions and connectivity properties of the communication network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' In our case, this network refers to the communication between dog robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Below, we present the distributed dual sub-gradient method of [10,11] adapted to the costs and constraints of (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' This algorithm calculates an estimate of dog Dk’s ve- locity ˆuDk which, given large enough iterations Kmax, matches with the kth velocity component in the optimal velocities u∗all D returned by (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Ak ∈ RnS×2 refers to those columns of AH that correspond to uDk in uall D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Algorithm 1 Distributed Dual Subgradient for (12) (based on sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='2 in [11]) Initialize Lagrange Multiplier: µ0 k = 0 ∈ RnS Evolution: t = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Kmax Gather Multipliers µt r from Dr ∀r ∈ {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' nD}\\k Average Multipliers: vt+1 k = 1 nD � r∈{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='nD}\\k µt r Local Solution: ut+1 Dk = arg min u ∥u∥2 + (vt+1 k )T (Aku − 1 nD bH) = − 1 2AT k vt+1 k Update Multiplier: µt+1 k = � vt+1 k + γt � Akut+1 Dk − 1 nD b �� + Return Average: ˆuDk = (1/Kmax) �Kmax t=1 ut Dk 8 Nishant Mohanty∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Jaskaran Grover∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Changliu Liu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Katia Sycara 5 Results In this section,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' we provide simulation and real-world experimental results demon- strating our proposed distributed algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='1 Simulation Results We first validate the first distributed algorithm and the feasibility proof given in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' For this, we model the sheep with the Reynolds-Boids dynamics (1) with gains kS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='5, kG = 1 and kD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' The dogs use (14) to compute their velocities, where hyperparameters α and β are computed following (5) and (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' We chose a circular protected zone of radius Rp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='6m and center xP at origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' The sheep are initialized outside of the protected zone, and their goal location xG is chosen such that their nominal trajectory would make them breach the zone, thus necessitating intervention from dogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' The positions of dogs are initialized randomly within a certain range of the protected zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' In figures 1(a) and 1(b), we show two examples involving a) two dog robots vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' two sheep robots and b) three dog robots vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' three sheep robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' To demonstrate the compositionality of our approach, we consider two protected zones in figure 1(c) where we have four dogs defending both zones from four sheep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' In all these simulations, none of the sheep breach any zone, thus demonstrating the correctness of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' In the interest of space, we skip the simulation results for the algorithm in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='2 but do provide experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' (a) Two dogs v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' two sheep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' (b) Three dogs v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' three sheep (c) Four dogs v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' four sheep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' 1: Preventing the breaching of the protected zone using our proposed dis- tributed algorithm in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Here dogs are shown in blue and sheep in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' The green disc represents the protected zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' The nominal task of the sheep is to go straight towards goal xG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' However, since this would result in infiltration of the protected zone, the dog intervenes using the control algorithm presented in (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' 1(c), we defend two protected zones from four sheep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='5 1 01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='5 1 01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='5 0 1Distributed Herding 9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='2 Robot Experiments In this section, we show the results obtained by performing robot experiments by implementing the distributed algorithms of section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='1 and section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Ad- ditionally, we also present more experimental results for our prior centralized algorithm from [9] (because at the time, we did not have as many robots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' We conduct these experiments in our lab’s multi-robot arena, which consists of a 14ft × 7ft platform with multiple Khepera IV robots and eight Vicon cameras for motion tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Although Khepera robots have unicycle dynamics, [9] con- sists of a technique to convert the single-integrator dynamics (assumed for dogs and sheep) to linear and angular velocity commands for the robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' First of all, to build upon our previous work, we show additional experiments using centralized velocity computation of the dog robots (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Figure 2 shows a case with 2 dog and 4 sheep robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' The dog robots have a green tail, and the sheep robots have an orange tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' The tails are pointing in the opposite direction of the robot’s heading angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' The protected zone is the green-colored circular re- gion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' This figure shows the performance in the case of an underactuated system, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='e, there are more sheep against less number of dogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Another example is shown in figure 3 where 3 dogs successfully prevent breaching against 5 sheep robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Following that, multiple experiments were conducted using the distributed algorithm presented in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='1, which requires equal numbers of dogs and sheep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Figure 4 shows 4 dog robots against 4 sheep robots scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Here we take two protected zones and show that the dogs can protect both of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' This highlights the compositional nature of our proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' We conducted experiments with 5 dog robots and 5 sheep robots, as shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Here we can see some dog robots did not require to move as the assigned sheep were being prevented from entering the protected zone due to the configuration of the flock itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Finally, we test our distributed algorithm presented in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Figure 6 shows a case where 2 dogs prevent the breaching of protected zone against three dogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' This highlights that our distributed approach can handle under-actuated scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Figure 7 and figure 2 can be compared to see both centralized and distributed algorithm handling a similar scenario of 2 dogs against 4 sheep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' 6 Conclusions In this paper, we developed a novel optimization-based distributed control tech- niques to enable multiple dog robots to prevent the breaching of protected zones by sheep agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' We provided proof of feasibility of the controller when n dog robots face an equal number of sheep robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Additionally, we developed another distributed algorithm that iteratively computes a solution that agrees with the solution returned by the centralized problem without requiring equal number of dogs and sheep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' We experimentally validated both distributed algorithms in ad- dition to validating our previously developed centralized control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' We show that multiple dog robots can prevent breaching of protected zone in both simulation and real-world experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' In future work, we aim for the dog robots to learn the dynamics of the sheep robots online while preventing them from breaching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' 10 Nishant Mohanty∗, Jaskaran Grover∗, Changliu Liu, Katia Sycara (a) t = 0s (b) t = 5s (c) t = 12s (d) t = 30s Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' 2: Experiments for Centralized Control: Two dogs defending the pro- tected zone from four sheep using centralized control algorithm (12) from our prior work [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Video at https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='ly/3OTAnOu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' (a) t = 0s (b) t = 5s (c) t = 30s (d) t = 50s Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' 3: Experiment for Centralized Control: Three dogs (green-tailed robots) defending a protected zone from five sheep (orange-tailed robots) us- ing centralized control (12) from our prior work [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Video at https://youtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='be/ 2 Xuxnd9jZw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' leepGoaleepGoalleepGoaleepGoaleepGoaleepooaeepGoaieep GoalDistributed Herding 11 (a) t = 0s (b) t = 6s (c) t = 12s (d) t = 20s Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' 4: Experiment for the distributed algorithm in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='1 : Four dogs (green-tailed robots) defending two protected zone from four sheep (orange- tailed robots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' The goal position xG (red disc) is in extreme left that would encourage sheep to breach both zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' However, our proposed algorithm moves the dogs so that none of the zones get breached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Video at https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='ly/3yo9ziC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' (a) t = 0s (b) t = 12s (c) t = 25s (d) t = 40s Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' 5: Experiment for the distributed algorithm in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='1) : Five dogs (green-tailed robots) defending the protected zone from five sheep (orange- tailed robots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' The sheep’s goal (red disc) is in the center of the protected zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Eventually, in this scenario a deadlock occurs where all sheep come to a stop outside the protected zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Video at https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='ly/3o51Cu1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' leenGnaeereepGoal12 Nishant Mohanty∗, Jaskaran Grover∗, Changliu Liu, Katia Sycara (a) t = 0s (b) t = 4s (c) t = 15s (d) t = 30s Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' 6: Experiment for distributed algorithm in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='2) : Two dogs (green-tailed robots) defending the protected zone from three sheep (orange- tailed robots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' The goal position xG (red disc) is at the center of the zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Video at https://youtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='be/IbCjkR1ye0c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' (a) t = 0s (b) t = 4s (c) t = 15s (d) t = 30s Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' 7: Experiment for distributed algorithm in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content='2) : Two dogs (green-tailed robots) defending the protected zone from four sheep (orange-tailed robots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' This case is similar to the one shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Video at https://youtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' be/51FoHZWFYC4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' leepGoalheepGoalheepGoalheepGoalheepGoalheepGoalheepGoalDistributed Herding 13 7 Appendix: Proof of feasibility for Approach 1 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' In a scenario with ‘n’ dogs and ‘n’ sheep, with each dog assigned a unique sheep, the herding constraint (13) for a given dog is always feasible, provided assumptions 3 are met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Our strategy to guarantee feasibility of constraint (13) relies on ruling out situations in which it is infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' (13) can become infeasible – either when AH i = 0 and bH i < 0 (possibility 1) – or when bH i = −∞ (possibility 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' To determine the conditions in which possibility 1 occurs, we calculate the de- terminant of JD ki as det(JD ki) = −2k2 D ∥xDk − xSi∥3 The determinant det(JD ki) is non-zero as long as the distance between dog Dk and sheep Si is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Therefore, JD ki will have no null space, implying that AH i ̸= 0 ∀xSi ∈ R2, xDk ∈ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' This rules out possibility 1 for infeasibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' To rule out possibility 2, we need to check for condition when bH i −→ −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Given bH i in (13), we find its worst case lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Here f T i f i ≥ 0 and as we assume that at the current time step, the sheep is outside P, this ensures β h 2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' By removing these terms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' the lower bound of bH i can be given as bH i ≥ � j∈S\\i (xSi − xP )T JS jif j + (xSi − xP )T JS iif i + � l∈D\\k (xSi − xP )T JD li uDl + α(xSi − xP )T f i (1) Using the triangle inequality on the RHS and Cauchy-Schwarz inequality on individual terms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' we get bH i ≥ � j∈S\\i � −σmax � JS ji � ∥xSi − xP ∥ ∥f j∥ � − σmax � JS ii � ∥xSi − xP ∥ ∥f i∥ (2) + � l∈D\\k � −σmax � JD li � ∥xSi − xP ∥ ∥uDl∥ � − α∥xSi − xP ∥∥f i∥ where σmax is the largest singular value of a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Further, using the fact that the largest singular value of a matrix (σmax) is upper bounded by its Frobenius norm (σF ), we obtain bH i ≥ � j∈S\\i � −σF � JS ji � ∥xSi − xP ∥ ∥f j∥ � − σF � JS ii � ∥xSi − xP ∥ ∥f i∥ (3) � l∈D\\k � −σF � JD ki � ∥xSi − xP ∥ ∥uDl∥ � − α∥xSi − xP ∥∥f i∥ Now to compute this lower bound we make use of assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' We use the dynamics in (1) to compute JS ii and obtain the upper bound on σF � JS ii � and use the bounds on distances from assumption 3 to get following upper bound: 14 Nishant Mohanty∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Jaskaran Grover∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Changliu Liu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Katia Sycara σF � JS ii � ⩽ � j∈S\\i kS � √ 2 + (3 + √ 2)R3 ∥xSi − xSj∥3 � + √ 2kG + � l∈D\\k � 3 + √ 2 � kD ∥xSi − xDl∥3 ⩽ (n − 1) � √ 2kS + (3 + √ 2)kSR3 L3 S � + √ 2kG + n �� 3 + √ 2 � kD L3 D � := λM We omit the proof of this computation in the interest of space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' using the dynamics in (1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' we compute an expression for JS ji and obtain an upper bound on σF � JS ji � as follows: σF � JS ji � ⩽ √ 2kS + (3 + √ 2)kSR3 ∥xS1 − xSj∥3 ⩽ √ 2kS + (3 + √ 2)kSR3 L3 S := λS Likewise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' an upper bound of σF � JS li � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' is given by σF � JS li � ⩽ (3 + √ 2)kSR3 ∥xS1 − xDl∥3 ⩽ (3 + √ 2)kSR3 L3 D := λD Lastly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' we use obtain an upper bound on the dynamics of each sheep f i as: ∥f i∥ ⩽ � j∈S\\i kS � ∥xSi − xSj∥ + R3 ∥xSi − xSj∥2 � + kG∥xG − xSi∥ + � l∈D kD ∥xSi − xDl∥ ∥xSi − xDl∥3 (4) Now we need to compute the maximum possible value of the RHS to get the upper bound of the sheep dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' The first term has a local minima at ∥xSi − xSj∥ = (2)1/3R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Therefore the maximum value can occur at either the lower bound or upper bound of ∥xSi − xSj∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Thus the maximum value of the first term can be given as Fmax := max(kSLS + kS R3 L2 S , kSMS + kS R3 M 2 S ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Second term is maximum when ∥xG − xSi∥ = MG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' The last term is maximum when distance of the sheep to the dogs are minimum, ∥xSi −xDk∥ = LD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Using these the upper bound on the sheep dynamics is computed as: ∥f i∥ ⩽ (n − 1)Fmax + kGMG + nkD � 1 L2 D � Assuming that the velocity of the dog robots have an upper bound, and by taking the upper bound on the dynamics of all the sheep to be equal, the lower bound on bH i from 3 is (taking γ = −(α + λM + (n − 1)λS)Mp) bH i ⩾ γ � (n − 1)Fmax + kGMG + nkD L2 D � − (n − 1)λDMP ∥uD∥max This shows that bH i has a finite lower bound, thus ruling out possibility 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' Thus, the herding constraint (13) for a one dog to repel 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' New York, NY, USA: Association for Computing Machinery, 1987, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} +page_content=' 25–34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfmASv/content/2301.03293v1.pdf'} diff --git a/69E1T4oBgHgl3EQfTgNE/content/tmp_files/2301.03078v1.pdf.txt b/69E1T4oBgHgl3EQfTgNE/content/tmp_files/2301.03078v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..47ec537728436a86c867cd09c5985b33673164a4 --- /dev/null +++ b/69E1T4oBgHgl3EQfTgNE/content/tmp_files/2301.03078v1.pdf.txt @@ -0,0 +1,822 @@ +Structural tuning magnetism and topology in a magnetic +topological insulator +Christopher Eckberg,1, 2, 3, 4, ∗ Gang Qiu,4, ∗ Tao Qu,4 Sohee Kwon,5 Yuhang +Liu,5 Lixuan Tai,4 David Graf,6 Su Kong Chong,4 Peng Zhang,4 Kin L. +Wong,4 Roger K. Lake,5 Mahesh R. Neupane,2, 5, 7 and Kang L. Wang4, † +1Fibertek Inc., Herndon, Virginia 20171, USA +2DEVCOM Army Research Laboratory, Adelphi, Maryland 20783, USA +3DEVCOM Army Research Laboratory, +Playa Vista, California 90094, USA +4Department of Electrical and Computer Engineering, +University of California, Los Angeles, California 90095, USA +5Department of Electrical and Computer Engineering, +University of California, Riverside, CA, 92521, US +6National High Magnetic Field Laboratory, +Florida State University, Tallahassee, Florida, 32310, USA. +7Materials Science and Engineering Program, +University of California, Riverside, CA, 92521, US +(Dated: January 10, 2023) +∗ These two authors contributed equally +† wangkl@ucla.edu +1 +arXiv:2301.03078v1 [cond-mat.mes-hall] 8 Jan 2023 + +To date, the most widely-studied quantum anomalous Hall insulator (QAHI) +platform is achieved by dilute doping of magnetic ions into thin films of the al- +loyed tetradymite topological insulator (TI) (Bi1−xSbx)2Te3 (BST) [1–4]. In these +films, long-range magnetic ordering of the transition metal substituants opens +an exchange gap ∆ in the topological surface states, stabilizing spin-polarized, +dissipationless edge channels with a nonzero Chern number C. The long-range +ordering of the spatially separated magnetic ions is itself mediated by electronic +states in the host TI, leading to a sophisticated feedback between magnetic and +electronic properties. Here we present a study of the electronic and magnetic +response of a BST-based QAHI system to structural tuning via hydrostatic pres- +sure. We identify a systematic closure of the topological gap under compressive +strain accompanied by a simultaneous enhancement in the magnetic ordering +strength. Combining these experimental results with first-principle calculations +we identify structural deformation as a strong tuning parameter to traverse a +rich topological phase space and modify magnetism in the magnetically doped +BST system. +Time-reversal invariant Z2 TIs feature gapless edge and surface states protected by time- +reversal symmetry. Due to this time-reversal symmetry requirement, electronic band struc- +tures of Z2 TIs respond strongly to magnetic perturbation [5–7]. This relationship is notably +exemplified in the realization of the quantum anomalous Hall effect in magnetic TI systems. +In QAHIs, long-range magnetic order gaps the otherwise mass-less topological surface state. +When the chemical potential is positioned within the exchange gap, the 2D density of states +vanishes, and a chiral edge state represents the lone channel for electrical transport. The +resulting phase is characterized by a combination of long-range magnetic order, quantized +Hall conductivity, and vanishing longitudinal resistance; all of which persist in the ab- +sence of an applied magnetic field. The promise of technologically significant phenomena in +QAHI materials including dissipationless, non-reciprocal electrical transport [8, 9], quantized +magneto-electric dynamics [10, 11], and exotic quasiparticle excitations [12, 13] to name a +few, has stimulated a tremendous research effort in QAHI systems and magnetic topological +matter in general. +Though recent discoveries have notably expanded the landscape of known QAHI hosts +[14–16], the most mature and widely studied QAHI platform is the Cr substituted +2 + +(Bi1−x−ySbxCry)2Te3 (CBST) system grown using molecular beam epitaxy (MBE). Due to +the greatly reduced spin-orbit coupling strength of Cr compared with the Bi/Sb atoms it +substitutes, in quantized CBST the concentration of magnetic dopants must be left rela- +tively dilute lest they promote a topological phase transition to a trivial insulating state +[6, 17]. The dilute magnetism and disordered, dual-doped crystal structure of CBST im- +bues an unfortunate fragility onto the quantum anomalous Hall effect in CBST at elevated +temperatures [18–21]. This fragility of the quantum anomalous Hall state presents a major +hurdle limiting the technical applicability of these compounds. Operational temperatures +may be improved to a degree by varying dopant concentrations and profiles [22, 23]. How- +ever, in practice chemical optimization of these materials is a delicate and imperfect process, +as chemical composition simultaneously impacts the positioning of the chemical potential, +electronic band structure, disorder profile, and magnetic ordering strength. A cleaner tuning +parameter to more directly engineer CBST band structures is therefore essential to improve +CBST QAHI operating temperatures. +Here we report the magnetic and electronic evolution of CBST in response to a continuous +deformation of the crystal lattice via hydrostatic pressure. Pressure dependent experiments +were performed on gated Hall bar devices at pressures up to 1.6 GPa and temperatures +as low as 20 mK. Our experiments demonstrate the electronic and magnetic properties +of CBST to be highly responsive to strain, with lattice compression both suppressing the +QAHI phase and enhancing the magnetic order. First-principle calculations confirm these +effects emerge from a structural driven evolution of the CBST band structure, and indicate +a rich topological phase space may be addressed through the application of even larger +pressures. Together, these experimental and theoretical results demonstrate crystal strain +as an effective tuning parameter to selectively modify the low energy electronic structure +in BST based magnetic topological matter, establishing structural engineering as a viable +pathway to control critical material properties in the future. +Experiments are conducted on 6 quintuple layer (QL) thick MBE grown CBST films with +a magnetic Curie temperature TC of roughly 20 K. Data are presented for three different +photolithographically defined Hall bar devices, labelled P1, P2, and P3. Samples P2 and +P3 were fabricated simultaneously from the same wafer in field-effect transistor geometries, +where an approximately 20 nm thick HfOx layer serves as the gate dielectric. These two +devices display virtually identical behaviors over a wide range of temperature and magnetic +3 + +field [24], and, in the following, their properties will frequently be compared directly. P1 +meanwhile was fabricated from a separate wafer. The impact of pressure on the topological +transport signatures and magnetism of these devices was studied using a piston cell equipped +for electrical transport experiments. During experiment, P1 was measured in a dilution +refrigerator while P2 and P3 were primarily studied in a 3He sorption cryostat. +Taken +together, data gathered on these different devices span nearly three decades in temperature +ranging in regime from kbT << ∆ to kbT ≈ TC. +We begin by presenting the ambient pressure properties of device P2 (Fig. 1). At TC, +the systems develops a magnetization when subjected to a small external field. +When +cooled to lower temperatures, this magnetic order manifests a rapidly increasing anomalous +Hall signal, and at temperatures well below TC ρxx begins to rapidly decrease as seen in +Fig. 1 (a). At dilution refrigerator conditions, ρxx approaches zero while ρyx approaches the +quantized value of h/e2 ≈ 25.8 kΩ. Field dependent hysteresis loops in a dilution refrigerator +environment are presented in Fig. 1 (b). In these data, transitions between ρyx plateaus +accompany the switching of the magnetic order in the system between the down and up +states and mark a topological transition between C = 1 and C = −1. The ρxx peaks and ρyx +zero crossings observed in the magnetic hysteresis loops occur at the magnetic coercive field +µ0Hc and correspond with an Mz = 0 condition (Fig. 1 (b)). +Figure 1 (c) demonstrates the gate response of device P2 at a temperature of T = 500 mK. +At this relatively elevated temperature thermal excitations into dissipative states precludes +the high quality quantization observed in the dilution cooled regime. Nevertheless, a clear +ρxx (ρyx) minimum (maximum) is observed at an optimized gate voltage of roughly −1.5 V; +indicative of the incipient QAHI phase. The magnetic coercive field µ0Hc, a rough avatar +for the magnetic ordering strength, is also measured as a function of the gate voltage. We +observe an enhanced magnetic order when the system is driven away from the charge neutral +point. Such an enhancement in magnetism with the addition of carriers into the system has +been previously reported, and is commonly attributed to itinerant carrier mediated RKKY +interactions strengthening the coupling between Cr-ions [25–27]. In a narrow range near V c +g , +however, µ0Hc exhibits very little if any response to the changing gate voltage (gray regime +in Fig. 1 (c)). In this region the carrier concentration is minimized, and the Cr-Cr magnetic +coupling is sustained by the van Vleck mechanism [5, 28, 29]. +Following ambient pressure characterization, QAHI devices were loaded into a piston +4 + +pressure cell (Fig. 2(a)) and studied at hydrostatic pressures up to 1.6 GPa (Fig. 2 (b)). +Comparison to the related Sb2Te3 system suggests a nearly isotropic compression of roughly +1% may be expected in our CBST films at the maximal pressure [30, 31], though clamping +by the more rigid GaAs substrate may slightly mute the lattice compression experienced by +our thin film devices [31, 32]. Despite the modest compression that may be anticipated in +these experiments, our QAHI devices are nevertheless quite responsive to lattice tuning in +the range of pressure studied. Figure 2 presents a summary of basic transport data collected +at 1.5 K, indicating that the system trends away from quantized transport with shrinking +unit cell size. This is evidenced by an increase in ρxx and concomitant reduction in ρyx +seen in both gate voltage traces (Fig. 2 (c), 2 (d)) and field hysteresis loops (Fig. 2 (e), 2 +(f)). Despite the trend away from quantized transport, a ρyx (ρxx) maxima (minima) is still +seen near V c +g at all pressures. That the gate cannot recover the same degree of transport +quantization at all pressures suggests the flow away from idealized QAHI behavior is due to +a pressure driven modification of the electronic band structure rather than a rigid shift of the +chemical potential, as the latter scenario could possibly be compensated for by sweeping out +carriers using the gate. In fact, the observation of a consistent V c +g at all pressures suggests +any shifting of the Fermi energy within the pressure range explored is minimal. +Comparison of the temperature and voltage dependent ρxx and ρyx at pressures of 0, +0.7, and 1.6 GPa are shown as two-dimensional color plots in Figs. 3 (a-f), providing a +qualitative visualization of a closing topological gap with increasing pressure. By fitting the +temperature dependent ρxx at Vg = 0 V and µ0H = 2 T to an Arrhenius model (Fig. 3 g) the +value of this gap is quantified at all pressures. The pressure dependence of the topological +gap is summarized in Fig. 3 (i), indicating the gap remains intact, but decreases in a linear +fashion by almost exactly a factor of 2 from 1.2 K to 0.6 K over the pressure range explored. +Extrapolating the linear trend to zero suggests a critical pressure PC of approximately 3.3 +GPa, at which point we anticipate a topological phase transition away from the QAHI +state to occur. To demonstrate that the QAHI phase does persist to the highest pressures +measured in this study, in Fig. 3 (h) we present data collected on another device, sample P1, +at a temperature of 20 mK and pressure of 1.6 GPa. At these conditions, we still observe +conductivity values within 3% of the quantized expectation of e2/h, confirming that 1.6 GPa +is insufficient to drive these samples out of the QAHI ground state. +Intriguingly, while the transport gap closes in pressurized QAHI samples, we observe +5 + +a pressure driven enhancement in the magnetic order as demonstrated by an increasing +coercive field. This enhancement is visible in Fig. 4, where the field dependent ρxx hysteresis +loops collected from device P1 at a temperature of 20 mK are presented. The ρxx peaks +are clearly pushed to higher fields with increasing pressure, reflecting the growth in µ0Hc +from a value of 133 mT at 0.1 GPa to a notably larger 144 mT at 1.6 GPa. At the 20 +mK temperature where these data are collected kbT << ∆. Consequently, the strength of +magnetic interactions dependent upon itinerant charge carriers are vanishingly weak and +this enhanced magnetic order is presumably sustained through the van Vleck mechanism. +In addition to the magnetic enhancement observed at 20 mK, an increasing coercive field +under pressure is also observed in devices P2/P3 between 280 mK and 15 K, demonstrating +this effect to be consistent between samples and persistent over a wide temperature range +[24]. Gate-dependent data collected in P2/P3 demonstrate the magnetic enhancement can +be maximized by gate-tuning towards the valence band; indicating that pressure likely also +enhances the hole-mediated RKKY interaction in a manner consistent with previous reports +in more traditional dilute magnetic semiconductors [33]. +The strengthened magnetic ordering indicates that it is unlikely that the pressure driven +reduction in the transport gap is a consequence of a reduced exchange field at the Dirac sur- +face states. Alternative sources that may account for the suppressed gap include increasing +surface hybridization or occupation of delocalized, dissipative states. To understand what +effects are dominant in our system, we perform first principle band structure calculations +for a 6 QL thick slab of CBST host compound Sb2Te3. As Bi primarily functions as a +counter-dopant in CBST [34], and Cr d-electrons do not contribute to the density of states +at the Fermi energy [5], Sb2Te3 calculations capture the principal details of the CBST band +structure [29] while notably excluding the material magnetism and resulting exchange gap. +Thus, in the presented calculations, trends in the surface hybridization are observed directly +and are not obfuscated by magnetic gapping of the surface band structure. Following pre- +vious example [30, 35], pressure is simulated by isotropically compressing the boundaries +of the computational unit and allowing all interior atoms to relax to their lowest energy +configuration. +Band structure calculations were performed in 1% strain increments between 0% and 4%. +Calculations at 0% and -4% compressive strains are presented in Fig. 5 (a-d). Consistent +with previous reports, we find that increasing pressure widens the direct, bulk band gap at +6 + +Γ, while simultaneously raising the energy of the valence band valley Evb between Γ and M +[30]. At the computational thickness of 6 QL, we observe a small hybridization gap m in the +surface bands even in the zero-compression limit (Fig. 5 (e)). This feature is amplified as +the unit cell size is decreased, indicating increasingly pronounced hybridization between top +and bottom surfaces of the TI under pressure. Finally, using the calculated band structures, +we extract the van Vleck susceptibility χV V according to the relationship: +χV V = 1 +N +� +k +� +Enk<µ 0) delocalized states in the valence band will populate down to +lowest temperatures and the system will behave as a metal. In the case when Evb > 0 and +∆ex > m carrier conduction in the chiral channels will relax internally through the dissipative +valence band states, precluding transport quantization. The calculated evolutions of m, χV V , +and Evb with increasing pressure (Fig. 5 (e-g)) therefore imply a coalescence of metallic, +insulating, and QAHI electronic ground states in the pressure tuned CBST system. Based +upon these calculated band structures and the known evolutions of the hybridization gap, +bulk state quantum confinement, and exchange energy with decreasing CBST thickness +[18, 29, 36], we present a proposed topological phase diagram in layer thickness and pressure +dependent parameter space in Fig. 5(h). These phase boundaries could be further adjusted +by tuning along other axes such as external magnetic field [18, 29] or applied gate voltage. +We will also note that the QAHI/insulator phase boundary presented here assumes that +the hybridization gap is more responsive to pressure than the exchange gap, an assumption +7 + +supported by the relatively weak pressure effect on χvv (Fig. 5 (f)) compared with m (Fig. 5 +(e)). It is possible, however, that the exchange gap may feature its own pressure dependence, +altering the trajectory of the ∆ex = m boundary. +Having discussed the calculated pressure dependent band structure for this system, we +now consider how these calculations comport with the experimentally determined results. +Notably, our band structure calculations unambiguously indicate a trend away from the +QAHI state in compressed tetradymite TIs, consistent with the experimental reality. While +the calculations indicate that the electronic state beyond Pc may be either trivially insulat- +ing or metallic depending upon the details of the material, in our samples we believe the +topological phase transition occurring at Pc to be towards the metallic regime. We come to +this conclusion through the observations of reduced ρxx at temperatures above TC, as well +as reductions in the ρxx peaks at µ0Hc with increasing pressure; both of which suggest an +increasing density of states near the Fermi energy. Meanwhile, the enhanced χvv observed in +calculation captures the increasing magnetic ordering strength observed in our pressurized +QAHI films. +To conclude, these results establish lattice deformation as an effective, clean tuning pa- +rameter for modifying the electronic and magnetic properties of alloyed QAHI materials. +Though a significant material response is observed in the pressure range explored in this +study, we believe increasing pressure may evoke even more dramatic electronic and mag- +netic responses. Crucially, Pc is well below the 9+ GPa threshold at which a structural +phase transition from rhombehedral to monoclinic crystallographic point groups has been +previously reported in tetradymite TI systems [30, 37, 38], indicating future experiments +may explore a significantly larger pressure range without concern of interference from ad- +ditional structural phases. Finally, on the basis of these results, we propose that tensile +strain, as opposed to its compressive counterpart studied here, may present an exciting +tuning parameter to explore in future efforts to enhance QAHI behavior. +8 + +I. +METHODS +A. +Material Growth +All CBST films were grown in an ultra-high vacuum, Perkin-Elmer molecular beam epi- +taxy (MBE) system. Epi-ready semi-insulating GaAs (111)B substrates were used for the +growth. Before growth, the substrates were loaded into the MBE chamber and pre-annealed +at the temperature of 630 °C in a Te-rich environment in order to desorb the oxide on the +surface. During growth, the substrate was kept at 190 °C. High-purity Bi, Sb, Cr and Te +sources were evaporated simultaneously from standard Knudsen cells. The growth process +was monitored by the reflection high-energy electron diffraction (RHEED) in-situ, and the +digital RHEED images were captured using a KSA400 system built by K-space Associates, +Inc. +Sharp and streaky lines in the RHEED pattern indicate good epitaxial crystalline +quality. +B. +High pressure experiments +Pressure was applied using a standard piston pressure-cell. To fit within the active area +of the pressure cell, single devices were cut from pre-patterned wafers to dimensions of less +than 3.0 mm, and were fixed to a fiber optic using epoxy to orient the sample within the +pressure cell. Thin platinum wires were attached by hand to the contact pads of the device +under test with silver paint. A small ruby chip was fixed to the tip of the fiber optic, which +was used to calibrate the pressure at room temperature and again at low temperature. A +PTFE cup was filled with Daphne 7575 oil and fixed in place over the sample platform so +that the device was surrounded by the hydrostatic fluid. Once assembled, the cell was placed +in a hydraulic press where a piston fed through a hole in the threaded top screw of the cell +was used to add pressure. When the appropriate pressure was reached, the top screw was +clamped, locking in the pressure. +Transport measurements were collected using a low-frequency (< 10 Hz) lockin technique +with ac excitations of 10 nA. Gate swept data display a small hysteresis based upon the +gate history. To compensate for this effect and ensure consistency all data presented were +taken during sweeps from +3.25 to −5 V. +Cryogenic sample environments for ambient pressure experiments were maintained us- +9 + +ing a Quantum Design Physical Property Measurement System equipped with a dilution +refrigerator insert. High pressure measurements were conducted in high magnetic field cells +SCM-1 and SCM-2 at the National High Magnetic Field Laboratory in Tallahassee. SCM-1 +is equipped with a dilution refrigerator cryogenic environment while SCM-2 was operated +with pure 3He cooled using a sorption pump. Both SCM-1 and SCM-2 are equipped with +18 T superconducting magnets. To compensate for the remnant field of the superconduct- +ing magnet, the field dependent data were calibrated using a Hall sensor. Additionally, to +account for the magnetoresistance of the SCM-2 thermometers, the temperatures used in +Figs. 3 and 4 were calibrated using the strong temperature dependence of the QAHI mate- +rial itself. For details of the field and temperature calibration processes please refer to the +Supplemental Information [24]. +C. +First principle calculations +We perform first-principles calculations as implemented in the Vienna Ab Initio Simu- +lation Package (VASP) [39].The Perdew, Burke, Ernzerhof (PBE) form of the generalized +gradient approximation is used as the exchange-correlation functional [40]. The computa- +tional cell employed is constructed from six QL Sb2Te3 slabs stacked with a 40 ˚A thick +vacuum region. We apply an energy cutoff of 500 eV and a 8x8x1 Γ-centered k-grid to +optimize cell structure and atomic positions. The optimized lattice constant of the slab is +a = b = 4.3307 ˚A and c = 31.09 ˚A. Tri-axial compressive strains between -4.0% and 0.0% +are applied by shrinking the perimeter of the Sb2Te3 slab. At each strain, atomic positions +inside the unit cell are allowed to relax in all directions. Spin-orbit coupling is included +during the charge density relaxation for electronic band structures. +Based upon the band structure, van Vleck susceptibilities were calculated according to +Eq. 2 [5]. +χV V = 1 +N +� +k +� +Enk<µ { +4 +console.log(stdout); }); var a={ hello: 'world`; +5 +search(opts) +Listing 3: Exploit for code injection (npm advisory 315). +rule designed to detect calls to the eval function using user- +controlled inputs. In order to create this CodeQL rule, one +starts by specifying the appropriate configuration, that is, a +code description of the targeted sources and sinks. In this +case, we are interested in code flows from remote flow sources, +described by the predicate isSource (lines 3 to 5), to the eval +sink, described by the predicate isSink (lines 6 to 8). Then +the main query (lines 11 to 13) states that, using the specified +configuration (EvalTaint cfg), CodeQL should find code paths +from the specified source to the specified sink. The output of +this query is a string with a description of the source-sink pairs +that match the query. This particular rule is a simplified version +of one of the CodeQL rules [69] executed inside VulcaN. +In general, graph-based analysis works well for taint-tracking, +but it requires every source and sink to be explicitly encoded +into rules. These sources and sinks change over time as +languages evolve and new popular third-party packages are +created. This is why the community has started to work on +automatically generating such taint-tracking specifications [71]. +Using syntax-based analysis: NodeJsScan helps us showcase +syntax-based analysis. Listing 5 lists an excerpt of the Node- +JsScan rule that detects potentially vulnerable uses of eval. To +be applied, this rule must match two related patterns. First, eval +must occur inside a function receiving two or more arguments +(line 2). Then, eval must be called with a parameter computed +using one of the given arguments (line 4). NodeJsScan includes +analogous rules for other vulnerability types [73]. +Similarly to the previous technique (graph-based), syntax- +based analysis also suffers from the source-sink specification +limitation. Additionally, there are some other limitations +specific to syntax-based analysis. For example, it may lead +to a high number of false positives, as it is not expressive +enough to capture the dependencies of the variables occurring +in the patterns; e.g., it will detect all calls to eval regardless +of whether or not their given input can be controlled by +the user. Furthermore, it commonly leads to rule overfitting, +resulting in over-specific rules that match known examples +of vulnerabilities, but are not general enough to capture other +instances of the same vulnerability. In Listing 5, the eval call is +7 + +1 +class EvalTaint extends TaintTracking::Configuration { +2 +EvalTaint() { this = "EvalTaint" } +3 +override predicate isSource(Node node) { +4 +node instanceof RemoteFlowSource +5 +} +6 +override predicate isSink(Node node) { +7 +node = globalVarRef("eval").getACall().getArgument(0) +8 +} +9 +} +10 +11 +from EvalTaint cfg, Node source, Node sink +12 +where cfg.hasFlow(source, sink) +13 +select sink, "Eval with user input from \$@.", source +Listing 4: CodeQL rule for eval taint-tracking [70]. +1 +patterns: +2 +- pattern-inside: function $FUNC($REQ, $RES, ...) {...} +3 +- pattern-either: +4 +- pattern: eval(..., <... $REQ.$QUERY ...>, ...) +Listing 5: NodeJsScan rule for eval detection [72]. +only detected when it occurs inside the body of a specific type +of function declaration. Besides ignoring eval calls at the top +level, this pattern also ignores calls to eval which occur inside +the body of JavaScript functions declared using alternative +syntactic constructs (e.g. function constructor and lambdas). +Using keyword-based analysis: The tool we use to illustrate +keyword-based analysis is Graudit. The following is an excerpt +of a Graudit rule that detects the use of eval: +eval[[:space:]]*\( +Here, we see a regular expression pattern that simply detects +any call to the eval function. This technique suffers from several +limitations such as the source-sink specification problem and +a high number of false positives. Graudit includes many other +rules for dangerous sinks in Node.js applications [74]. +V. EFFECTIVENESS OF THE TOOLS (RQ3) +In this section, we focus on our third research question +(RQ3). To perform a quantitative and qualitative assessment +of the selected tools, we begin by specifying the evaluation +metrics and methodology we used to rank the tools. Then +we present our findings, relying on the result of running the +selected tools across all 957 advisories of our curated dataset. +A. Evaluation Methodology +Tool evaluation metrics: To evaluate the selected tools, we use +two main metrics: true positive rate (TPR) and precision (P). +The TPR represents the proportion of the total vulnerabilities +that are correctly detected by a given tool, i.e. the true positives +(TP): TPR=TP/|vulnerabilities|. The TPR is useful to assess the +raw detection rate of a tool without considering the influence +of false positives (FP), i.e., its results that do not match +the reported advisory. Precision represents the proportion of +correctly classified positive cases: P=TP/(TP+FP). This metric +is useful to assess if a tool produces too many false positives +that can unnecessarily consume analysts’ resources. +Tool classification score: To compute the evaluation metrics +for a given tool, we need to analyze the output that it generates + +/src/lib/drivers/search/pouch.js +Possible code injection + +Command injection is an attack in which the goal is +execution of arbitrary commands on the host +operating system via a vulnerable application. +�→ +�→ + +High +eval\s?\(|setTimeout|setInterval +Line: 20 - eval(opts.filter); + +Listing 6: Snippet of Mosca output classified with Score A. +/src/lib/drivers/search/pouch.js-19- } +/src/lib/drivers/search/pouch.js:20: eval(opts.filter); +/src/lib/drivers/search/pouch.js-21- opts.filter = filter; +Listing 7: Snippet of Graudit output classified with Score B. +when applied to analyzing a specific vulnerability. Given that +each tool outputs the vulnerability analysis results in its own +specific, unstandardized format, we characterize a tool’s output +according to a common discrete classification score: +• Score A: The tool correctly detects and classifies the +vulnerability reported in the advisory (true positive). +• Score B: The tool shows a warning for the vulnerable +code, but does not explicitly classify the finding as a +vulnerability (true positive). +• Score C: The tool only shows results that do not match +the vulnerability in the advisory report (false positives). +• Score D: The tool produces no output (false negative). +We split the TP results according to two distinct classes: A +means an explicit vulnerability notification, and B a security +warning notification. The tools ranked with score A provide a +richer output to the user and, thus, more information about the +detected vulnerability. As an example, consider the output of +two tested tools, Mosca and Graudit, with regards to advisory +315 shown in Listings 6 and 7, respectively. Although both +outputs flag the vulnerable eval call reported by the review file +of Listing 1, Mosca’s output clearly identifies a possible code +injection, provides a description, a severity level, and the line +of code containing the vulnerability. On the other hand, Graudit +only shows the vulnerable line of code without explaining how +or why it flags that particular snippet. For this reason, the +output of Mosca is classified with Score A while the output of +Graudit is classified with Score B. +This discrete classification is also important to account for +tools that might flag for the vulnerability at a place that is only +close to it (textually, or on the AST). Considering this, we +require that tools must clearly identify the vulnerable statement +for some vulnerabilities, e.g., code injections and others that +can typically be pinpointed to a single statement, while for +other vulnerability types multiple lines-of-code are acceptable. +Two authors performed a cross-check of all tools’ outputs to +guarantee fairness of the tool classification in these cases. +B. Analysis Performance +We gauge analysis performance by measuring tools’ execu- +tion time for all 957 advisories on a machine with an Intel +8 + +Tool +Min +Max +Mean +St. Dev. +Q-90 +Total +ODGen +1.236 +3653.043 +148.757 +385.629 +370.969 +110823.8 +CodeQL +1.712 +736.546 +119.570 +98.755 +177.550 +28696.8 +NodeJsScan +20.023 +984.453 +99.562 +121.246 +230.216 +23795.4 +ESLint SSC +0.592 +3556.665 +29.871 +237.799 +25.465 +7139.1 +Graudit +0.042 +14.632 +0.550 +1.624 +0.704 +131.3 +InsiderSec +0.000 +243.000 +5.749 +25.186 +7.000 +1374.0 +MS DevSkim +0.276 +186.338 +6.393 +23.262 +8.044 +1527.9 +Drek +0.300 +6.649 +1.022 +0.865 +1.949 +244.3 +Mosca +0.005 +245.498 +7.408 +25.166 +12.216 +1770.5 +Table V: Summary statistics of the analysis times (in seconds) taken +by the tested tools across all 957 reviewed advisories. +ODGen +CodeQL +NodeJsScan +ESLint SSC +Graudit +InsiderSec +MS DevSkim +Drek +Mosca +0 +20 +40 +60 +80 +100 +Percentage of Advisories +146 +286 +98 +103 +75 +294 +212 +288 +231 +324 +414 +513 +158 +376 +210 +456 +515 426 +530 +146 225 +791 +500 +732 +475 +A +B +C +D +Figure 4: Score distribution for each tool. +Xeon E3-1220 v3 @ 3.10GHz processor and 32GB of memory. +Table V shows several statistics of the execution times taken +by each tool to analyze our dataset. When compared to all other +tools, ODGen, CodeQL, NodeJsScan and ESLint SSC require +considerably more time. To analyze all 957 packages, ODGen +took over 30 hours (110k seconds), CodeQL took nearly 8 +hours (27k seconds), NodeJsScan took nearly 7 hours (24k +seconds), while ESLint SSC took nearly 2 hours (7k seconds). +All other tools are considerably more efficient, taking at most +30 minutes to analyze all packages. +The mean execution time of ODGen, CodeQL and Node- +JsScan is 148.8, 119.6 and 99.6 seconds, respectively. ODGen, +CodeQL and NodeJsScan tools are slower because their detec- +tion techniques involve modeling statically computed structures. +These operations are more complex than performing keyword- +based matching searches (see Section IV-B). Depending on +the size of the package and on the CI/CD pipeline restrictions, +these tools may end up being exceedingly slow. +C. Results Across the Entire Dataset +Figure 4 displays the score distribution for each tool across +our entire dataset, and Table VI shows the evaluation metrics +for each tool. Globally, the tested tools perform rather poorly. +We can draw the following main observations: +1. Some tools have very low TPR: Counting A and B +scores as successful detections, we see that InsiderSec, Drek +and Mosca only detect 7 (0.7%), 15 (1.6%) and 25 (2.6%) +vulnerabilities, respectively. Hence, these tools fail to detect +most vulnerabilities of the dataset. +2. The tools with best TPR have very low precision: The +tools that have higher TPR are: ODGen, Graudit, ESLint SSC, +Graph +Syntax +Keyword +0 +25 +50 +75 +100 +Percentage of Advisories +320 +196 +104 +226 +197 +356 +443 +516 +262 +92 +140 +A +B +C +D +Figure 5: Score distribution for each detection technique. +and CodeQL. Unfortunately, Graudit and ESLint SSC also +have a considerable number of false positives, which tends to +erode the confidence of application developers in vulnerability +detection tools. Graudit detects 219 vulnerabilities (22.9%), +but it also reports over 109k FPs, giving it an overall precision +of just 0.2%. A higher number of FPs is expected from a +keyword-based tool like Graudit, as many of its string signatures +often match non-vulnerable code snippets. ESLint SSC has +the highest TPR (41.5%). However, it is also the tool with the +highest number of reported FPs (over 389k) and, consequently, +the lowest precision (0.1%). This is because ESLint SSC +includes many rules from different ESLint plugins, some of +which are simple matches (akin to keyword-based analysis) +with greedy behaviour, leading to a higher number of FPs. +3. Graph-based analysis has the best detection capability: +Figure 5 shows the scores according to a particular detection +technique. These results show that graph-based analysis reports +a significantly larger number of results with score A (explicit +vulnerability notifications). Syntax and keyword-based analysis +look fairly similar, with reasonable detection rates, but also a +high number of reports containing only false positive results. +When considering the results of both tools in this category, +i.e., ODGen and CodeQL, they strike a better balance between +true positives and precision. CodeQL detects 300 vulnerabilities +(31.3%) and has a significantly higher precision (7.8%), when +compared to most other tools, while ODGen detects 154 +vulnerabilities (16.1%). This number is significantly lower +than CodeQL’s, but it represents a much higher precision +(23.8%) than any other tool tested. Although both these +tools do not have the highest TPR, most of their detected +vulnerabilities were classified with the A score, meaning that +the reported information is richer and more meaningful to the +user. Consequently, CodeQL and ODGen are the most balanced +tools, achieving a reasonable detection rate (TPR) and less +FPs, when compared to other tools with similar TPR. We also +note that both these tools have the potential for being further +improved by extending them with additional rules. +4. Combining multiple tools increases TPR, but also lowers +the overall precision: The combination of the two best tools +(CodeQL and ESLint SSC) detects 508 vulnerabilities (53.1%), +albeit with only 0.12% precision. If we add the third best +tool (Graudit), we detect more vulnerabilities (551/57.6%), but +the precision further decreases to 0.11%. Finally, combining +9 + +Scope +ODGen +CodeQL +NodeJsScan +ESLint SSC +Graudit +InsiderSec +MS DevSkim +Drek +Mosca +TP (%) FP (P%) TP (%) FP (P%) TP (%) FP (P%) TP (%) FP (P%) TP (%) FP (P%) TP (%) FP (P%) TP (%) FP (P%) TP (%) FP (P%) TP (%) FP (P%) +CWE-22 +70 +136 +104 +416 +56 +257 +110 +25467 +122 +3101 +2 +401 +0 +368 +0 +1057 +0 +241 +(47.9) +(34.0) +(71.2) +(20.0) +(38.4) +(17.9) +(75.3) +(0.4) +(83.6) +(3.8) +(1.4) +(0.5) +(0.0) +(0.0) +(0.0) +(0.0) +(0.0) +(0.0) +CWE-79 +1 +33 +26 +843 +6 +924 +29 +123477 +11 +23634 +0 +28 +0 +3990 +0 +6353 +1 +1664 +(1.0) +(2.9) +(26.3) +(3.0) +(6.1) +(0.6) +(29.3) +(0.0) +(11.1) +(0.0) +(0.0) +(0.0) +(0.0) +(0.0) +(0.0) +(0.0) +(1.0) +(0.1) +CWE-400 +4 +40 +13 +212 +2 +374 +39 +21936 +4 +2795 +0 +22 +0 +1026 +0 +74 +1 +271 +(4.5) +(9.1) +(14.6) +(5.8) +(2.2) +(0.5) +(43.8) +(0.2) +(4.5) +(0.1) +(0.0) +(0.0) +(0.0) +(0.0) +(0.0) +(0.0) +(1.1) +(0.4) +CWE-78 +22 +40 +43 +416 +2 +121 +29 +7405 +4 +1567 +0 +15 +0 +269 +3 +380 +3 +190 +(29.3) +(35.5) +(57.3) +(9.4) +(2.7) +(1.6) +(38.7) +(0.4) +(5.3) +(0.3) +(0.0) +(0.0) +(0.0) +(0.0) +(4.0) +(0.8) +(4.0) +(1.6) +CWE-818 +0 +11 +16 +57 +0 +97 +1 +6990 +3 +1431 +1 +8 +64 +1093 +0 +393 +0 +150 +(0.0) +(0.0) +(21.3) +(21.9) +(0.0) +(0.0) +(1.3) +(0.0) +(4.0) +(0.2) +(1.3) +(11.1) +(85.3) +(5.5) +(0.0) +(0.0) +(0.0) +(0.0) +CWE-471 +11 +19 +13 +54 +0 +295 +38 +7466 +0 +2632 +0 +0 +0 +249 +0 +220 +0 +438 +(22.9) +(36.7) +(27.1) +(19.4) +(0.0) +(0.0) +(79.2) +(0.5) +(0.0) +(0.0) +(0.0) +(0.0) +(0.0) +(0.0) +(0.0) +(0.0) +(0.0) +(0.0) +CWE-20 +5 +19 +4 +25 +0 +116 +19 +7947 +4 +911 +0 +13 +0 +181 +1 +26 +2 +294 +(12.2) +(20.8) +(9.8) +(13.8) +(0.0) +(0.0) +(46.3) +(0.2) +(9.8) +(0.4) +(0.0) +(0.0) +(0.0) +(0.0) +(2.4) +(3.7) +(4.9) +(0.7) +CWE-1321 +3 +12 +5 +78 +0 +92 +31 +18130 +0 +4468 +0 +9 +0 +0 +0 +301 +0 +465 +(8.3) +(20.0) +(13.9) +(6.0) +(0.0) +(0.0) +(86.1) +(0.2) +(0.0) +(0.0) +(0.0) +(0.0) +(0.0) +(0.0) +(0.0) +(0.0) +(0.0) +(0.0) +CWE-94 +4 +18 +5 +79 +2 +159 +16 +17930 +10 +7159 +1 +23 +0 +358 +8 +858 +8 +199 +(12.1) +(18.2) +(15.2) +(6.0) +(6.1) +(1.2) +(48.5) +(0.1) +(30.3) +(0.1) +(3.0) +(4.2) +(0.0) +(0.0) +(24.2) +(0.9) +(24.2) +(3.9) +CWE-77 +8 +11 +11 +90 +1 +32 +9 +5390 +0 +892 +0 +1 +0 +1 +0 +52 +0 +97 +(34.8) +(42.1) +(47.8) +(10.9) +(4.3) +(3.0) +(39.1) +(0.2) +(0.0) +(0.0) +(0.0) +(0.0) +(0.0) +(0.0) +(0.0) +(0.0) +(0.0) +(0.0) +Other CWE +26 +154 +60 +1283 +34 +2548 +76 +147552 +61 +60895 +3 +104 +17 +7930 +3 +9159 +10 +2868 +(8.9) +(14.4) +(20.5) +(4.5) +(11.6) +(1.3) +(26.0) +(0.1) +(20.9) +(0.1) +(1.0) +(2.8) +(5.8) +(0.2) +(1.0) +(0.0) +(3.4) +(0.3) +Dataset +154 +493 +300 +3553 +103 +5015 +397 +389690 +219 +109485 +7 +624 +81 +15465 +15 +18873 +25 +6877 +(16.1) +(23.8) +(31.3) +(7.8) +(10.8) +(2.0) +(41.5) +(0.1) +(22.9) +(0.2) +(0.7) +(1.1) +(8.5) +(0.5) +(1.6) +(0.1) +(2.6) +(0.4) +Table VI: TP, TPR (%), FP and Precision (P%) for each tool by CWE. TPR highlights: green (TPR ≥ 50%) or yellow (50% > TPR ≥ +15%). When TPR is highlighted we also highlight the FP and P columns: yellow (50% > P ≥ 15%), light red (15% > P ≥ 2.5%) and dark +red (P < 2.5%). The CWEs are: Path Traversal (CWE-22), Cross-site Scripting (CWE-79), Resource Exhaustion (CWE-400), Insufficient +Transport Layer Protection (CWE-818), OS Command Injection (CWE-78), Modification of Assumed-Immutable Data (CWE-471), Improper +Input Validation (CWE-20), Code Injection (CWE-94), Improper Neutralization of Special Elements used in a Command (CWE-77), and +Improperly Controlled Modification of Object Prototype Attributes (CWE-1321). +both graph-based tools, CodeQL and ODGen, allows for the +detection of 339 vulnerabilities (35.4%) with a precision of +7.7%. This shows that combining the best tools can increase +the TPR, but at the cost of also increasing the number FPs, +which limits the advantage of such an approach. +D. Results Across Specific Vulnerability Types +We now assess the performance of the tools when focusing +on particular types of vulnerabilities. We concentrate on two +main aspects: i) studying the types of vulnerabilities that the +tools detect more frequently, and ii) analyzing which types of +vulnerabilities can be detected simultaneously by several tools. +1. Most frequently detected vulnerability types: From the +analysis of Table VI, we highlight seven CWEs that are detected +most often regardless of the used tool: CWE-22, CWE-471, +CWE-78, CWE-79, CWE-94, CWE-77, and CWE-1321. These +are colored in yellow and green in Table VI. +CWE-22 (path traversal) is the only type clearly detected by +all four best performing tools (ODGen, ESLint SSC, Graudit, +and CodeQL). This is because path traversal can be found +statically by searching for well-known dangerous sinks in +the Node.js API, e.g., the functions readFile, writeFile and +createReadStream. The difference in precision between these +four tools lies in that ESLint SSC and Graudit simply match +these function calls, while ODGen and CodeQL report only +cases where the path is tainted by user input. +CWE-78 and CWE-77 (OS command injection), CWE-79 +(cross-site scripting) and CWE-94 (code injection) correspond +to classic injection vulnerabilities. Detecting these vulnerabil- +ities depends on the sets of sinks considered by each tool. +CodeQL detects more OS command injections, while ESLint +SSC detects more code injections because each have more +extensive rulesets for those particular vulnerabilities. Both +tools detect about the same number of XSS vulnerabilities. +Both CWE-471 (Modification of Assumed-Immutable Data) +and CWE-1321 (Improperly Controlled Modification of Object +Prototype Attributes) are umbrella CWEs for several prototype +tampering and prototype pollution vulnerabilities, for which +both CodeQL and ESLint SSC have various rules. We expected +ODGen to perform better at detecting prototype pollution vul- +nerabilities (CWE-471), as this is one of its central goals [15]. +Although ODGen’s results for this vulnerability (22.9%) fall +short of those by CodeQL (27.1%) and ESLint SSC (79.2%), +ODGen does achieve a much higher precision (36.7%) than +CodeQL (19.4%) and, especially, ESLint SSC (0.5%). +2. Vulnerability types detected by the three best performing +tools: Figure 6 shows the intersections of TPs for the top-10 +CWEs. We can see a substantial intersection for CWE-22, +where all three tools detect the same 85 vulnerabilities. This +happens because path traversals are easy to find statically +using a limited set of known dangerous sinks from the Node.js +API, which all tools share. For CWE-79, both CodeQL and +ESLint SSC detect about the same number of vulnerabilities, +but only about half intersect with each other. This is due to +differences in the rules regarding XSS sources and sinks. CWE- +471 shows a significant intersection, but ESLint SSC detects +several vulnerabilities that CodeQL misses. This is because +ESLint SSC’s rules have a wider range of sinks. Other CWEs +have less intersections because their rulesets differ. For example, +CodeQL is the only tool with specific rules to detect resource +downloads over HTTP, hence the results for CWE-818. +10 + +2 +4 +4 +7 +13 +17 +85 +CWE-22 +12 +13 +11 +6 +0 +2 +3 +CWE-79 +2 +26 +9 +2 +0 1 +1 +CWE-400 +24 +7 +18 +3 +1 +CWE-78 +14 +1 +1 2 +CWE-818 +2 +26 +11 +CWE-471 +2 +12 +1 +1 3 +CWE-20 +26 +5 +CWE-1321 +3 +7 +0 +2 +0 +7 +1 +CWE-94 +5 +4 +2 +CWE-77 +CodeQL +ESLint SSC +Graudit +Figure 6: Intersections of TPs of the 3 best tools for top-10 CWEs. +5 +10 +15 +20 +Precision (%) +0 +200 +400 +600 +# Lines of Code (LOC) +CWE-22 +CWE-78 +CWE-77 +CWE-471 +CWE-79 +CWE-818 +CWE-94 +CWE-400 +CWE-1321 +CWE-20 +Figure 7: Correlation between Query LOC and Precision. +E. Results as a Function of Queries and Ruleset +To study the relationship between ruleset/queries and vulner- +ability detection results, we take CodeQL as an example and +show, in Figure 7, how the precision of this tool compares with +the number of lines of code of the specific queries CodeQL uses +to detect the vulnerabilities in the Top-10 CWE categories in +the dataset. Although this cannot be taken as a general rule, we +can see that, in most cases, the precision is higher for smaller +queries. Taking into account each CWE, simpler vulnerabilities, +like CWE-22, can be detected using smaller queries with higher +precision, while more complex vulnerabilities to detect, like +CWE-1312, are harder to detect even when using larger (more +complex) queries. +It is then clear that vulnerability detection results can be +influenced by the ruleset/queries executed by the tool. Using a +small (specific) ruleset may improve the precision in detecting +a specific vulnerability. However, in the context of a CI/CD +pipeline, application developers do not know beforehand which +specific ruleset to select to detect the (unknown) vulnerability. +Consequently, it is reasonable to apply the most comprehensive +ruleset available for the tool. This is the approach we used in +this paper. For every tool, we selected the most comprehensive +and complete ruleset, by either combining rules into a single +tool execution or executing the tool multiple times using a +different rule for every execution and combining the results. We +also used the rules available off-the-shelf instead of developing +or customizing rules. This allows us to reflect how developers +will use these tools as most of them are not technically versed +to improve the ruleset specified by the tool developers. +VI. REASONS FOR MISSED DETECTION (RQ4) +In RQ4, we study why existing tools fail to detect certain +vulnerabilities. Table VII shows the number of undetected +CWE +CWE Description +OWASP +Undetected +% +CWE-79 +Cross-site Scripting +50 / 99 +50.5% +CWE-400 +Resource Exhaustion +- +44 / 89 +49.4% +CWE-78 +OS Command Injection +20 / 75 +26.7% +CWE-20 +Improper Input Validation +16 / 41 +39.0% +CWE-22 +Path Traversal +12 / 146 +8.2% +CWE-94 +Code Injection +10 / 33 +30.3% +CWE-818 +Insecure Transport Layer +10 / 75 +13.3% +CWE-287 +Improper Authentication +9 / 9 +100.0% +CWE-471 +MAID +8 / 48 +16.7% +CWE-200 +Information Exposure +8 / 14 +57.1% +Others +- +- +137 / 286 +47.9% +Total +- +- +324 / 957 +33.9% +Table VII: Number of vulnerabilities undetected by any tool. +Limitation +Advisory +CWE +Vulnerability +L1 +63 +CWE-730 +CVE-2015-9241 +567 +CWE-287 +CVE-2017-11429 +L2 +165 +CWE-818 +CVE-2016-10583 +305 +CWE-22 +CVE-2016-1000249 +L3 +26 +CWE-287 +CVE-2014-10067 +92 +CWE-200 +CVE-2016-10533 +L4 +113 +CWE-89 +CVE-2016-10554 +43 +CWE-79 +CVE-2014-9772 +L5 +1469 +CWE-471 +CVE-2017-1000048 +313 +CWE-502 +CVE-2017-5954 +Table VIII: Examples of undetected vulnerabilities by cause (Lx). +vulnerabilities grouped by CWE and their mapping to OWASP +Top 10 Web Security Risks (2021) [29]. Of the 957 known +vulnerabilities in the dataset, 324 vulnerabilities (33.9%) were +not detected by any of the selected tools. To understand the +underlying reasons, we have manually analyzed a sample +of undetected vulnerabilities. So far, we have identified the +following five main tool limitations. In Table VIII, we map +these limitations with some vulnerability categories (CWE) and +provide specific examples of undetected vulnerabilities (CVE). +L1. Cross-package vulnerabilities: The selected tools come +with pre-defined sets of manually written rules, typically +focusing solely on popular APIs. We noticed that some +undetected vulnerabilities exist in code that invokes functions of +third-party packages that map directly to known dangerous code, +e.g., wrappers to OS-level commands. These vulnerabilities +could have been found by testing all package dependencies +(can be thousands of other packages [3]), or by using a more +complete set of rules and queries (covering additional sources +and sinks). However, the manual maintenance of such lists +of sources and sinks is impractical as the Node.js ecosystem +expands. Existing work [71] tries to automatically extract taint +specifications (sources and sinks) from JavaScript libraries, +which partially solves the issue of incomplete rules, but requires +the constant dynamic testing of every new npm package. +For instance, Listing 8 shows a code snippet of a command +injection vulnerability for advisory 1440. This problem exists +because user-controlled data reaches an exec sink inside the +third-party package comandante, a package meant to ease the +execution of OS-level commands. The tools fail to recognize +this vulnerability because the usual command injection sinks +are not directly present in the analyzed code, but are instead +inside a third-party dependency that is not modelled by the +11 + +1 +// Snippet of ./gnuplot.js: +2 +var run = require('comandante'); +3 +4 +module.exports = function () { +5 +var plot = run('gnuplot', []); +6 +plot.print = function (data, options) { +7 +plot.write(data); +8 +// (...) +9 +}; +10 +// (...) +11 +} +Listing 8: Command Injection (advisory 1440) - NPM and Github +Advisories [75, 76]. +{ +"scripts": { +"preinstall": +"""wget http://s.qdcdn.com/17mon/17monipdb.zip && +unzip -p 17monipdb.zip 17monipdb.dat > 17monipdb.dat""" +} +} +Listing 9: Insecure Transport Layer in package.json of ipip-coffee +package (advisory 279) - CVE-2016-10673. +vulnerability detection rules of each tool, i.e., they failed to +include the write function as a potential dangerous sink. +L2. Limited analysis scope: In addition to JavaScript code +files, Node.js projects depend on several other components, +such as configuration files, front-end template code, testing +frameworks, etc. However, by analyzing only the JavaScript +code in isolation, certain vulnerabilities can be missed. As an +example, npm packages contain a package.json file which may +include bootstrap scripts. In several analyzed packages, these +scripts are used to download resources over HTTP. As it turns +out, using HTTP allows for man-in-the-middle attacks, where +resources are replaced by malicious payloads. While some +tools can detect insecure downloads if they are performed by +the main JavaScript code (e.g., by searching for HTTP URLs), +they cannot detect downloads issued from package.json. +Listing 9 shows a snippet of the package.json file for the ipip- +coffee package, in which an external resource is downloaded +over HTTP. This allows for man-in-the-middle attacks that +might compromise the server. In this particular example, this +vulnerability can only be detected if the package.json file is +also considered when performing the vulnerability analysis. +L3. Lack of contextual knowledge: Packages may expose +sensitive information, e.g., by logging plaintext passwords to a +file. These vulnerabilities are application-specific and require +contextual knowledge of which data is sensitive. The analyzed +tools, however, are not designed to gain contextual knowl- +edge and thus miss vulnerabilities that depend upon it, e.g., +application-specific leaks. To help detect such vulnerabilities, +a possible approach is to annotate application inputs, objects, +or data flows with sensitivity levels, and check which system +resources handle the annotated features during the execution. +Listing 10 shows an example of a Credential Exposure +vulnerability, in which plaintext passwords are logged to the +console. The code snippet itself seems benign until one becomes +aware that the key variable holds security-critical information. +1 +// Snippet of ./lib/odbc.js: +2 +if(exports.debug) { +3 +console.log("""%s odbc.js : pool[%s] : +4 +pool.close() - processing pools %s - connections: %s""", +5 +getElapsedTime(), self.index, key, connections.length); +6 +} +Listing 10: Credential Exposure (advisory 1185) - SNYK-JS-IBMDB- +459762 [77]. +1 +// Snippet of ./protect/lib/rules/xss.js +2 +const xssSimple = new +RegExp('((%3C)|<)((%2F)|/)*[a-z0-9%]+((%3E)|>)', 'i') +�→ +3 +const xssImgSrc = new RegExp('((%3C)|<)((%69)|i|(%49))((%6D) +4 +|m|(%4D))((%67)|g|(%47))[^\n]+((%3E)|>)', 'i') +5 +6 +function isXss(value) { +7 +return xssSimple.test(value) || xssImgSrc.test(value) +8 +} +9 +// Example attack payload: +10 +// +Listing 11: XSS (advisory 1116) - CVE-2018-1000160. +This contextual knowledge is needed to detect the vulnerability +but is difficult to extract using automated tools. +L4. Incorrect sanitization: Application developers often use +regular expressions to detect malicious inputs. However, regular +expressions are complex, and developers usually do not test +them thoroughly, allowing sanitization bypasses to occur. +Sanitization errors are often hard to detect statically, as they +require dynamically testing each regular expression ensuring +that they generate semantically valid inputs that can both bypass +the validation and effectively trigger the vulnerability. +For instance, Listing 11 shows a code fragment containing +two regular expressions that aim to prevent potential XSS +vulnerabilities. However, these regular expressions are not +entirely correct as there still exist some specially crafted inputs, +such as the one shown in the comment of Listing 11, that can +bypass this validation and launch an XSS attack. +L5. Inability to cope with JavaScript dynamicity: Specific +features of JavaScript can lead to vulnerabilities that are hard +to detect by static analysis tools. For example, object-based +inheritance, extensible objects, and dynamic typing are key +features of JavaScript, which can lead to prototype pollution, +authentication bypass, and business logic vulnerabilities. +Listing 12 shows a type of Prototype Pollution vulnerability +present in the qs package, which is a querystring parsing library +that allows developers to create objects within query strings. +For example, the string ’foo[bar]=baz’ is converted to the +object {foo:{bar:’baz’}}. Usually, this package protects +against attacks that try to overwrite the existing prototype +properties of an object. However, in this vulnerable version, +the protection can be circumvented by prefixing the name of +the parameter with character [ or ], as shown in the proof- +of-concept exploit code shown in Listing 12. Consequently, +calling toString() on the object will throw an exception. This +can subvert the application logic, potentially allowing attackers +to work around security controls, modify data, and make the +application unstable. The selected tools miss this example +because they fail to model how objects change depending on +the instructions applied to them, specially the object prototype. +12 + +1 +// Snippet of ./lib/parse.js: +2 +module.exports = function (str, opts) { +3 +var options = opts || {}; +4 +var tempObj = typeof str === 'string' ? parseValues(str, +options) : str; +�→ +5 +var obj = options.plainObjects ? Object.create(null) : {}; +6 +7 +var keys = Object.keys(tempObj); +8 +for (var i = 0; i < keys.length; ++i) { +9 +var key = keys[i]; +10 +var newObj = parseKeys(key, tempObj[key], options); +11 +obj = Utils.merge(obj, newObj, options); +12 +} +13 +return Utils.compact(obj); +14 +}; +15 +// Proof-of-Concept exploit code: +16 +qs.parse("]=toString", { allowPrototypes: false }) +// {toString = true} <== prototype overwritten +�→ +Listing 12: Prototype Override (advisory 1469) - CVE-2017-1000048. +From these limitations, we can extract actionable insights on +the applicability of static code analysis tools for vulnerability +detection in Node.js code. On one hand, these tools can +potentially overcome limitations L1 and L2 by both employing +improved strategies for maintaining taint specifications, and by +considering all the appropriate analysis scopes for Node.js +code. On the other hand, every static analysis tool will +struggle to overcome limitations L3, L4, and L5, because +they fail to capture behavioral and contextual information that +is only available at runtime when the package is executed +with appropriate, and application-specific, test inputs. To this +end, it seems that the approaches employed by current static +vulnerability detection tools can mainly be used successfully to +detect classic injection-style vulnerabilities even if all the tools +tested in this paper cannot do so with reasonable precision. +VII. THREATS TO VALIDITY +1. Even though our dataset is composed of real known- +vulnerable npm packages, there may be an implicit bias towards +vulnerabilities that are easier to analyze and more common +across different programming languages (i.e., not specific to +JavaScript code). Thus, since our curated dataset may not be +fully representative of all vulnerabilities in Node.js applications, +a tool that can detect all the vulnerabilities of our dataset may +still miss other unreported ones. +2. We may have missed some relevant tool, failed to evaluate +an analyzer that excels above all tested tools in our study, +or overlooked third-party detection rules that produce better +results. To reduce this risk, we will promote the reproducibility +of our evaluation by providing both the source code of VulcaN +and our curated dataset. +3. Both the labeling of vulnerable packages and identification of +their vulnerable code snippets were performed manually. Given +the challenges of manual code inspection, these annotations +could be mislabeled. To mitigate this risk, all vulnerabilities +were analysed by at least two authors at separate times and +we will make our dataset available for public scrutiny. +4. A potential concern is whether our study is susceptible +to survivor bias. For instance, assuming hypothetically that +all the packages that we analyze had already been analyzed +using CodeQL during the code development phase, and that +the vulnerabilities reported by CodeQL had been accordingly +fixed by the developer prior to package release on npm, then +the number of vulnerabilities effectively detected by CodeQL +could be higher than those reported in our study. This would +misleadingly suggest that the quality of CodeQL is worse than +what it is in reality. Note, however, that such a comprehensive +characterization of each tool is beyond the scope of this work. +In our study, we concentrate on evaluating tools’ ability to +detect, not all possible vulnerabilities, but only those that have +been officially reported in npm packages already in production. +VIII. RELATED WORK +The literature covers many tools for detecting vulnerabilities +in Web applications, including static [78, 79, 80], dynamic [81, +82], and hybrid analysis tools [83, 84, 85, 86], often combining +different types of program analysis techniques, such as fuzzing +(e.g. [81, 82]), control-flow and data-flow analysis (e.g. [57, 79, +83, 85]), and symbolic execution (e.g. [80, 84, 86]). The great +majority of these tools is, however, aimed at PHP-based Web +applications, with considerably fewer tools targeting JavaScript +applications. Most of the existing tools for JavaScript are aimed +at client-side JavaScript code and its specific vulnerabilities: for +instance, DOM-based XSS [87, 88], unrestricted inclusion of +third-party cross-origin scripts [89], and potentially malicious +flows via client-side persistent storage [8]. +Graph-based vulnerability scanners: State-of-the-art static +vulnerability analysis techniques often work by first computing +a static model describing the dynamic behaviour of the +application to be analyzed. Most notably, code property graphs +(CPGs) [57] were proposed as a compact representation +of an application’s behaviour. With CPGs, one can encode +specific vulnerability types as simple graph traversals, which +can, in turn, be expressed using graph query languages and +then executed on top of off-the-shelf graph databases (e.g. +Neo4J [90]). Code property graphs have successfully been +applied to find SQL injection, XSS, and CSRF vulnerabilities +in PHP applications [79, 85]. Furthermore, they are at the +core of CodeQL [14]. For JavaScript, code property graphs +were employed by JAW [56] and ODGen [15], for client-side +and server-side JavaScript respectively. In our work, we have +extensively evaluated CodeQL and ODGen as representative +state-of-the-art, graph-based vulnerability scanners. +Vulnerability studies & analyzers for Node.js applications: +Unlike client-side JavaScript applications, which run in the +browser, Node.js application code is not sandboxed. Recent +empirical studies [3, 91] have shown that, contrary to popular +belief, npm applications are often poorly maintained and +tested, with a significant percentage (up to 40%) of all +packages depending on code with at least one publicly known +vulnerability. Furthermore, after reviewing more than 200K +npm applications, Staicu et al. [4] concludes that 20% of the +analyzed applications either directly or indirectly make use +of an injection API. Despite this security-critical situation, +there is only a small number of research tools for detecting +vulnerabilities in Node.js applications and their underlying +13 + +infrastructure, most of which based on dynamic code analysis. +For instance, Synode [4] aims to prevent injection attacks +in Node.js applications, and NodeSec [9] aims to detect +vulnerabilities in Node.js applications. The authors of [5] +and [92] design specific dynamic analysis for finding regular +expression denial of service (ReDoS) vulnerabilities. The +authors of [93] also apply dynamic analysis and symbolic +execution to detect attacks that leverage hidden properties in +client- and server-side JavaScript. There are also academic +works that employ static analysis techniques for detecting +vulnerabilities in Node.js, but most focus on detecting prototype +pollution vulnerabilities [94, 95]. ODGen [15] is the only purely +static code analysis tool developed by the academia that aims +to detect several types of vulnerabilities in Node.js. +Empirical studies of vulnerability analyzers: Several em- +pirical studies aim at characterizing the efficacy of existing +white-box vulnerability detection tools (e.g. [88, 96, 97]). +Durieux et al. [96] evaluated 9 automated analysis tools for +Ethereum Smart Contracts. The authors created a curated +dataset consisting of 69 annotated vulnerable smart contracts, as +well as a raw dataset consisting of 47,518 smart contracts. They +report that only 42% of the vulnerabilities on the annotated +dataset were detected, with the highest ranking tool having an +accuracy of 21%. Melicher et al. [88] evaluated 3 automated +static analysis tools for detecting DOM-based XSS in client- +side JavaScript code (Esflow [98], ScanJS [54], and Burp +Suite Pro [99]). They created a dataset with 3219 confirmed +vulnerabilities. However, many security flaws in server-side +code for Node.js do not exist on the client-side (e.g., SQL +injections), and vice-versa. As such, the dataset from [88] is +not representative enough of server-side vulnerabilities. Finally, +Nunes et al. [97] evaluate five free static analysis tools for +detecting SQL injection and XSS vulnerabilities in PHP web +applications using a dataset comprising 134 WordPress plugins. +In contrast to the studies referenced above, our paper presents +the first empirical study targeting fully automated vulnerability +detection tools for npm packages. Our study comes with a +comprehensive manually-annotated dataset based on confirmed +real-world vulnerabilities. +IX. CONCLUSIONS +This paper presented an empirical study of static analysis +tools for detecting vulnerabilities in Node.js packages. To +conduct this study, we built VulcaN, an automated analysis +framework, using which we created the largest known curated +dataset of Node.js packages with well-characterized security +vulnerabilities. Currently, our curated dataset includes 745 +reviews that accurately identify the exact location of known +vulnerabilities inside affected npm packages. We found that +the nine evaluated tools fail to detect many vulnerabilities and +exhibit high false positive rates. Additionally, we show that +many important vulnerabilities appearing in the OWASP Top- +10 are not detected by any evaluated tool or even when using +the combination of all tools. +We believe that our curated dataset will substantially con- +tribute to enabling future research on automatic vulnerability +detection tools for server-side JavaScript applications. 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Neves, M. Correia, +and M. Vieira, “An empirical study on combining diverse +static analysis tools for web security vulnerabilities based +on development scenarios,” Computing, vol. 101, 2019. +[98] “Esflow,” https://www.npmjs.com/package/esflow. +[99] P. W. Security, https://portswigger.net/burp. +16 + diff --git a/CNE4T4oBgHgl3EQfeQ0p/content/tmp_files/load_file.txt b/CNE4T4oBgHgl3EQfeQ0p/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6189d9793d72ab7b39b70d2b5968e22a5a8fc86b --- /dev/null +++ b/CNE4T4oBgHgl3EQfeQ0p/content/tmp_files/load_file.txt @@ -0,0 +1,1608 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf,len=1607 +page_content='Study of JavaScript Static Analysis Tools for Vulnerability Detection in Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js Packages Tiago Brito∗, Mafalda Ferreira, Miguel Monteiro, Pedro Lopes, Miguel Barros, José Fragoso Santos, Nuno Santos {∗tiago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='de.' metadata={'source': 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miguel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='barros, jose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='fragoso, nuno.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='santos}@tecnico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='ulisboa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='pt INESC-ID / IST, Universidade de Lisboa, Portugal Abstract—With the emergence of the Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js ecosystem, JavaScript has become a widely-used programming language for implementing server-side web applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' In this paper, we present the first empirical study of static code analysis tools for detecting vulnerabilities in Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' To conduct a comprehen- sive tool evaluation, we created the largest known curated dataset of Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js code vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' We characterized and annotated a set of 957 vulnerabilities by analyzing information contained in npm advisory reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' We tested nine different tools and found that many important vulnerabilities appearing in the OWASP Top-10 are not detected by any tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' The three best performing tools combined only detect up to 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='6% of all vulnerabilities in the dataset, but at a very low precision of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='11%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Our curated dataset offers a new benchmark to help characterize existing Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js code vulnerabilities and foster the development of better vulnerability detection tools for Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' INTRODUCTION JavaScript has become one of the most popular programming languages for implementing server-side web applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' A driving factor in this trend has been the emergence of Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js is a cross-platform, back-end runtime environment that executes JavaScript code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Essentially, it can be used as a web container, housing JavaScript code that handles HTTP requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Pivoted around Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js, there is also an ecosystem of third-party packages managed by the Node Package Manager (npm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Currently, npm stores thousands of packages that web developers can readily import into their code, either for writing web applications or other packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' The widespread adoption of Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js makes the development of effective JavaScript vulnerability scanners a pressing matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' For one, the JavaScript [2] language features various constructs that display subtle behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' When employed by inexperienced code developers, these constructs may all too easily lead to the introduction of vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' In addition, the manual detection of code vulnerabilities is complicated by the intricate npm inter-package dependency system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' In some cases, correct packages may become the source of security bugs as a result of ill-use by other packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' In others, buggy packages may end up propagating vulnerabilities up in the dependency chain to correct packages [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' This combination of factors opens up the path for serious security breaches in web applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' By exploiting security bugs, an attacker may be able to take over the entire server and/or affect many users through SQL injection, remote code execution, and other attacks [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' An effective technique to prevent security vulnerabilities from creeping into production code is to integrate security analysis tools as part of Continuous Integration/Continuous Deployment (CI/CD) pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Using automatic vulnerability detection tools, developers can seamlessly receive prompt feedback about potentially existing security flaws in their code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' This enables them to apply the necessary fixes at an early code development stage, thus helping them to improve the reliability of their software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' In the same vein, JavaScript developers can benefit from code analysis tools that allow them to detect and fix security flaws inside npm packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Ideally, such tools should have high detection quality (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=', low false-positive rate), and high coverage (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=', low false-negative rate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Motivated by this need, we set out to evaluate the effec- tiveness of existing JavaScript vulnerability detection tools at analyzing Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' We found a large body of work on client-side JavaScript security [6, 7, 8], and some recent work in the study of vulnerabilities in npm packages [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' However, no prior work has focused on evaluating tools that analyze server-side JavaScript code vulnerabilities, let alone on studying their effectiveness at finding security flaws in npm packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' As it turns out, performing this task is rather involved, given the absence of a gold standard for classifying such tools, and the lack of a comprehensive vulnerability dataset that can be used for benchmarking purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' In this paper, we present the first empirical study aimed at evaluating existing JavaScript vulnerability detection tools on Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' We focus exclusively on fully automatic, static code analysis tools that can be used in CI/CD pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' This excludes tools [4, 9, 10] that expect additional inputs, such as test suites, or tools that perform simple checks on known vulnerable dependencies [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' In total, we screened 40 analysis tools for JavaScript and selected nine that can detect vulnerabilities at continuous integration time: NodeJsScan [13], CodeQL [14], ODGen [15], Graudit [16], InsiderSec [17], ESLint SSC [18], Microsoft’s DevSkim [19], Mosca [20] and Drek [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' We executed them against a curated dataset created by us containing npm packages with annotated vulnerabilities, mainly: path traversal, cross-site scripting, insecure transfer using HTTP, resource exhaustion/denial-of-service, prototype pollution, OS command injection, code injection, and improper input validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Then, we checked whether these tools can correctly identify these vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Given that there is no curated dataset of Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js vulnerabili- ties, our first step was to develop our own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Building this dataset was in itself a challenging endeavor because we needed to 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='05097v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='CR] 12 Jan 2023 identify real vulnerabilities in a large dataset of npm packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Our starting point was the npm system itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' The npm system runs a vulnerability report service that results in the generation of the so-called advisory reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' These consist of textual descriptions of security vulnerabilities identified inside specific packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' These are real security vulnerabilities collectively identified by the Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js developer community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Reports may also include an advice to upgrade the package to a fixed version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' As such, advisory reports provide a reliable source for building our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Unfortunately, these reports are not represented in a format that allows for automatic processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Moreover, some of them may contain errors and therefore cannot be used unless a thorough analysis and verification are performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' To overcome these difficulties, we manually analyzed 1359 advisory reports covering an equal number of vulnerable npm package versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' These advisories represent 74% of all the vulnerabilities officially reported inside benign npm package versions until June 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' In this process, we identified several anomalies in the advisory reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' We have then generated a curated dataset covering 957 of these advisories extended with annotations that specify the precise location of the reported code vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' We found that the location of a large fraction of existing vulnerabilities can be fully expressed through source-sink pair annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Our dataset can help the research community to i) characterize the vulnerabilities already detected within the npm ecosystem, and ii) benchmark vulnerability detection tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Our dataset is publicly available1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' We tested the pre-selected tools against our dataset and found they perform rather poorly, missing many vulnerabilities (low true positive rate/recall) and showing a high false positive rate (low precision).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' On average, they were able to correctly identify only 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='1% of the total number of vulnerabilities in our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' The combination of the three best-performing tools detects 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='6% of all vulnerabilities, albeit with only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='11% precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' The best performing tools, ESLint SSC and CodeQL, manage to detect 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='5% and 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='3% across all types of vulnerabilities and reach their peaks when it comes to identifying prototype pollution (79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='2% for CWE-471 and 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='1% for CWE-1321) and path traversal (71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='2%) vulnerabilities, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Of the 957 known vulnerabilities in the dataset, 324 (33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='8%) were not detected by any of the selected tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Some of the causes are tied to fundamental limitations of state-of-the-art code analysis techniques when it comes to analyzing server-side, JavaScript code vulnerabilities in the npm ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Addressing these limitations is an interesting research direction for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' In summary, our paper makes four contributions: (i) a curated dataset with 957 real-world vulnerabilities in npm package versions, which will be fundamental to evaluate future advancements in static analysis tools for detection of vulnerabilities in Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js applications, (ii) a survey of existing vulnerability detection tools for JavaScript / Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js code, (iii) a quantitative assessment of the vulnerability detection toolset against our curated dataset, and (iv) a study of the main causes 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='com/VulcaN-Study/Supplementary-Material 2016 2017 2018 2019 2020 2021 2022 0 1000 2000 Severity Low Moderate High Critical Figure 1: Evolution of published advisories over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' of missing important vulnerabilities in npm packages, which opens up several research avenues in this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' STUDY DESIGN A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Background Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js features a package manager system named Node Package Manager (npm), which currently stores thousands of third-party packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Presently, it is difficult to guarantee the absence of security vulnerabilities in packages uploaded to npm because there is no systematic code triage in place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Consequently, the npm community mainly relies on third- party vulnerability reports to identify the potentially vulnerable packages that have already been included in the ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' The npm system alerts JavaScript developers whenever they use a package version reported as vulnerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' These vulnerability alerts are called npm advisories, as their purpose is to advise developers to update the vulnerable dependencies to either a fixed package version or to select another package entirely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' When someone identifies a potential vulnerability in an npm package, they produce a vulnerability report and submit it to the npm security team, which checks the report, notifies package maintainers, and publishes the advisory, either when the package maintainers release a fix or if they remain unresponsive for longer than 45 days [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Typically, an advisory report includes: package name, affected versions, description of the vulnerability, effects and references, commits, and/or code examples that help trigger the vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' This information is then made available to developers in the advisory page (see example page for advisory 315 in [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Figure 1 displays the evolution of the number of published advisories since npm’s inception until October 11th 2022, broken down according to their severity level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' This number has steadily grown at nearly 40 advisories per month;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' about 70% cover vulnerabilities considered to pose risks of high/critical severity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Research Questions and Scope In this work, we investigate four main research questions: RQ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' How to obtain an annotated dataset of vulnerabilities in server-side JavaScript code?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' To evaluate JavaScript vul- nerability detection tools, we require an annotated vulnerability dataset to compare the output of a given tool against ground truth data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' The npm repository provides an excellent source for retrieving both (i) an extensive collection of vulnerable Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js applications (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=', vulnerable npm package versions), and (ii) information about real-world vulnerabilities (documented by the advisory database).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Unfortunately, this information cannot 2 be used as-is from existing advisories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' First, advisories often lack relevant information about the reported vulnerability (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=', the exact code location of the vulnerability within the package).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Second, in many cases, most of the explanations regarding the reported vulnerability are given in external references, where information tends to be inconsistent and unstructured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Third, some advisories may be incorrect in places (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=', the classification of the vulnerability type), which may lead to the mischaracterization of existing JavaScript vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' These obstacles preclude an automated advisory analysis approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' RQ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' What is the state-of-the-art of existing security-orien- ted static analysis tools for Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js code?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' We are interested in assessing which static vulnerability detection tools are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' We need to distinguish between a broader set of code analysis tools which can serve many different purposes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=', detecting programming malpractices) from those that are specifically oriented toward the detection of vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Furthermore, we aim to analyze which code analysis techniques are employed by these tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' This will help us to better assess the strengths and weaknesses of each technique when evaluating each tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' RQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' How effective are available detection tools in uncov- ering vulnerabilities in JavaScript code?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' We are interested in empirically determining and characterizing how precise the publicly available vulnerability detection tools are at identifying vulnerabilities in known vulnerable JavaScript code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' RQ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' What are the main reasons for missing the detection of vulnerabilities?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' We aim to understand the key limitations of existing static vulnerability detection tools that explain their failure to detect known vulnerabilities in JavaScript code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' The research questions above have a twofold goal: (i) characterize vulnerabilities in npm packages in the wild, and (ii) evaluate the effectiveness of existing static JavaScript vul- nerability detection tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' To better set the reader’s expectations about our study, we further clarify these subgoals and discuss other relevant directions we left outside the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Firstly, our study is narrowed toward the characterization of vulnerabilities reported inside npm packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' As such, our results cannot be extrapolated to characterize the typical programming flaws introduced by JavaScript developers during the code development stage;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' nor is this our goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Developers may use vulnerability detection tools in their continuous integration frameworks that may capture some vulnerabilities before the code is deployed to npm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' This means that some security flaws may have been fixed prior to deploying the code into production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Also note that, given that we only analyze formerly identified vulnerabilities, our curated dataset may not be representative of all existing JavaScript vulnerabilities lingering inside npm packages;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' this is also not our purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' In contrast, we intend that our curated dataset contains a large set of confirmed real-world vulnerabilities that can be used for assessing existing (and future) vulnerability detection tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Secondly, we focus exclusively on fully automatic vulnerabil- ity analysis tools that can be easily integrated into existing code Internet npm website & registry Advisory Collection Engine Dockerfiles Tool Configs 1) Tool Execution Engine 2) 3) Analyst 4) 5) 6) Tool Outputs Package Code Advisory Metadata VulcaN Reviews API Analysis Engine VulcaN Figure 2: VulcaN architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' review pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' This excludes the only three existing dynamic analysis tools for detecting vulnerabilities in Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js code: SyNode [4], NodeSec[9] and Affogato [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' These tools require a set of unit tests covering all security sensitive behaviors of the package to be analysed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Such test suites must be manually written as the automatic generation of high-coverage test suits for Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js applications is still an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Nevertheless, our curated dataset can still be used as a benchmark for evaluating the effectiveness of these excluded tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Study Methodology Our approach to answering the questions above is to perform an empirical study consisting of the following three tasks: Task 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Manual analysis of advisory reports and building the annotated dataset: We manually analyzed all advisories, filling in the missing information, namely by identifying the code location that triggers the vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' This information is part of our curated dataset used in the following tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Task 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Selection and execution of code analysis tools for vulnerability detection in JavaScript code: We searched both the literature and the web for code analysis tools that can be used for vulnerability detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' We executed all of the selected tools on all the vulnerable packages in our curated dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Task 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Evaluation of tool output according to the ground truth extracted from the annotated dataset: We automati- cally compare a tool’s result with the corresponding dataset annotations, considering the verbosity levels of each tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' To support our methodology, we developed VulcaN, a testbed for analyzing vulnerability detection tools for Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' VulcaN is an execution and analysis framework to collect vulnerable versions of npm packages, run vulnerability detection tools over all collected packages, and help security analysts perform the vulnerability analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Currently, VulcaN supports nine tools (see Section IV) and has these main features: an interface to download the latest published advisories;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' a Docker-based extension for adding more analysis tools;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' an interface for accessing both the metadata and the output of the analysis tools for each advisory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' 3 an interface for annotating each advisory with a review file, which contains the code location of the vulnerability and the classification of the results of the analysis tools;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' an interface for automatically comparing the output of the tools with the corresponding review in our curated dataset, this includes a parser for the ouput of each tool;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' a web API exposing the curated dataset and the classifi- cation of the analysis tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Figure 2 shows the architecture of VulcaN and how each component is used in the context of our empirical study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' First, the Advisory Collection Engine crawls the npm advisory website [24] and collects the available metadata about all published advisories saving it in a MongoDB database (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' The latest vulnerable version for each advisory, referenced in the collected metadata, is then downloaded via the npm registry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Second, the Tool Execution Engine builds a docker image for each tool, according to the specified Dockerfile, and runs the respective container for all downloaded packages, storing the output of each tool locally (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Third, the advisory metadata, code, and tool outputs are fed to the Analysis Engine, which generates an environment for advisory report analysis (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' The analyst accesses the environment (4) and produces a review file comprising an accurate description of the vulnerable code location, which is then submitted to the framework (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Finally, the Analysis Engine processes the submitted information, automatically compares the tools’ outputs with the analyst reviews (dataset) using a dedicated parser for each tool, and exposes it through a web API (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' DATASET OF VULNERABILITIES (RQ1) In this section, we address RQ1, explaining how we created our curated dataset of JavaScript code vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Selection and Validation of Reports To create our dataset, we collected a snapshot of the existing npm advisories until the end of June 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Then, through manual analysis, we excluded some advisories and fixed inconsistencies in the remaining ones (see Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Out of the 1828 advisories from the original snapshot, we excluded 469, keeping 1359 for further analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Next, we present our exclusion criteria and discuss the detected inconsistencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Excluded advisories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' As of June 30th 2021, there were 1828 advisories published in npm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Of these 1828 advisories, 416 are categorized by npm as Embedded Malicious Code (CWE-506): these are packages designed with malicious intent, named very similarly to real legitimate packages so as to deceive developers into installing them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' These packages are not relevant for our study, which focuses only on unintentional vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' From the remaining 1412 advisories, we excluded 31 for lacking available code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Lastly, out of the resulting 1381 vulnerable package versions, 22 were excluded for not including JavaScript code;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' instead, they had pre-transpiled variants such as CoffeeScript [25] and TypeScript [26], which prevented us from analyzing their source code directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Consequently, in the end, VulcaN successfully collected 1359 advisories and their corresponding package versions for further manual analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Exclusions & Inconsistencies # of Advisories Malware Packages 416 Missing Package Code 31 Missing JavaScript Code 22 Incorrect Vulnerable Version 42 Missing External References 291 Imprecise CWE 101 Lack of Analysis Information 402 Table I: Number of advisories excluded or inconsistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Detected inconsistencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' During the manual analysis, we noticed several inconsistencies in the collected advisories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Most notably, only a small minority of advisories comes with the ex- act code locations that trigger their corresponding vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Furthermore, some advisories provide an incorrect vulnerable package version, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=', the advisory metadata points to a package version that does not contain the described vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' When the advisory does not come with additional external references, which is the case for 21% of the analyzed advisories, correcting the incorrect vulnerable package version anomaly can be quite challenging, as the advisory metadata alone is generally insufficient for pinpointing the correct package version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Another detected anomaly is the imprecise classification of vulnerability type/category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Most of the times this imprecision is subjective, as a Common Weakness Enumeration (CWE) [27] class can be a subcategory (child) of another more general CWE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' This is particularly common for Path Traversals (CWE-22) and Code Injections (CWE-94), to which more precise classes can be attributed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' in particular, CWE-23 (Relative Path Traversal) and CWE-24 (another specific Path Traversal variant) to CWE-22 and CWE-95 (Eval Injection) to CWE-94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Some vulnerabilities are simply miscategorised;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' for instance, sometimes Code Injection (CWE-94) vulnerabilities are categorized as Cross- Site Scripting (CWE-79).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' During the analysis, we detected 63 cases of vulnerability miscategorization (different CWE) and 21 cases of incorrect vulnerable package version referenced in the advisory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' The remaining cases lack the CWE categorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Analysis of Reported Vulnerabilities We analyzed the vulnerabilities in the selected 1359 npm advisories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Our goal was to characterize the vulnerability landscape of the npm package ecosystem by studying the dis- tribution of existing vulnerabilities according to their category and assess the potential security risks posed by the affected packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' From the 1359 advisories manually analyzed, we managed to verify the vulnerability for 957 advisories: these are the ones included in our dataset and characterized in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' The remaining advisories (402) did not include sufficient information to successfully verify the vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Figure 3 displays the cumulative distribution function (CDF) of the number of vulnerabilities of our dataset ranked by their CWE category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' This distribution is heavily skewed toward a relatively small number of CWE categories, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=', a large fraction of vulnerabilities pertains to a restricted set of categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' In particular, the top-10 CWEs cover 665 advisories, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=', 69% of the total number of verified vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' 4 0 25 50 75 100 Number of CWE categories (ordered by occurrence) 200 400 600 800 Number of advisories 665 (69%) 957 (100%) Top 10 CWE All Reviewed Advisories Figure 3: CDF of # of reviewed advisories ranked by CWE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' To estimate the potential security risks of such vulnerabilities, we mapped each of the top-10 CWE categories to the latest OWASP ranking (from 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' OWASP is an organization that works to raise awareness about web security and ultimately improve it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' The OWASP Top 10 [28] list is a popular document representing a broad consensus about the most critical security risks to web applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Updated every few years, this document describes risks, such as injection attacks, broken authentication, and known vulnerable dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' As shown in Table II, most vulnerability types can be mapped to a top-10 web security risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' This means that npm packages have well- known risks that security professionals are familiar with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Most notably, 9 out of 10 CWE categories (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=', 576 advisories) ap- pear in the top-3 OWASP list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' This translates to approximately 60% of the total number of analyzed vulnerabilities in our dataset (957).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' In other words, many reported vulnerabilities can introduce serious security flaws in web applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Our Curated Dataset Based on the selected advisories, we created a curated dataset aimed at providing a baseline for assessing the effectiveness of vulnerability detection tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' It comprises: (i) the code for the vulnerable version of the npm package indicated in the advisory, and (ii) a corresponding review file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' This file contains ground truth information that allows us to validate the output of a given tool when analyzing that specific package version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Review example: Listing 1 shows the created review file for advisory 315 [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' This advisory reports the presence of a code injection vulnerability in package summit-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' A review is a JSON object that contains several fields that describe: i) the advisory identifier (id), ii) the vulnerability type as per the CWE taxonomy (cwe), iii) the affected package version (package_link), and iv) the vulnerability location expressed as a source/sink pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' A source/sink is specified as a JSON object with fields denoting: the file name (file);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' the line number (lineno);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' and the corresponding line of code (code).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Vulnerability location: The location fields are used to deter- mine if the output of a given vulnerability detection tool is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Besides using (one or multiple) source/sink pairs to locate a vulnerability, we can also employ (one or multiple) block patterns, which indicate contiguous code regions in which the flaw exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' This form of representation is necessary for vulnerability types that cannot be expressed as source/sink # CWE Security Risk # Occurrences 1 CWE-22 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Broken Access Control 146 2 CWE-79 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Injection 99 3 CWE-400 89 4 CWE-78 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Injection 75 5 CWE-818 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Cryptographic Failures 75 6 CWE-471 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Injection 48 7 CWE-20 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Injection 41 8 CWE-1321 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Injection 36 9 CWE-94 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Injection 33 10 CWE-77 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Injection 23 Table II: Possible mapping of most occurring vulnerabilities in dataset with OWASP Top 10 Web Security Risks (2021) [29]: Path Traversal (CWE-22),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Cross-site Scripting (CWE-79),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Resource Exhaustion (CWE-400),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Insufficient Transport Layer Protection (CWE-818),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' OS Command Injection (CWE-78),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Modification of Assumed-Immutable Data (CWE-471),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Improper Input Validation (CWE-20),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Improperly Controlled Modification of Object Prototype Attributes (CWE-1321),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Code Injection (CWE-94),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' and Improper Neutralization of Special Elements used in a Command (CWE-77).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' { "advisory": { "id": 315, "cwe": "CWE-94" }, "package_link": "registry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='npmjs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='org/summit/-/summit-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='tgz", "vulnerability": [ { "source": { "file": "lib/drivers/search/pouch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js", "lineno": 4, "code": "return function search (opts) {" }, "sink": { "file": "lib/drivers/search/pouch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js", "lineno": 20, "code": "eval(opts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='filter);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='" } } ] } Listing 1: Example of VulcaN review file for advisory 315.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' pairs, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=', usage of HTTP instead of HTTPS which allows for MITM attacks (CWE-818).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Interestingly, we found that, in most cases (78%), the exact location of a vulnerability is very clear, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=', a call to eval in a code injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' These cases can be represented by source/sink pairs, while the remaining cases (22%) can be represented using code blocks that cover all vulnerability-relevant code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Although the block size depends on the vulnerability, in our dataset the average block size is six lines-of-code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Moreover, since the review files can be processed automatically, we believe that our curated dataset will be useful for benchmarking purposes beyond the scope of our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Methodology: Vulnerability locations were identified manually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' To account for their possible mislabeling, each advisory was analyzed by two authors at separate times and their results cross- checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Two authors performing cross-validation disagreed in 84 reviews (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='8% of 957 reviews).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Most inconsistencies were differences in source/sink pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' These cases were resolved by selecting the correct source/sink pair or specifying a superset of source/sink pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' In rare cases, one author failed to locate the vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' These cases were handled by jointly reviewing the location identified by the other author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='Included Tools ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='Only Package ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='Source Code (C0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='Not Available ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='or Proprietary (C1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='Not Scriptable ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='Interface (C2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='Not Security ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='Oriented (C3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='Other Exclusionary ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='Reasons ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='NodeJsScan [13] 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='CodeQL [14] 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='ODGen [15] 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='Graudit [16] 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='InsiderSec [17] 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='ESLint SSC [18] 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='MS DevSkim [19] 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='Mosca [20] 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='Drek [21] 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='SyNode [4] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='NodeSec [9] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='Affogato [10] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='Beyond Security [30] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='Checkmarx [31] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='Fortify [32] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='Veracode [33] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='Kiuwan [34] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='CodeSonar [35] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='Thunderscan [36] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='WhiteHat [37] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='JsPrime [38] 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='Codeburner [39] 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='SonarQube [40] 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='CodeWarrior [41] 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='WALA [42] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='PMD [43] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='Aether [44] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='Coala [45] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='JsHint [46] 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='EsComplex [47] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='Coverty Scan [48] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='DeepScan [49] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='AppInspector [50] 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='TAJS [51] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='SAFE [52] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='ESLint SP [53] 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='Mozilla ScanJs [54] 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='SemGrep [55] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='JAW [56] 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='Joern [57,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' 58] 3 Table III: Tools included in / excluded from this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Excluded for reasons beyond C0, C1, C2, & C3: ESLint Security Plugin and Mozilla ScanJs use ESLint rules subsumed by ESLint SSC’s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' SemGrep requires specially-crafted rules for security purposes and is the backbone of NodeJsScan;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' JAW only implements rules for detecting client-side CSRF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Joern supports JavaScript inspection, but does not support default rules for detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Some tools excluded due to C1 were tested using free trials, but failed to comply with additional criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Dataset size: From the 1359 analyzed advisories, we were able to manually verify 957 review files (70%) at the time of this paper submission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' For each review file we confirmed the exact location of the reported vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' For this reason, we are confident to include these reviews in our curated dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' VULNERABILITY DETECTION TOOLS (RQ2) We now focus on RQ2, explaining how we selected the tools considered in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' We specify our eligibility criteria, survey the existing tools that satisfy them, and classify these tools according to the detection technique that they employ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Tool Selection Criteria and Selection Process We focus specifically on fully automatic tools for analysis of npm packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' In particular, to select a given tool, it must: C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Depend only on the package source code: The tool requires only the source code of the package to analyze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' This excludes tools that require a test suite to guide the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Be available and transparent: The tool is publicly available and implements a technique that is non-proprietary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Its source code does not need to be open as long as the tool’s code analysis techniques can be clearly characterized, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=', through available documentation, rule set, and usage examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Have a scriptable interface: The tool must support a command-line interface (CLI), or similar interaction, allowing it to be executed and its output analyzed via a script.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' This facilitates the scalability and automation of the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Be security-oriented: The tool must identify vulnerabil- ities or security bad practices in JavaScript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' This excludes tools that only construct artifacts, such as control-flow graphs, produce warnings about coding styles and conventions, or produce statistical information about the code, such as code metrics, that might be irrelevant from a security standpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Based on the above criteria, we ended up selecting nine tools for our testing purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' We started by examining the academic literature [4, 9, 10, 15, 56, 57] and searching the Internet, including OWASP lists [59, 60], repository collections [61, 62, 63] and other websites [64, 65, 66], for suitable tools for vulnerability detection in server-side JavaScript code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Most tools we screened were developed by the industry and the open- source community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' In total, we first collected 40 JavaScript analysis tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' This full list is presented in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' After inspecting all 40 analysis tools, we found that we needed to manually test 19 tools, which are annotated with the symbol 3 in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' These 19 tools were tested against the Damn Vulnerable Node Application (DVNA) [67], a web application written in JavaScript that was purposely built with a range of vulnerabilities matching the OWASP Top 10 Web Security Risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' After executing each of the 19 tools against the vulnerable application, we excluded those that do not allow the analysis to be automated via a script and also those that fail to show security-oriented results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Out of the remaining 19 tools, we selected 14 tools that are available, transparent, can be automated, and show security- oriented results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Out of these 14 tools, we also excluded ESLint Security Plugin, Mozilla ScanJs, SemGrep, JAW, and Joern as explained in the caption of Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Consequently, we ended up with 9 distinct candidates that represent proper fully automatic vulnerability detection tools for Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Detection Techniques By manually analyzing the source code of the nine selected tools, we characterized them according to the employed tech- nique for finding vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' It is important to understand how these techniques work as they can have a significant impact on the effectiveness of the tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' We categorize these techniques using three classes: graph-based analysis, syntax- based analysis and keyword-based analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Table IV maps every selected tool to its corresponding detection technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Graph-based analysis tools work by first constructing a graph- based model of the program to be analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Such models usually coalesce into a single graph-like data structure various types of statically computed program artefacts, including: abstract syntax trees, control-flow graphs, and dependency graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' The obtained data structure can be then inspected using queries written in domain-specific languages (DSL) especially designed for specifying vulnerable code patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' For instance, typical queries aim at identifying code-flow paths through which user-controllable inputs can reach dangerous 6 Technique Tool Version Graph-based Analysis CodeQL 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='6 ODGen – Syntax-based Analysis NodeJsScan 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='8 ESLint SSC ** Keyword-based Analysis Graudit 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='8 InsiderSec 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='5 MS DevSkim 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='109 Mosca 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='8 Drek 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='3 Table IV: Vulnerability detection technique employed by each tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' **ESLint SSC was used with eslint@7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' sinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' CodeQL is one of such tools that models source code as database records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' These records can be queried using SQL-like statements that are specified in the form of rules/queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Syntax-based analysis tools employ a technique that searches the code to be analyzed for insecure syntax-aware patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Patterns can express simple control-flow conditions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=', calling a function with a particular variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' In contrast to graph-based analysis, this technique operates directly on source code and does not typically cater for more intricate dependency analysis or for matching patterns across multiple files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' NodeJsScan is an example of such tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' It is used in GitLab’s CI/CD [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Keyword-based analysis tools employ a code analysis tech- nique that searches the code to be analyzed for strings associated with potentially insecure code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' This search is typically performed through the use of regular expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Note that keyword-based analysis does not model the AST of the program to analyze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Consequently, it is considerably less expressive than graph- and syntax-based analysis, as it cannot reason about fine-grained control-flow interactions, often operating on a single line of code at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' How Different Detection Techniques Work To better understand how the aforementioned vulnerability detection techniques work, we examine, as an example, the advisory 315 of our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' According to the information available in the advisory page [23], this example consists of a code injection vulnerability (CWE-94), which allows a malicious user to run arbitrary code on the targeted execution platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' In this type of attack the adversary is only limited by the expressiveness of the injected language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' JavaScript code injections at the server can have more significant impact than those at the client-side, given that Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js has fewer security barriers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=', no sandbox), and a larger and privileged API facilitating access to critical system resources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=', file system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' The vulnerable code snippet of the advisory 315 is given in Listing 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' In this example, the variable opts is bound to a user-controlled object with the properties filter and collection, which can be trivially tainted with a maliciously crafted input to produce valid JavaScript code that reaches the eval function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' One can leverage this vulnerability to execute arbitrary JavaScript code, including OS-level commands by using a payload like the one given in Listing 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Using graph-based analysis: Both CodeQL and ODGen adopt graph-based analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Listing 4 shows an example of a CodeQL 1 function search (opts) { 2 if (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='opts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='filter && opts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='collection) { 3 if (typeof opts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content="collection === 'string') { 4 opts." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='filter = "function filter (doc) { return doc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='type === \'" + opts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='collection + "\'}";' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' �→ �→ 5 } else { .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' } 6 eval(opts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='filter);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' 7 opts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='filter = filter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' 8 } 9 } Listing 2: Code injection vulnerability (npm advisory 315).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' 1 opts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content="collection = `'};" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' 2 const exec = require("child_process").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='exec;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' 3 exec("cat /etc/passwd", (err, stdout, stderr) => { 4 console.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='log(stdout);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' });' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=" var a={ hello: 'world`;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' 5 search(opts) Listing 3: Exploit for code injection (npm advisory 315).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' rule designed to detect calls to the eval function using user- controlled inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' In order to create this CodeQL rule, one starts by specifying the appropriate configuration, that is, a code description of the targeted sources and sinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' In this case, we are interested in code flows from remote flow sources, described by the predicate isSource (lines 3 to 5), to the eval sink, described by the predicate isSink (lines 6 to 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Then the main query (lines 11 to 13) states that, using the specified configuration (EvalTaint cfg), CodeQL should find code paths from the specified source to the specified sink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' The output of this query is a string with a description of the source-sink pairs that match the query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' This particular rule is a simplified version of one of the CodeQL rules [69] executed inside VulcaN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' In general, graph-based analysis works well for taint-tracking, but it requires every source and sink to be explicitly encoded into rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' These sources and sinks change over time as languages evolve and new popular third-party packages are created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' This is why the community has started to work on automatically generating such taint-tracking specifications [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Using syntax-based analysis: NodeJsScan helps us showcase syntax-based analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Listing 5 lists an excerpt of the Node- JsScan rule that detects potentially vulnerable uses of eval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' To be applied, this rule must match two related patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' First, eval must occur inside a function receiving two or more arguments (line 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Then, eval must be called with a parameter computed using one of the given arguments (line 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' NodeJsScan includes analogous rules for other vulnerability types [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Similarly to the previous technique (graph-based), syntax- based analysis also suffers from the source-sink specification limitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Additionally, there are some other limitations specific to syntax-based analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' For example, it may lead to a high number of false positives, as it is not expressive enough to capture the dependencies of the variables occurring in the patterns;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=', it will detect all calls to eval regardless of whether or not their given input can be controlled by the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Furthermore, it commonly leads to rule overfitting, resulting in over-specific rules that match known examples of vulnerabilities, but are not general enough to capture other instances of the same vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' In Listing 5, the eval call is 7 1 class EvalTaint extends TaintTracking::Configuration { 2 EvalTaint() { this = "EvalTaint" } 3 override predicate isSource(Node node) { 4 node instanceof RemoteFlowSource 5 } 6 override predicate isSink(Node node) { 7 node = globalVarRef("eval").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='getACall().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='getArgument(0) 8 } 9 } 10 11 from EvalTaint cfg, Node source, Node sink 12 where cfg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='hasFlow(source, sink) 13 select sink, "Eval with user input from \\$@.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' ", source Listing 4: CodeQL rule for eval taint-tracking [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' 1 patterns: 2 pattern-inside: function $FUNC($REQ, $RES, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=') {.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='} 3 pattern-either: 4 pattern: eval(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=', <.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' $REQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='$QUERY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='>, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=') Listing 5: NodeJsScan rule for eval detection [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' only detected when it occurs inside the body of a specific type of function declaration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Besides ignoring eval calls at the top level, this pattern also ignores calls to eval which occur inside the body of JavaScript functions declared using alternative syntactic constructs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' function constructor and lambdas).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Using keyword-based analysis: The tool we use to illustrate keyword-based analysis is Graudit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' The following is an excerpt of a Graudit rule that detects the use of eval: eval[[:space:]]*\\( Here, we see a regular expression pattern that simply detects any call to the eval function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' This technique suffers from several limitations such as the source-sink specification problem and a high number of false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Graudit includes many other rules for dangerous sinks in Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js applications [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' EFFECTIVENESS OF THE TOOLS (RQ3) In this section, we focus on our third research question (RQ3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' To perform a quantitative and qualitative assessment of the selected tools, we begin by specifying the evaluation metrics and methodology we used to rank the tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Then we present our findings, relying on the result of running the selected tools across all 957 advisories of our curated dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Evaluation Methodology Tool evaluation metrics: To evaluate the selected tools, we use two main metrics: true positive rate (TPR) and precision (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' The TPR represents the proportion of the total vulnerabilities that are correctly detected by a given tool, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' the true positives (TP): TPR=TP/|vulnerabilities|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' The TPR is useful to assess the raw detection rate of a tool without considering the influence of false positives (FP), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=', its results that do not match the reported advisory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Precision represents the proportion of correctly classified positive cases: P=TP/(TP+FP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' This metric is useful to assess if a tool produces too many false positives that can unnecessarily consume analysts’ resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Tool classification score: To compute the evaluation metrics for a given tool, we need to analyze the output that it generates /src/lib/drivers/search/pouch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js Possible code injection Command injection is an attack in which the goal is execution of arbitrary commands on the host operating system via a vulnerable application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' �→ �→ High eval\\s?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='\\(|setTimeout|setInterval Line: 20 - eval(opts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='filter);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Listing 6: Snippet of Mosca output classified with Score A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' /src/lib/drivers/search/pouch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js-19- } /src/lib/drivers/search/pouch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js:20: eval(opts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='filter);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' /src/lib/drivers/search/pouch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js-21- opts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='filter = filter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Listing 7: Snippet of Graudit output classified with Score B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' when applied to analyzing a specific vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Given that each tool outputs the vulnerability analysis results in its own specific, unstandardized format, we characterize a tool’s output according to a common discrete classification score: Score A: The tool correctly detects and classifies the vulnerability reported in the advisory (true positive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Score B: The tool shows a warning for the vulnerable code, but does not explicitly classify the finding as a vulnerability (true positive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Score C: The tool only shows results that do not match the vulnerability in the advisory report (false positives).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Score D: The tool produces no output (false negative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' We split the TP results according to two distinct classes: A means an explicit vulnerability notification, and B a security warning notification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' The tools ranked with score A provide a richer output to the user and, thus, more information about the detected vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' As an example, consider the output of two tested tools, Mosca and Graudit, with regards to advisory 315 shown in Listings 6 and 7, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Although both outputs flag the vulnerable eval call reported by the review file of Listing 1, Mosca’s output clearly identifies a possible code injection, provides a description, a severity level, and the line of code containing the vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' On the other hand, Graudit only shows the vulnerable line of code without explaining how or why it flags that particular snippet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' For this reason, the output of Mosca is classified with Score A while the output of Graudit is classified with Score B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' This discrete classification is also important to account for tools that might flag for the vulnerability at a place that is only close to it (textually, or on the AST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Considering this, we require that tools must clearly identify the vulnerable statement for some vulnerabilities, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=', code injections and others that can typically be pinpointed to a single statement, while for other vulnerability types multiple lines-of-code are acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Two authors performed a cross-check of all tools’ outputs to guarantee fairness of the tool classification in these cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Analysis Performance We gauge analysis performance by measuring tools’ execu- tion time for all 957 advisories on a machine with an Intel 8 Tool Min Max Mean St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Q-90 Total ODGen 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='236 3653.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='043 148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='757 385.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='629 370.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='969 110823.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='8 CodeQL 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='712 736.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='546 119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='570 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='755 177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='550 28696.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='8 NodeJsScan 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='023 984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='453 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='562 121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='246 230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='216 23795.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='4 ESLint SSC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='592 3556.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='665 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='871 237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='799 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='465 7139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='1 Graudit 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='042 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='632 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='550 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='624 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='704 131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='3 InsiderSec 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='000 243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='000 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='749 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='186 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='000 1374.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='0 MS DevSkim 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='276 186.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='338 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='393 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='262 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='044 1527.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='9 Drek 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='300 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='649 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='865 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='949 244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='3 Mosca 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='005 245.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='498 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='408 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='166 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='216 1770.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='5 Table V: Summary statistics of the analysis times (in seconds) taken by the tested tools across all 957 reviewed advisories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' ODGen CodeQL NodeJsScan ESLint SSC Graudit InsiderSec MS DevSkim Drek Mosca 0 20 40 60 80 100 Percentage of Advisories 146 286 98 103 75 294 212 288 231 324 414 513 158 376 210 456 515 426 530 146 225 791 500 732 475 A B C D Figure 4: Score distribution for each tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Xeon E3-1220 v3 @ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='10GHz processor and 32GB of memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Table V shows several statistics of the execution times taken by each tool to analyze our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' When compared to all other tools, ODGen, CodeQL, NodeJsScan and ESLint SSC require considerably more time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' To analyze all 957 packages, ODGen took over 30 hours (110k seconds), CodeQL took nearly 8 hours (27k seconds), NodeJsScan took nearly 7 hours (24k seconds), while ESLint SSC took nearly 2 hours (7k seconds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' All other tools are considerably more efficient, taking at most 30 minutes to analyze all packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' The mean execution time of ODGen, CodeQL and Node- JsScan is 148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='8, 119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='6 and 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='6 seconds, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' ODGen, CodeQL and NodeJsScan tools are slower because their detec- tion techniques involve modeling statically computed structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' These operations are more complex than performing keyword- based matching searches (see Section IV-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Depending on the size of the package and on the CI/CD pipeline restrictions, these tools may end up being exceedingly slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Results Across the Entire Dataset Figure 4 displays the score distribution for each tool across our entire dataset, and Table VI shows the evaluation metrics for each tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Globally, the tested tools perform rather poorly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' We can draw the following main observations: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Some tools have very low TPR: Counting A and B scores as successful detections, we see that InsiderSec, Drek and Mosca only detect 7 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='7%), 15 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='6%) and 25 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='6%) vulnerabilities, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Hence, these tools fail to detect most vulnerabilities of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' The tools with best TPR have very low precision: The tools that have higher TPR are: ODGen, Graudit, ESLint SSC, Graph Syntax Keyword 0 25 50 75 100 Percentage of Advisories 320 196 104 226 197 356 443 516 262 92 140 A B C D Figure 5: Score distribution for each detection technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' and CodeQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Unfortunately, Graudit and ESLint SSC also have a considerable number of false positives, which tends to erode the confidence of application developers in vulnerability detection tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Graudit detects 219 vulnerabilities (22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='9%), but it also reports over 109k FPs, giving it an overall precision of just 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' A higher number of FPs is expected from a keyword-based tool like Graudit, as many of its string signatures often match non-vulnerable code snippets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' ESLint SSC has the highest TPR (41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='5%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' However, it is also the tool with the highest number of reported FPs (over 389k) and, consequently, the lowest precision (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='1%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' This is because ESLint SSC includes many rules from different ESLint plugins, some of which are simple matches (akin to keyword-based analysis) with greedy behaviour, leading to a higher number of FPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Graph-based analysis has the best detection capability: Figure 5 shows the scores according to a particular detection technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' These results show that graph-based analysis reports a significantly larger number of results with score A (explicit vulnerability notifications).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Syntax and keyword-based analysis look fairly similar, with reasonable detection rates, but also a high number of reports containing only false positive results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' When considering the results of both tools in this category, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=', ODGen and CodeQL, they strike a better balance between true positives and precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' CodeQL detects 300 vulnerabilities (31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='3%) and has a significantly higher precision (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='8%), when compared to most other tools, while ODGen detects 154 vulnerabilities (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='1%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' This number is significantly lower than CodeQL’s, but it represents a much higher precision (23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='8%) than any other tool tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Although both these tools do not have the highest TPR, most of their detected vulnerabilities were classified with the A score, meaning that the reported information is richer and more meaningful to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Consequently, CodeQL and ODGen are the most balanced tools, achieving a reasonable detection rate (TPR) and less FPs, when compared to other tools with similar TPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' We also note that both these tools have the potential for being further improved by extending them with additional rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Combining multiple tools increases TPR, but also lowers the overall precision: The combination of the two best tools (CodeQL and ESLint SSC) detects 508 vulnerabilities (53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='1%), albeit with only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='12% precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' If we add the third best tool (Graudit), we detect more vulnerabilities (551/57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='6%), but the precision further decreases to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='11%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Finally, combining 9 Scope ODGen CodeQL NodeJsScan ESLint SSC Graudit InsiderSec MS DevSkim Drek Mosca TP (%) FP (P%) TP (%) FP (P%) TP (%) FP (P%) TP (%) FP (P%) TP (%) FP (P%) TP (%) FP (P%) TP (%) FP (P%) TP (%) FP (P%) TP (%) FP (P%) CWE-22 70 136 104 416 56 257 110 25467 122 3101 2 401 0 368 0 1057 0 241 (47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='9) (34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='0) (71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='2) (20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='0) (38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='4) (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='9) (75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='3) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='4) (83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='6) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='8) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='4) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='5) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='0) 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+page_content='0) (39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='1) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='2) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='0) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='0) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='0) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='0) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='0) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='0) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='0) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='0) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='0) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='0) Other CWE 26 154 60 1283 34 2548 76 147552 61 60895 3 104 17 7930 3 9159 10 2868 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='9) (14.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='9) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='1) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='0) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='8) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='8) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='2) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='0) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='0) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='4) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='3) Dataset 154 493 300 3553 103 5015 397 389690 219 109485 7 624 81 15465 15 18873 25 6877 (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='1) (23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='8) (31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='3) (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='8) (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='8) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='0) (41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='5) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='1) (22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='9) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='2) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='7) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='1) (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='5) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='5) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='6) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='1) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='6) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='4) Table VI: TP, TPR (%), FP and Precision (P%) for each tool by CWE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' TPR highlights: green (TPR ≥ 50%) or yellow (50% > TPR ≥ 15%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' When TPR is highlighted we also highlight the FP and P columns: yellow (50% > P ≥ 15%), light red (15% > P ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='5%) and dark red (P < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='5%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' The CWEs are: Path Traversal (CWE-22), Cross-site Scripting (CWE-79), Resource Exhaustion (CWE-400), Insufficient Transport Layer Protection (CWE-818), OS Command Injection (CWE-78), Modification of Assumed-Immutable Data (CWE-471), Improper Input Validation (CWE-20), Code Injection (CWE-94), Improper Neutralization of Special Elements used in a Command (CWE-77), and Improperly Controlled Modification of Object Prototype Attributes (CWE-1321).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' both graph-based tools, CodeQL and ODGen, allows for the detection of 339 vulnerabilities (35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='4%) with a precision of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' This shows that combining the best tools can increase the TPR, but at the cost of also increasing the number FPs, which limits the advantage of such an approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Results Across Specific Vulnerability Types We now assess the performance of the tools when focusing on particular types of vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' We concentrate on two main aspects: i) studying the types of vulnerabilities that the tools detect more frequently, and ii) analyzing which types of vulnerabilities can be detected simultaneously by several tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Most frequently detected vulnerability types: From the analysis of Table VI, we highlight seven CWEs that are detected most often regardless of the used tool: CWE-22, CWE-471, CWE-78, CWE-79, CWE-94, CWE-77, and CWE-1321.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' These are colored in yellow and green in Table VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' CWE-22 (path traversal) is the only type clearly detected by all four best performing tools (ODGen, ESLint SSC, Graudit, and CodeQL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' This is because path traversal can be found statically by searching for well-known dangerous sinks in the Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js API, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=', the functions readFile, writeFile and createReadStream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' The difference in precision between these four tools lies in that ESLint SSC and Graudit simply match these function calls, while ODGen and CodeQL report only cases where the path is tainted by user input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' CWE-78 and CWE-77 (OS command injection), CWE-79 (cross-site scripting) and CWE-94 (code injection) correspond to classic injection vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Detecting these vulnerabil- ities depends on the sets of sinks considered by each tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' CodeQL detects more OS command injections, while ESLint SSC detects more code injections because each have more extensive rulesets for those particular vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Both tools detect about the same number of XSS vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Both CWE-471 (Modification of Assumed-Immutable Data) and CWE-1321 (Improperly Controlled Modification of Object Prototype Attributes) are umbrella CWEs for several prototype tampering and prototype pollution vulnerabilities, for which both CodeQL and ESLint SSC have various rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' We expected ODGen to perform better at detecting prototype pollution vul- nerabilities (CWE-471), as this is one of its central goals [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Although ODGen’s results for this vulnerability (22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='9%) fall short of those by CodeQL (27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='1%) and ESLint SSC (79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='2%), ODGen does achieve a much higher precision (36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='7%) than CodeQL (19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='4%) and, especially, ESLint SSC (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='5%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Vulnerability types detected by the three best performing tools: Figure 6 shows the intersections of TPs for the top-10 CWEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' We can see a substantial intersection for CWE-22, where all three tools detect the same 85 vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' This happens because path traversals are easy to find statically using a limited set of known dangerous sinks from the Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js API, which all tools share.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' For CWE-79, both CodeQL and ESLint SSC detect about the same number of vulnerabilities, but only about half intersect with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' This is due to differences in the rules regarding XSS sources and sinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' CWE- 471 shows a significant intersection, but ESLint SSC detects several vulnerabilities that CodeQL misses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' This is because ESLint SSC’s rules have a wider range of sinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Other CWEs have less intersections because their rulesets differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' For example, CodeQL is the only tool with specific rules to detect resource downloads over HTTP, hence the results for CWE-818.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' 10 2 4 4 7 13 17 85 CWE-22 12 13 11 6 0 2 3 CWE-79 2 26 9 2 0 1 1 CWE-400 24 7 18 3 1 CWE-78 14 1 1 2 CWE-818 2 26 11 CWE-471 2 12 1 1 3 CWE-20 26 5 CWE-1321 3 7 0 2 0 7 1 CWE-94 5 4 2 CWE-77 CodeQL ESLint SSC Graudit Figure 6: Intersections of TPs of the 3 best tools for top-10 CWEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' 5 10 15 20 Precision (%) 0 200 400 600 # Lines of Code (LOC) CWE-22 CWE-78 CWE-77 CWE-471 CWE-79 CWE-818 CWE-94 CWE-400 CWE-1321 CWE-20 Figure 7: Correlation between Query LOC and Precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Results as a Function of Queries and Ruleset To study the relationship between ruleset/queries and vulner- ability detection results, we take CodeQL as an example and show, in Figure 7, how the precision of this tool compares with the number of lines of code of the specific queries CodeQL uses to detect the vulnerabilities in the Top-10 CWE categories in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Although this cannot be taken as a general rule, we can see that, in most cases, the precision is higher for smaller queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Taking into account each CWE, simpler vulnerabilities, like CWE-22, can be detected using smaller queries with higher precision, while more complex vulnerabilities to detect, like CWE-1312, are harder to detect even when using larger (more complex) queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' It is then clear that vulnerability detection results can be influenced by the ruleset/queries executed by the tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Using a small (specific) ruleset may improve the precision in detecting a specific vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' However, in the context of a CI/CD pipeline, application developers do not know beforehand which specific ruleset to select to detect the (unknown) vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Consequently, it is reasonable to apply the most comprehensive ruleset available for the tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' This is the approach we used in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' For every tool, we selected the most comprehensive and complete ruleset, by either combining rules into a single tool execution or executing the tool multiple times using a different rule for every execution and combining the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' We also used the rules available off-the-shelf instead of developing or customizing rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' This allows us to reflect how developers will use these tools as most of them are not technically versed to improve the ruleset specified by the tool developers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' REASONS FOR MISSED DETECTION (RQ4) In RQ4, we study why existing tools fail to detect certain vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Table VII shows the number of undetected CWE CWE Description OWASP Undetected % CWE-79 Cross-site Scripting 50 / 99 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='5% CWE-400 Resource Exhaustion 44 / 89 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='4% CWE-78 OS Command Injection 20 / 75 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='7% CWE-20 Improper Input Validation 16 / 41 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='0% CWE-22 Path Traversal 12 / 146 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='2% CWE-94 Code Injection 10 / 33 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='3% CWE-818 Insecure Transport Layer 10 / 75 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='3% CWE-287 Improper Authentication 9 / 9 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='0% CWE-471 MAID 8 / 48 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='7% CWE-200 Information Exposure 8 / 14 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='1% Others 137 / 286 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='9% Total 324 / 957 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='9% Table VII: Number of vulnerabilities undetected by any tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Limitation Advisory CWE Vulnerability L1 63 CWE-730 CVE-2015-9241 567 CWE-287 CVE-2017-11429 L2 165 CWE-818 CVE-2016-10583 305 CWE-22 CVE-2016-1000249 L3 26 CWE-287 CVE-2014-10067 92 CWE-200 CVE-2016-10533 L4 113 CWE-89 CVE-2016-10554 43 CWE-79 CVE-2014-9772 L5 1469 CWE-471 CVE-2017-1000048 313 CWE-502 CVE-2017-5954 Table VIII: Examples of undetected vulnerabilities by cause (Lx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' vulnerabilities grouped by CWE and their mapping to OWASP Top 10 Web Security Risks (2021) [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Of the 957 known vulnerabilities in the dataset, 324 vulnerabilities (33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='9%) were not detected by any of the selected tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' To understand the underlying reasons, we have manually analyzed a sample of undetected vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' So far, we have identified the following five main tool limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' In Table VIII, we map these limitations with some vulnerability categories (CWE) and provide specific examples of undetected vulnerabilities (CVE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Cross-package vulnerabilities: The selected tools come with pre-defined sets of manually written rules, typically focusing solely on popular APIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' We noticed that some undetected vulnerabilities exist in code that invokes functions of third-party packages that map directly to known dangerous code, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=', wrappers to OS-level commands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' These vulnerabilities could have been found by testing all package dependencies (can be thousands of other packages [3]), or by using a more complete set of rules and queries (covering additional sources and sinks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' However, the manual maintenance of such lists of sources and sinks is impractical as the Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js ecosystem expands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Existing work [71] tries to automatically extract taint specifications (sources and sinks) from JavaScript libraries, which partially solves the issue of incomplete rules, but requires the constant dynamic testing of every new npm package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' For instance, Listing 8 shows a code snippet of a command injection vulnerability for advisory 1440.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' This problem exists because user-controlled data reaches an exec sink inside the third-party package comandante, a package meant to ease the execution of OS-level commands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' The tools fail to recognize this vulnerability because the usual command injection sinks are not directly present in the analyzed code, but are instead inside a third-party dependency that is not modelled by the 11 1 // Snippet of .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='/gnuplot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content="js: 2 var run = require('comandante');" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' 3 4 module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content="exports = function () { 5 var plot = run('gnuplot', []);" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' 6 plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='print = function (data, options) { 7 plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='write(data);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' 8 // (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=') 9 };' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' 10 // (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=') 11 } Listing 8: Command Injection (advisory 1440) - NPM and Github Advisories [75, 76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' { "scripts": { "preinstall": """wget http://s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='qdcdn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='com/17mon/17monipdb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='zip && unzip -p 17monipdb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='zip 17monipdb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='dat > 17monipdb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='dat""" } } Listing 9: Insecure Transport Layer in package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='json of ipip-coffee package (advisory 279) - CVE-2016-10673.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' vulnerability detection rules of each tool, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=', they failed to include the write function as a potential dangerous sink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Limited analysis scope: In addition to JavaScript code files, Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js projects depend on several other components, such as configuration files, front-end template code, testing frameworks, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' However, by analyzing only the JavaScript code in isolation, certain vulnerabilities can be missed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' As an example, npm packages contain a package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='json file which may include bootstrap scripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' In several analyzed packages, these scripts are used to download resources over HTTP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' As it turns out, using HTTP allows for man-in-the-middle attacks, where resources are replaced by malicious payloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' While some tools can detect insecure downloads if they are performed by the main JavaScript code (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=', by searching for HTTP URLs), they cannot detect downloads issued from package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='json.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Listing 9 shows a snippet of the package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='json file for the ipip- coffee package, in which an external resource is downloaded over HTTP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' This allows for man-in-the-middle attacks that might compromise the server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' In this particular example, this vulnerability can only be detected if the package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='json file is also considered when performing the vulnerability analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' L3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Lack of contextual knowledge: Packages may expose sensitive information, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=', by logging plaintext passwords to a file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' These vulnerabilities are application-specific and require contextual knowledge of which data is sensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' The analyzed tools, however, are not designed to gain contextual knowl- edge and thus miss vulnerabilities that depend upon it, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=', application-specific leaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' To help detect such vulnerabilities, a possible approach is to annotate application inputs, objects, or data flows with sensitivity levels, and check which system resources handle the annotated features during the execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Listing 10 shows an example of a Credential Exposure vulnerability, in which plaintext passwords are logged to the console.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' The code snippet itself seems benign until one becomes aware that the key variable holds security-critical information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' 1 // Snippet of .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='/lib/odbc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js: 2 if(exports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='debug) { 3 console.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='log("""%s odbc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js : pool[%s] : 4 pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='close() - processing pools %s - connections: %s""", 5 getElapsedTime(), self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='index, key, connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='length);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' 6 } Listing 10: Credential Exposure (advisory 1185) - SNYK-JS-IBMDB- 459762 [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' 1 // Snippet of .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='/protect/lib/rules/xss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content="js 2 const xssSimple = new RegExp('((%3C)|<)((%2F)|/)*[a-z0-9%]+((%3E)|>)', 'i') �→ 3 const xssImgSrc = new RegExp('((%3C)|<)((%69)|i|(%49))((%6D) 4 |m|(%4D))((%67)|g|(%47))[^\\n]+((%3E)|>)', 'i') 5 6 function isXss(value) { 7 return xssSimple." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='test(value) || xssImgSrc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='test(value) 8 } 9 // Example attack payload: 10 // Listing 11: XSS (advisory 1116) - CVE-2018-1000160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' This contextual knowledge is needed to detect the vulnerability but is difficult to extract using automated tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' L4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Incorrect sanitization: Application developers often use regular expressions to detect malicious inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' However, regular expressions are complex, and developers usually do not test them thoroughly, allowing sanitization bypasses to occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Sanitization errors are often hard to detect statically, as they require dynamically testing each regular expression ensuring that they generate semantically valid inputs that can both bypass the validation and effectively trigger the vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' For instance, Listing 11 shows a code fragment containing two regular expressions that aim to prevent potential XSS vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' However, these regular expressions are not entirely correct as there still exist some specially crafted inputs, such as the one shown in the comment of Listing 11, that can bypass this validation and launch an XSS attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' L5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Inability to cope with JavaScript dynamicity: Specific features of JavaScript can lead to vulnerabilities that are hard to detect by static analysis tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' For example, object-based inheritance, extensible objects, and dynamic typing are key features of JavaScript, which can lead to prototype pollution, authentication bypass, and business logic vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Listing 12 shows a type of Prototype Pollution vulnerability present in the qs package, which is a querystring parsing library that allows developers to create objects within query strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' For example, the string ’foo[bar]=baz’ is converted to the object {foo:{bar:’baz’}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Usually, this package protects against attacks that try to overwrite the existing prototype properties of an object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' However, in this vulnerable version, the protection can be circumvented by prefixing the name of the parameter with character [ or ], as shown in the proof- of-concept exploit code shown in Listing 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Consequently, calling toString() on the object will throw an exception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' This can subvert the application logic, potentially allowing attackers to work around security controls, modify data, and make the application unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' The selected tools miss this example because they fail to model how objects change depending on the instructions applied to them, specially the object prototype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' 12 1 // Snippet of .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='/lib/parse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js: 2 module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='exports = function (str, opts) { 3 var options = opts || {};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=" 4 var tempObj = typeof str === 'string' ?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' parseValues(str, options) : str;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' �→ 5 var obj = options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='plainObjects ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='create(null) : {};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' 6 7 var keys = Object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='keys(tempObj);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' 8 for (var i = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' i < keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='length;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' ++i) { 9 var key = keys[i];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' 10 var newObj = parseKeys(key, tempObj[key], options);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' 11 obj = Utils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='merge(obj, newObj, options);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' 12 } 13 return Utils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='compact(obj);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' 14 };' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' 15 // Proof-of-Concept exploit code: 16 qs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='parse("]=toString", { allowPrototypes: false }) // {toString = true} <== prototype overwritten �→ Listing 12: Prototype Override (advisory 1469) - CVE-2017-1000048.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' From these limitations, we can extract actionable insights on the applicability of static code analysis tools for vulnerability detection in Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' On one hand, these tools can potentially overcome limitations L1 and L2 by both employing improved strategies for maintaining taint specifications, and by considering all the appropriate analysis scopes for Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' On the other hand, every static analysis tool will struggle to overcome limitations L3, L4, and L5, because they fail to capture behavioral and contextual information that is only available at runtime when the package is executed with appropriate, and application-specific, test inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' To this end, it seems that the approaches employed by current static vulnerability detection tools can mainly be used successfully to detect classic injection-style vulnerabilities even if all the tools tested in this paper cannot do so with reasonable precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' THREATS TO VALIDITY 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Even though our dataset is composed of real known- vulnerable npm packages, there may be an implicit bias towards vulnerabilities that are easier to analyze and more common across different programming languages (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=', not specific to JavaScript code).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Thus, since our curated dataset may not be fully representative of all vulnerabilities in Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js applications, a tool that can detect all the vulnerabilities of our dataset may still miss other unreported ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' We may have missed some relevant tool, failed to evaluate an analyzer that excels above all tested tools in our study, or overlooked third-party detection rules that produce better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' To reduce this risk, we will promote the reproducibility of our evaluation by providing both the source code of VulcaN and our curated dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Both the labeling of vulnerable packages and identification of their vulnerable code snippets were performed manually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Given the challenges of manual code inspection, these annotations could be mislabeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' To mitigate this risk, all vulnerabilities were analysed by at least two authors at separate times and we will make our dataset available for public scrutiny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' A potential concern is whether our study is susceptible to survivor bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' For instance, assuming hypothetically that all the packages that we analyze had already been analyzed using CodeQL during the code development phase, and that the vulnerabilities reported by CodeQL had been accordingly fixed by the developer prior to package release on npm, then the number of vulnerabilities effectively detected by CodeQL could be higher than those reported in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' This would misleadingly suggest that the quality of CodeQL is worse than what it is in reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Note, however, that such a comprehensive characterization of each tool is beyond the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' In our study, we concentrate on evaluating tools’ ability to detect, not all possible vulnerabilities, but only those that have been officially reported in npm packages already in production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' RELATED WORK The literature covers many tools for detecting vulnerabilities in Web applications, including static [78, 79, 80], dynamic [81, 82], and hybrid analysis tools [83, 84, 85, 86], often combining different types of program analysis techniques, such as fuzzing (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' [81, 82]), control-flow and data-flow analysis (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' [57, 79, 83, 85]), and symbolic execution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' [80, 84, 86]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' The great majority of these tools is, however, aimed at PHP-based Web applications, with considerably fewer tools targeting JavaScript applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Most of the existing tools for JavaScript are aimed at client-side JavaScript code and its specific vulnerabilities: for instance, DOM-based XSS [87, 88], unrestricted inclusion of third-party cross-origin scripts [89], and potentially malicious flows via client-side persistent storage [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Graph-based vulnerability scanners: State-of-the-art static vulnerability analysis techniques often work by first computing a static model describing the dynamic behaviour of the application to be analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Most notably, code property graphs (CPGs) [57] were proposed as a compact representation of an application’s behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' With CPGs, one can encode specific vulnerability types as simple graph traversals, which can, in turn, be expressed using graph query languages and then executed on top of off-the-shelf graph databases (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Neo4J [90]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Code property graphs have successfully been applied to find SQL injection, XSS, and CSRF vulnerabilities in PHP applications [79, 85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Furthermore, they are at the core of CodeQL [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' For JavaScript, code property graphs were employed by JAW [56] and ODGen [15], for client-side and server-side JavaScript respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' In our work, we have extensively evaluated CodeQL and ODGen as representative state-of-the-art, graph-based vulnerability scanners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Vulnerability studies & analyzers for Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js applications: Unlike client-side JavaScript applications, which run in the browser, Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js application code is not sandboxed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Recent empirical studies [3, 91] have shown that, contrary to popular belief, npm applications are often poorly maintained and tested, with a significant percentage (up to 40%) of all packages depending on code with at least one publicly known vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Furthermore, after reviewing more than 200K npm applications, Staicu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' [4] concludes that 20% of the analyzed applications either directly or indirectly make use of an injection API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Despite this security-critical situation, there is only a small number of research tools for detecting vulnerabilities in Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js applications and their underlying 13 infrastructure, most of which based on dynamic code analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' For instance, Synode [4] aims to prevent injection attacks in Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js applications, and NodeSec [9] aims to detect vulnerabilities in Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' The authors of [5] and [92] design specific dynamic analysis for finding regular expression denial of service (ReDoS) vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' The authors of [93] also apply dynamic analysis and symbolic execution to detect attacks that leverage hidden properties in client- and server-side JavaScript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' There are also academic works that employ static analysis techniques for detecting vulnerabilities in Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js, but most focus on detecting prototype pollution vulnerabilities [94, 95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' ODGen [15] is the only purely static code analysis tool developed by the academia that aims to detect several types of vulnerabilities in Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Empirical studies of vulnerability analyzers: Several em- pirical studies aim at characterizing the efficacy of existing white-box vulnerability detection tools (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' [88, 96, 97]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Durieux et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' [96] evaluated 9 automated analysis tools for Ethereum Smart Contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' The authors created a curated dataset consisting of 69 annotated vulnerable smart contracts, as well as a raw dataset consisting of 47,518 smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' They report that only 42% of the vulnerabilities on the annotated dataset were detected, with the highest ranking tool having an accuracy of 21%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Melicher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' [88] evaluated 3 automated static analysis tools for detecting DOM-based XSS in client- side JavaScript code (Esflow [98], ScanJS [54], and Burp Suite Pro [99]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' They created a dataset with 3219 confirmed vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' However, many security flaws in server-side code for Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js do not exist on the client-side (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=', SQL injections), and vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' As such, the dataset from [88] is not representative enough of server-side vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Finally, Nunes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' [97] evaluate five free static analysis tools for detecting SQL injection and XSS vulnerabilities in PHP web applications using a dataset comprising 134 WordPress plugins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' In contrast to the studies referenced above, our paper presents the first empirical study targeting fully automated vulnerability detection tools for npm packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Our study comes with a comprehensive manually-annotated dataset based on confirmed real-world vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' CONCLUSIONS This paper presented an empirical study of static analysis tools for detecting vulnerabilities in Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' To conduct this study, we built VulcaN, an automated analysis framework, using which we created the largest known curated dataset of Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='js packages with well-characterized security vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Currently, our curated dataset includes 745 reviews that accurately identify the exact location of known vulnerabilities inside affected npm packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' We found that the nine evaluated tools fail to detect many vulnerabilities and exhibit high false positive rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' Additionally, we show that many important vulnerabilities appearing in the OWASP Top- 10 are not detected by any evaluated tool or even when using the combination of all tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' We believe that our curated dataset will substantially con- tribute to enabling future research on automatic vulnerability detection tools for server-side JavaScript applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' To this end, we have made this dataset publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' REFERENCES [1] “Node.' metadata={'source': 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+page_content=' Security, https://portswigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content='net/burp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} +page_content=' 16' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE4T4oBgHgl3EQfeQ0p/content/2301.05097v1.pdf'} diff --git a/CtE1T4oBgHgl3EQfDwP-/content/tmp_files/2301.02883v1.pdf.txt b/CtE1T4oBgHgl3EQfDwP-/content/tmp_files/2301.02883v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..375c579c98021a6bec40e686ffab4ce84a2a51ef --- /dev/null +++ b/CtE1T4oBgHgl3EQfDwP-/content/tmp_files/2301.02883v1.pdf.txt @@ -0,0 +1,491 @@ +arXiv:2301.02883v1 [hep-ph] 7 Jan 2023 +One-Loop Electron Mass and QED Trace Anomaly +Michael I. Eides∗ +Department of Physics and Astronomy, +University of Kentucky, Lexington, KY 40506, USA +Abstract +Electron mass is considered as a matrix element of the energy-momentum trace in the rest frame. +The one-loop diagrams for this matrix element are different from the textbook diagrams for the +electron mass renormalization. We clarify connection between the two sets of diagrams and explain +analytically and diagrammatically why the results of both calculations coincide. +∗ Email address:meides@g.uky.edu +Typeset by REVTEX +1 + +I. +INTRODUCTION +Hadron energy-momentum tensor (EMT), its matrix elements, anomalous trace, form +factors and multipole expansion is now a vibrant field of research. EMT form factors describe +interaction of particles with weak external gravitational field [1, 2]. For a long time there +was no way to measure form factors of hadron EMT, but the situation changed when it was +realized that they are connected to the generalized parton distribution functions which can +be measured in deeply virtual Compton scattering and other hard exclusive reactions, see, +e.g., [3–8]. +Due to nonperturbative nature of the low-energy QCD theoretical studies of hadron grav- +itational form factors and other hadron EMT properties use general gauge theory principles, +lattice QCD, QCD inspired low-energy models and model theories which admit quantitative +analysis, see e.g., [9–16] and numerous other papers. +A new insight into the EMT properties could arise from consideration of EMT in theories +which allow perturbative treatment. While perturbative approach is clearly impossible in +QCD, one can consider a simpler gauge theory, namely QED, and hope to acquire some +experience which would be useful for the hadronic world. One-loop QED contributions to +the EMT form factors, matrix elements and trace were calculated for a free electron and +electron in the Coulomb field in a number of old and recent papers [17–28]. +One-loop electron mass renormalization in the mass-shell renormalization scheme is a +textbook problem discussed in every introductory quantum field theory textbook, see, e.g., +[29]. Consideration of EMT suggests another perspective on this classical problem. One can +calculate electron mass as a matrix element of the EMT trace. The diagrams describing this +matrix element do not coincide with the well known diagrams for the one-loop corrections +to the electron mass. We will calculate electron mass with the help of both sets of different +contributions and explain why they produce coinciding results. +II. +MATRIX ELEMENTS OF EMT AND MASS OF PARTICLES +General formulae for EMT T µν(x) follow from its definition as a conserved two-index +symmetric tensor. Due to translational invariance +2 + +⟨p′|T µν(x)|p⟩ = ei(p′−p)·x⟨p′|T µν(0)|p⟩, +(1) +where |p⟩ is a particle eigenstate with momentum p and states here are normalized rela- +tivistically, ⟨p′|p⟩ = 2Ep(2π)3δ(3)(p − p′). +The Hamiltonian is a three-dimensional integral, H = +� +d3xT 00(x), and on the one hand +� +d3x⟨p′|T 00(x)|p⟩ = 2E2 +p(2π)3δ(3)(p − p′), +(2) +and on the other hand (see Eq. (1)) +� +d3x⟨p′|T 00(x)|p⟩ = (2π)3δ(3)(p − p′)⟨p′|T 00(0)|p⟩. +(3) +Hence, +⟨p|T 00(0)|p⟩ = 2E2 +p, +(4) +and due to Lorentz invariance +⟨p|T µν(0)|p⟩ = 2pµpν, +⟨p|T µ +µ(0)|p⟩ = 2m2. +(5) +In the rest frame and with the nonrelativistic normalization of states ⟨p′|p⟩ = (2π)3δ(3)(p − +p′) +⟨0|T µ +µ(0)|0⟩ = m, +⟨0|T 00(0)|0⟩ = m. +(6) +These relations hold both elementary particles and for bound states, and are obviously valid +in any relativistic field theory. Below we will consider the first of these equations for an +electron in QED. +Symmetric EMT tensor is conserved in a translationally invariant relativistic field theory +and it is not renormalized as any conserved operator T µν +0 += [T µν]R. It is well known that +EMT trace in gauge theories acquires an anomalous contribution [30–32] and has the form +T µ +0 µ = (1 + γm(e0)) ¯ψ0m0ψ0 + β(e0) +2e0 +F 2 +0 = (1 + γm(e))[ ¯ψmψ]R + β(e) +2e [F 2]R, +(7) +where β(e)/2e = α/6π, γm(e) = 3α/2π. +The left hand side in Eq. (7) is renorminvariant and then the sum of the operators on +the right hand side (RHS) is also renorminvariant. There are subtleties with separation of +3 + +the terms on the right hand side in a sum of renorminvariant operators beyond one loop, +see [15, 33–38]. +We are going to consider matrix element of the anomalous trace in Eq. (7) for an electron +at rest in the one-loop approximation. We will be working in the renormalized perturbation +theory and use the mass-shell renormalization scheme. Then, according to Eq. (6), this +matrix element should be equal to the physical electron mass m and describe one-loop mass +renormalization. At the same time the diagrams which contribute to this matrix element do +not coincide with the well known mass renormalization diagrams. Our goal is to clarify from +the diagrammatic and analytic perspectives, why two different diagrammatic descriptions +lead to the identical results1. +III. +ONE-LOOP MASS RENORMALIZATION AND EMT ANOMALOUS TRACE +FOR A FREE ELECTRON +Let us recall one-loop electron mass renormalization in the mass-shell scheme with dimen- +sional regularization. We collected the well known relevant formulae in the Appendix. In +the mass-shell renormalization scheme the counterterm δm(2) kills the ultraviolet divergence +in the regularized but not renormalized self-energy diagram Σ(p) and preserves the physical +mass at m += +m ++ +− +m +δm +i +Σ(m) +FIG. 1. One-loop mass renormalization. +m = m + Σ(m) − δm(2), +(8) +where δm(2) = Σ(m). This expression is illustrated in Fig. 1. The factor i before the self- +energy diagram in Fig. 1 is included in the standard diagrammatic definition of Σ(p), see +e.g., [29]. +1 One-loop matrix element of the EMT trace in the electron state was calculated in the mass-shell renor- +malization scheme with the momentum cutoff in [23], but the relationship between two sets of diagrams +was not addressed there. +4 + +Next we turn to the matrix element of the EMT trace in Eq. (6) for the electron. The +leading contribution to the matrix element of the term (β(e0)/2e0)F 2 +0 in Eq. (7) is of order +α2 and we ignore it. It is sufficient to calculate the one-loop matrix element +T = ⟨0|m0(1 + γm) ¯ψ0ψ0|0⟩. +(9) +In the one-loop approximation +T = ⟨0|(1 + γm)m0 ¯ψ0ψ0|0⟩ ≈ ⟨0|(1 + γm)(m − δm(2)) ¯ψ0ψ0|0⟩ +(10) += m + mδZ2 − δm(2) + mγm + Γm(m) +where Γm(m) is the one-loop diagram for the scalar vertex m ¯ψψ, see Fig. 2. +mδZ2 +Γm(m) +δm(2) +− +T = ++ ++ +m ++ +γmm +FIG. 2. One-loop matrix element of the EMT trace. +All terms on the RHS in Eq. (10) and in Fig. 2, except Γm(m), are known from the one-loop +mass-shell renormalization scheme (see Eq. (A7) and Eq. (A8)) and only Γm(m) requires +calculation. After an easy one-loop calculation we obtain +Γm(m) = α +4πm +�8 +ǫ − 4γ + 2 ln λ2 +m2 + 4 ln µ2 +m2 + 2 + 4 ln(4π) +� +. +(11) +Next we use Γm(m), the renormalization constants in Eq. (A7) and Eq. (A8), and γm = +3α/2π to calculate the sum in Eq. (10) +T = m + mδZ2 − δm + mγm + Γm(m) = m. +(12) +Thus we confirmed that the matrix element T of the anomalous EMT trace in the one-loop +approximation is equal the physical electron mass, as it should be. Comparing Eq. (8) and +Eq. (12) (and the respective Figs. 1 and 2) we see that +Σ(m) = Γm(m) + mδZ2 + γmm. +(13) +5 + +At this stage it is unclear why the different sets of diagrams in Fig. 1 and Fig. 2 produce +coinciding results. To figure out a deeper reason why this happens we expand the unrenor- +malized electron self-energy Σ(/p) in the Taylor series near the physical mass +Σ(/p) = Σ(/p = m) + (/p − m)Σ′(/p = m) + O((/p − m)2) +(14) += δm(2) + (/p − m)δZ2 + O((/p − m)2). +Differentiating with respect to m we obtain at /p = m +mdΣ(/p) +dm +|/p=m = mdΣ(/p = m) +dm +− mΣ′(/p = m). +(15) +Notice that (see Fig. 3) +mdΣ(p) +dm += Γm(p), +(16) +i +Γm(p) +Σ(p) +m d +dm +� +� += +FIG. 3. Logarithmic mass derivative of Σ(p). +which holds due to the identity +m d +dm +� +1 +/p − m +� += +1 +/p − mm +1 +/p − m. +(17) +Then Eq. (15) can be written in the form +Γm(m) + mδZ2 = mdΣ(/p = m) +dm +, +(18) +and Eq. (13) turns into (δm(2) = Σ(/p = m)) +Σ(m) = mdΣ(/p = m) +dm ++ γmm ≡ md(δm(2)) +dm ++ γmm. +(19) +We calculate the derivative on the RHS using the explicit expression for δm(2) in Eq. (A7) +and obtain (see Fig. 4) +6 + +md(δm(2)) +dm += δm(2) − µdδm(2) +dµ += δm(2) − γmm, +(20) +where at the last step we used the definition of the electron mass anomalous dimension. +i +Σ(m) +Σ(m) +m d +dm +� +� += +− +γmm +i +FIG. 4. Logarithmic mass derivative of Σ(m). +Thus we proved by direct calculation that Eq. (19) holds and the expressions in Eq. (8) +and Eq. (12) (and the respective sets of diagrams in Fig. 1 and Fig. 2) coincide. +IV. +CONCLUSIONS +We have shown that the standard mass renormalization in Fig. 1 and the sum of the +diagrams for the matrix element of the EMT trace in Fig. 2 coincide. This happens due +to two important relationships. First, the one-loop diagram for a scalar vertex is equal to +the logarithmic derivative of the self-energy diagram, see Eq. (16) and Fig. 3. Second, the +mass renormalization counterterm (self-energy at /p = m) is equal to its own logarithmic +derivative plus the product of mass and its anomalous dimension, see Eq. (19) and Fig. 4. +The calculations above are made in the one-loop approximation, but we expect that they +can be generalized to any number of loops. Really, connection between an arbitrary diagram +and its logarithmic derivative with respect to the fermion mass, and the connection between +such derivative and the fermion mass anomalous dimension do not depend on the number +of loops, and these are the only essentail steps in the derivation above. +Appendix A: Standard one-loop electron mass renormalization +Some well known results are collected below. +We use dimensional regularization and +mass-shell renormalization. The QED Lagrangian in this scheme is +L0 = −1 +4F 2 +0 + ¯ψ0(i/∂ − m0)ψ0 − e0 ¯ψ0 /A0ψ0 = L + δL, +(A1) +7 + +where +L = −1 +4F 2 + ¯ψ(i/∂ − m)ψ − µ +ǫ +2e ¯ψ /Aψ, +(A2) +δL = −1 +4δZ3F 2 + ¯ψ(iδZ2/∂ − δm)ψ − µ +ǫ +2eδZ1 ¯ψ /Aψ, +(A3) +L + δL = −1 +4Z3F 2 + iZ2 ¯ψ/∂ψ − mZm ¯ψψ − eZ1µ +ǫ +2 ¯ψ /Aψ, +(A4) +and +Z1 = 1 + δZ1, +Z2 = 1 + δZ2, +Z3 = 1 + δZ3, +mZm = m(1 + δZm) = m + δm. +(A5) +We define δm = m−m0 = m−mZmZ−1 +2 . In the one-loop approximation δm = m−m0 = +m − mZmZ−1 +2 +≈ −mδZm + mδZ2 = −δm + mδZ2. +The renormalized one-loop self-energy Σr(p) is +Σr(p) = α +2π +� 1 +0 +dx +� +(2m − x/p) +�2 +ǫ − γ + ln(4π) + ln +µ2 +−x(1 − x)p2 + xλ2 + (1 − x)m2 +� +− (m − x/p) +� +− (/pδZ2 − δm)|/p→m +≈ Σ(m) + (/p − m)Σ′(/p = m) − (/p − m)δZ2 + (δm − mδZ2) += 3α +4π m +�2 +ǫ − γ + ln(4π) + ln µ2 +m2 + 4 +3 +� +− (/p − m) α +4π +�2 +ǫ − γ + ln(4π) + ln µ2 +m2 + 2 ln λ2 +m2 + 4 +� +− (/p − m)δZ2 + (δm − mδZ2), +(A6) +where Σ(p) is the dimensionally regularized self-energy, µ is the auxiliary dimensional reg- +ularization mass, λ is the IR photon mass and ǫ = 4 − d. +The one-loop counterterms are +δm(2) ≡ mδZ2 − δm = Σ(m) = 3α +4πm +�2 +ǫ − γ + ln(4π) + ln µ2 +m2 + 4 +3 +� +, +(A7) +δZ2 = Σ′(/p = m) = − α +4π +�2 +ǫ − γ + ln(4π) + ln µ2 +m2 + 2 ln λ2 +m2 + 4 +� +. +(A8) +8 + +ACKNOWLEDGMENTS +This work was supported by the NSF grant PHY- 2011161. +[1] I. Y. Kobzarev and L. B. 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Tanaka, “Twist-four gravitational form factor at NNLO QCD from trace anomaly con- +straints,” [arXiv:2212.09417 [hep-ph]]. +11 + diff --git a/CtE1T4oBgHgl3EQfDwP-/content/tmp_files/load_file.txt b/CtE1T4oBgHgl3EQfDwP-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a7c86345121b2dbdfb88e4040bc7799b29bf437e --- /dev/null +++ b/CtE1T4oBgHgl3EQfDwP-/content/tmp_files/load_file.txt @@ -0,0 +1,444 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf,len=443 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content='02883v1 [hep-ph] 7 Jan 2023 One-Loop Electron Mass and QED Trace Anomaly Michael I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' Eides∗ Department of Physics and Astronomy, University of Kentucky, Lexington, KY 40506, USA Abstract Electron mass is considered as a matrix element of the energy-momentum trace in the rest frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' The one-loop diagrams for this matrix element are different from the textbook diagrams for the electron mass renormalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' We clarify connection between the two sets of diagrams and explain analytically and diagrammatically why the results of both calculations coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' ∗ Email address:meides@g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content='uky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content='edu Typeset by REVTEX 1 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' INTRODUCTION Hadron energy-momentum tensor (EMT), its matrix elements, anomalous trace, form factors and multipole expansion is now a vibrant field of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' EMT form factors describe interaction of particles with weak external gravitational field [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' For a long time there was no way to measure form factors of hadron EMT, but the situation changed when it was realized that they are connected to the generalized parton distribution functions which can be measured in deeply virtual Compton scattering and other hard exclusive reactions, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=', [3–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' Due to nonperturbative nature of the low-energy QCD theoretical studies of hadron grav- itational form factors and other hadron EMT properties use general gauge theory principles, lattice QCD, QCD inspired low-energy models and model theories which admit quantitative analysis, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=', [9–16] and numerous other papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' A new insight into the EMT properties could arise from consideration of EMT in theories which allow perturbative treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' While perturbative approach is clearly impossible in QCD, one can consider a simpler gauge theory, namely QED, and hope to acquire some experience which would be useful for the hadronic world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' One-loop QED contributions to the EMT form factors, matrix elements and trace were calculated for a free electron and electron in the Coulomb field in a number of old and recent papers [17–28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' One-loop electron mass renormalization in the mass-shell renormalization scheme is a textbook problem discussed in every introductory quantum field theory textbook, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=', [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' Consideration of EMT suggests another perspective on this classical problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' One can calculate electron mass as a matrix element of the EMT trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' The diagrams describing this matrix element do not coincide with the well known diagrams for the one-loop corrections to the electron mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' We will calculate electron mass with the help of both sets of different contributions and explain why they produce coinciding results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' MATRIX ELEMENTS OF EMT AND MASS OF PARTICLES General formulae for EMT T µν(x) follow from its definition as a conserved two-index symmetric tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' Due to translational invariance 2 ⟨p′|T µν(x)|p⟩ = ei(p′−p)·x⟨p′|T µν(0)|p⟩, (1) where |p⟩ is a particle eigenstate with momentum p and states here are normalized rela- tivistically, ⟨p′|p⟩ = 2Ep(2π)3δ(3)(p − p′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' The Hamiltonian is a three-dimensional integral, H = � d3xT 00(x), and on the one hand � d3x⟨p′|T 00(x)|p⟩ = 2E2 p(2π)3δ(3)(p − p′), (2) and on the other hand (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' (1)) � d3x⟨p′|T 00(x)|p⟩ = (2π)3δ(3)(p − p′)⟨p′|T 00(0)|p⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' (3) Hence, ⟨p|T 00(0)|p⟩ = 2E2 p, (4) and due to Lorentz invariance ⟨p|T µν(0)|p⟩ = 2pµpν, ⟨p|T µ µ(0)|p⟩ = 2m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' (5) In the rest frame and with the nonrelativistic normalization of states ⟨p′|p⟩ = (2π)3δ(3)(p − p′) ⟨0|T µ µ(0)|0⟩ = m, ⟨0|T 00(0)|0⟩ = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' (6) These relations hold both elementary particles and for bound states, and are obviously valid in any relativistic field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' Below we will consider the first of these equations for an electron in QED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' Symmetric EMT tensor is conserved in a translationally invariant relativistic field theory and it is not renormalized as any conserved operator T µν 0 = [T µν]R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' It is well known that EMT trace in gauge theories acquires an anomalous contribution [30–32] and has the form T µ 0 µ = (1 + γm(e0)) ¯ψ0m0ψ0 + β(e0) 2e0 F 2 0 = (1 + γm(e))[ ¯ψmψ]R + β(e) 2e [F 2]R, (7) where β(e)/2e = α/6π, γm(e) = 3α/2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' The left hand side in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' (7) is renorminvariant and then the sum of the operators on the right hand side (RHS) is also renorminvariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' There are subtleties with separation of 3 the terms on the right hand side in a sum of renorminvariant operators beyond one loop, see [15, 33–38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' We are going to consider matrix element of the anomalous trace in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' (7) for an electron at rest in the one-loop approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' We will be working in the renormalized perturbation theory and use the mass-shell renormalization scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' Then, according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' (6), this matrix element should be equal to the physical electron mass m and describe one-loop mass renormalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' At the same time the diagrams which contribute to this matrix element do not coincide with the well known mass renormalization diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' Our goal is to clarify from the diagrammatic and analytic perspectives, why two different diagrammatic descriptions lead to the identical results1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' ONE-LOOP MASS RENORMALIZATION AND EMT ANOMALOUS TRACE FOR A FREE ELECTRON Let us recall one-loop electron mass renormalization in the mass-shell scheme with dimen- sional regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' We collected the well known relevant formulae in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' In the mass-shell renormalization scheme the counterterm δm(2) kills the ultraviolet divergence in the regularized but not renormalized self-energy diagram Σ(p) and preserves the physical mass at m = m + − m δm i Σ(m) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' One-loop mass renormalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' m = m + Σ(m) − δm(2), (8) where δm(2) = Σ(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' This expression is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' The factor i before the self- energy diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' 1 is included in the standard diagrammatic definition of Σ(p), see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=', [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' 1 One-loop matrix element of the EMT trace in the electron state was calculated in the mass-shell renor- malization scheme with the momentum cutoff in [23], but the relationship between two sets of diagrams was not addressed there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' 4 Next we turn to the matrix element of the EMT trace in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' (6) for the electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' The leading contribution to the matrix element of the term (β(e0)/2e0)F 2 0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' (7) is of order α2 and we ignore it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' It is sufficient to calculate the one-loop matrix element T = ⟨0|m0(1 + γm) ¯ψ0ψ0|0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' (9) In the one-loop approximation T = ⟨0|(1 + γm)m0 ¯ψ0ψ0|0⟩ ≈ ⟨0|(1 + γm)(m − δm(2)) ¯ψ0ψ0|0⟩ (10) = m + mδZ2 − δm(2) + mγm + Γm(m) where Γm(m) is the one-loop diagram for the scalar vertex m ¯ψψ, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' mδZ2 Γm(m) δm(2) − T = + + m + γmm FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' One-loop matrix element of the EMT trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' All terms on the RHS in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' (10) and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' 2, except Γm(m), are known from the one-loop mass-shell renormalization scheme (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' (A7) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' (A8)) and only Γm(m) requires calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' After an easy one-loop calculation we obtain Γm(m) = α 4πm �8 ǫ − 4γ + 2 ln λ2 m2 + 4 ln µ2 m2 + 2 + 4 ln(4π) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' (11) Next we use Γm(m), the renormalization constants in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' (A7) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' (A8), and γm = 3α/2π to calculate the sum in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' (10) T = m + mδZ2 − δm + mγm + Γm(m) = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' (12) Thus we confirmed that the matrix element T of the anomalous EMT trace in the one-loop approximation is equal the physical electron mass, as it should be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' Comparing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' (8) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' (12) (and the respective Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' 1 and 2) we see that Σ(m) = Γm(m) + mδZ2 + γmm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' (13) 5 At this stage it is unclear why the different sets of diagrams in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' 1 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' 2 produce coinciding results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' To figure out a deeper reason why this happens we expand the unrenor- malized electron self-energy Σ(/p) in the Taylor series near the physical mass Σ(/p) = Σ(/p = m) + (/p − m)Σ′(/p = m) + O((/p − m)2) (14) = δm(2) + (/p − m)δZ2 + O((/p − m)2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' Differentiating with respect to m we obtain at /p = m mdΣ(/p) dm |/p=m = mdΣ(/p = m) dm − mΣ′(/p = m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' (15) Notice that (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' 3) mdΣ(p) dm = Γm(p), (16) i Γm(p) Σ(p) m d dm � � = FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' Logarithmic mass derivative of Σ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' which holds due to the identity m d dm � 1 /p − m � = 1 /p − mm 1 /p − m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' (17) Then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' (15) can be written in the form Γm(m) + mδZ2 = mdΣ(/p = m) dm , (18) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' (13) turns into (δm(2) = Σ(/p = m)) Σ(m) = mdΣ(/p = m) dm + γmm ≡ md(δm(2)) dm + γmm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' (19) We calculate the derivative on the RHS using the explicit expression for δm(2) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' (A7) and obtain (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' 4) 6 md(δm(2)) dm = δm(2) − µdδm(2) dµ = δm(2) − γmm, (20) where at the last step we used the definition of the electron mass anomalous dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' i Σ(m) Σ(m) m d dm � � = − γmm i FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' Logarithmic mass derivative of Σ(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' Thus we proved by direct calculation that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' (19) holds and the expressions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' (8) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' (12) (and the respective sets of diagrams in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' 1 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' 2) coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' CONCLUSIONS We have shown that the standard mass renormalization in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' 1 and the sum of the diagrams for the matrix element of the EMT trace in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' 2 coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' This happens due to two important relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' First, the one-loop diagram for a scalar vertex is equal to the logarithmic derivative of the self-energy diagram, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' (16) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' Second, the mass renormalization counterterm (self-energy at /p = m) is equal to its own logarithmic derivative plus the product of mass and its anomalous dimension, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' (19) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' The calculations above are made in the one-loop approximation, but we expect that they can be generalized to any number of loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' Really, connection between an arbitrary diagram and its logarithmic derivative with respect to the fermion mass, and the connection between such derivative and the fermion mass anomalous dimension do not depend on the number of loops, and these are the only essentail steps in the derivation above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' Appendix A: Standard one-loop electron mass renormalization Some well known results are collected below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' We use dimensional regularization and mass-shell renormalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' The QED Lagrangian in this scheme is L0 = −1 4F 2 0 + ¯ψ0(i/∂ − m0)ψ0 − e0 ¯ψ0 /A0ψ0 = L + δL, (A1) 7 where L = −1 4F 2 + ¯ψ(i/∂ − m)ψ − µ ǫ 2e ¯ψ /Aψ, (A2) δL = −1 4δZ3F 2 + ¯ψ(iδZ2/∂ − δm)ψ − µ ǫ 2eδZ1 ¯ψ /Aψ, (A3) L + δL = −1 4Z3F 2 + iZ2 ¯ψ/∂ψ − mZm ¯ψψ − eZ1µ ǫ 2 ¯ψ /Aψ, (A4) and Z1 = 1 + δZ1, Z2 = 1 + δZ2, Z3 = 1 + δZ3, mZm = m(1 + δZm) = m + δm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' (A5) We define δm = m−m0 = m−mZmZ−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' In the one-loop approximation δm = m−m0 = m − mZmZ−1 2 ≈ −mδZm + mδZ2 = −δm + mδZ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' The renormalized one-loop self-energy Σr(p) is Σr(p) = α 2π � 1 0 dx � (2m − x/p) �2 ǫ − γ + ln(4π) + ln µ2 −x(1 − x)p2 + xλ2 + (1 − x)m2 � − (m − x/p) � − (/pδZ2 − δm)|/p→m ≈ Σ(m) + (/p − m)Σ′(/p = m) − (/p − m)δZ2 + (δm − mδZ2) = 3α 4π m �2 ǫ − γ + ln(4π) + ln µ2 m2 + 4 3 � − (/p − m) α 4π �2 ǫ − γ + ln(4π) + ln µ2 m2 + 2 ln λ2 m2 + 4 � − (/p − m)δZ2 + (δm − mδZ2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' (A6) where Σ(p) is the dimensionally regularized self-energy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' µ is the auxiliary dimensional reg- ularization mass,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' λ is the IR photon mass and ǫ = 4 − d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' The one-loop counterterms are δm(2) ≡ mδZ2 − δm = Σ(m) = 3α 4πm �2 ǫ − γ + ln(4π) + ln µ2 m2 + 4 3 � , (A7) δZ2 = Σ′(/p = m) = − α 4π �2 ǫ − γ + ln(4π) + ln µ2 m2 + 2 ln λ2 m2 + 4 � .' metadata={'source': 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NNLO QCD from trace anomaly con- straints,” [arXiv:2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content='09417 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} +page_content=' 11' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfDwP-/content/2301.02883v1.pdf'} diff --git a/F9E2T4oBgHgl3EQf-Qn4/content/tmp_files/load_file.txt b/F9E2T4oBgHgl3EQf-Qn4/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..808794890b648681a2793233d0a6664ba76762af --- /dev/null +++ b/F9E2T4oBgHgl3EQf-Qn4/content/tmp_files/load_file.txt @@ -0,0 +1,1719 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf,len=1718 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='04238v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='DG] 10 Jan 2023 MODIFIED CONFORMAL EXTENSIONS MATTHIAS HAMMERL, KATJA SAGERSCHNIG, JOSEF ˇSILHAN AND VOJTˇECH ˇZ´ADN´IK Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' We present a geometric construction and characterization of 2n-dimensional split-signature conformal structures endowed with a twistor spinor with integrable kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The construction is regarded as a modification of the conformal Patterson–Walker metric construction for n-dimensional projective manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The characterization is presented in terms of the twistor spinor and an integrability condition on the conformal Weyl curva- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' We further derive complete description of Einstein metrics and infinitesimal conformal symmetries in terms of suitable projective data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Finally, we obtain an explicit geometrically constructed Fefferman–Graham ambient metric and show vanishing of Q-curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Introduction Walker manifolds are pseudo-Riemannian manifolds admitting a parallel isotropic distri- bution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In the case of split-signature, there is an interesting subclass of Walker metrics that are induced on the cotangent bundle of a manifold M with a torsion-free affine connection D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' These are the standard Patterson–Walker metrics or (pseudo-)Riemannian extensions of affine structures, introduced in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' A Patterson–Walker (shortly, PW) metric on the total space of T ∗M is given by the natural pairing between the vertical distribution of the pro- jection T ∗M → M and the horizontal distribution corresponding to D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' It has the following coordinate expression �g = 2 dxA ⊙ dpA − 2 Γ C A B pC dxA ⊙ dxB , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1) where (xA) are local coordinates on M, (pA) the canonical fibre coordinates and Γ C A B the Christoffel symbols of the underlying affine connection D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Here, the vertical distribution V ⊂ TT ∗M is the isotropic distribution that is parallel with respect to the Levi-Civita connection of �g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' There are variants or modifications of the standard PW construction that still yield Walker metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' They appear with various motivations and applications in both early and more recent references, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=', [25], [28], [26], [1], [14], [9], [15], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' A tight relation between the underlying and the induced data is a common feature of all such constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In this article, we focus on modifications of the form g := �g + π∗Φ , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2) where �g is the standard PW metric and π∗Φ is the pullback of a symmetric 2-tensor Φ on M with respect to the projection π : T ∗M → M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' We call such metrics the Φ-modified or just modified PW metrics, noticing that other names can be found in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Date: January 12, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' 2000 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' 53A20, 53A30, 53B30, 53C07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Differential geometry, Patterson–Walker metric, Projective structure, Conformal structure, Conformal Killing field, Einstein metric, Fefferman–Graham ambient metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' 1 2 HAMMERL, SAGERSCHNIG, ˇSILHAN, ˇZ´ADN´IK Considering the relations between the initial affine connection on M and the induced PW metric on T ∗M, it is natural to analyze the effect of projective change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' It is observed al- ready in [1] that projectively flat affine connections correspond to conformally flat metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' A projective-to-conformal adaptation of the standard PW construction is made precise and its various aspects are explored in the series of papers [21], [22], [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' For a projective change of an affine connection to lead to a conformal change of the induced metric, several specifications are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In particular, M is supposed to be oriented and the cotangent bundle T ∗M is replaced by its appropriately weighted variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In this case, we speak of the conformal PW metric or the conformal extension of a projective structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The conformal PW metrics were characterized in conformally invariant terms and relationships between infinitesimal symme- tries and other objects on the respective structures were obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Our aim is to extend this research to the modified case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In the present article, we study the modified conformal PW metrics or the modified conformal extensions of projective structures, which are the conformal structures that can be represented by a modified PW metric (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' To keep the equivariance of the construction, the modification tensor Φ is also to be weighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Details and preliminary observations are in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Our first result concerns the characterization of the flatness of the modified conformal extension via the flatness of the initial projective structure and certain condition on the modification tensor, see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Here we encounter the role of invariant differential operators called BGG operators, which appear repeatedly in the entire article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' For the reader’s convenience, a quick introduction and details on all projective operators used in the article are collected in appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Conformal extensions of projective structures are characterized in [23, Theorem 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Strictly speaking, the characterization concerns conformal spin structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Since we discuss local pro- perties throughout this article, we may skip the spin assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Relaxing some of conditions in [23, Theorem 1], we obtain the characterization of modified conformal extensions in The- orem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='7: Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Locally, a conformal structure of split signature is a modified conformal exten- sion of a projective structure if and only if the following properties are satisfied: (a) there is a pure twistor spinor χ with integrable kernel ker χ, (b) the following integrability condition holds Wabcd vawd = 0, for all va, wd ∈ ker χ, where Wabcd is the conformal Weyl curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Here, of course, ker χ corresponds to the vertical distribution V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The theorem can be under- stood as a conformal version of known results in the (pseudo-)Riemannian setting, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' [1] and [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Another ingredient entering the proof, as well as many other places in this article, is the existence of a metric in the conformal class whose Levi-Civita connection leaves χ parallel, see [23, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The central part of the article deals with special properties and objects on modified con- formal extensions and their relations to underlying projective counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In particular, for a modified PW metric, we provide a complete description of Einstein metrics in its conformal class and the infinitesimal conformal symmetries in section 3 and 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The descrip- tion is given in terms of geometric data on the underlying projective manifold, the main results are collected in Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Although the technicalities may seem discouraging, the scheme is clear: both for Einstein metrics and infinitesimal conformal symmetries, there are MODIFIED CONFORMAL EXTENSIONS 3 just few building blocks, satisfying certain projectively invariant conditions, from which the objects of interest are constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In the standard (non-modified) case, the blocks are well separated and the conditions consist mainly of the BGG equations, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In general, the equations are twisted by additional differential operators involving the modification term Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' This is also why the current development is in many respects different from the non-modified situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The conditions characterizing Einstein scales and infinitesimal conformal symmetries sim- plify considerably in certain special cases that are discussed in section 4 and 6, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In such cases, the otherwise intertwined conditions can be (at least partly) separated and simplified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' For the sake of illustration, we pick Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1 that describes Einstein metrics of modified conformal extensions of projectively flat underlying structures: Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Consider a modified PW metric associated to a projectively flat affine connec- tion DA and a modification tensor ΦAB ∈ E(AB)(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Locally, there is a bijective correspondence between almost Einstein scales of the conformal class and pairs τ ∈ E(1) and ξA ∈ EA(−1) satisfying (DADB + PAB)τ = 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3) (DAξB)0 = 0, ξR B2(Φ)ABCR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='4) The dimension of the space of almost Einstein scales equals to d + n + 1, where d is the dimension of the space of solutions to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='4) and n is the dimension of the underlying projective manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Here and below, PAB is the projective Schouten tensor of DA and the numbers in parentheses are the projective weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The operator B2 : E(AB)(2) → E[AB][CD](2) is the projective second BGG operator in the sequence starting with EA(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The summand n+1 in the previous count is the dimension of the solution space to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3), which is a (first) BGG equation corresponding to so-called almost Ricci-flat scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' An analogous result for infinitesimal symmetries is presented in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1: Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Consider a modified PW metric associated to a projectively flat affine connec- tion DA and a modification ΦAB ∈ E(AB)(2) such that B2(Φ) is generic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Locally, there is a bijective correspondence between its conformal Killing fields and triples vA ∈ EA, αA ∈ EA(2) and ˚ ψ ∈ R satisfying � DADBvC + PAB vC� 0 = 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='5) Lv(B2(Φ)) = ˚ ψB2(Φ), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='6) D(AαB) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='7) The dimension of the space of conformal Killing fields equals to d + 1 2n(n + 1), where d is the dimension of the space of solutions to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='5) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='6) and n is the dimension of the underlying projective manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Here Lv denotes the Lie derivative in the direction of the vector field vA and the genericity of B2(Φ) means that this field, interpreted as a bundle map EA → E[AB]C, is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The summand 1 2n(n + 1) in the previous count is the dimension of the solution space to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='7), which is a (first) BGG equation corresponding to so-called projective Killing forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Another special cases treated in sections 4 and 6 concern modified conformal extensions of 2-dimensional projective structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In particular, we describe all possible dimensions of 4 HAMMERL, SAGERSCHNIG, ˇSILHAN, ˇZ´ADN´IK the spaces of almost Einstein scales and conformal infinitesimal symmetries in Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In this context, an example with submaximal algebra of infinitesimal conformal symmetries is analyzed in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In the last section 7, we show that any modified PW metric admits a global Fefferman– Graham ambient metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In particular, the Fefferman–Graham obstruction tensor (which is the Bach tensor in dimension four) vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In this part, the modification tensor Φ enters quite innocently and the arguments are very similar to the non-modified situation studied in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The main statement, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1, is reproduced as follows: Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Let g be a modified PW metric with the Schouten tensor P and let t, ρ be the additional coordinates on the ambient manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Then g = 2ρ dt ⊙ dt + 2t dt ⊙ dρ + t2� g + 2ρ P � (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='8) is a globally Ricci-flat Fefferman–Graham ambient metric of the conformal class of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In addition, it is shown in Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2 that the Q-curvature of any modified PW metric vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The authors are grateful to Maciej Dunajski, Boris Kruglikov and Arman Taghavi-Chabert for valuable discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' JS was supported by the Czech science foundation (GAˇCR) under the grant GA19-06357S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' KS acknowledges funding received from the Norwe- gian Financial Mechanism 2014-2021, project registration number UMO-2019/34/H/ST1/00636.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Construction and characterization In this section, we add details to the projective-to-conformal analogue of the modified PW construction sketched in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' After fixing notation and other conventions, we give auxiliary comparisons of various curvature quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' This leads to a characterization of the flatness of the induced conformal structure and the structure itself, see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='5 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='7, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Preliminaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Most of the following preliminaries is taken from [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' We start with a smooth manifold M and pass to its (weighted) cotangent bundle, which we denote as � M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' We use the standard abstract index notation so that the tensorial objects on M and � M are distinguished by the type of indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' For instance, the tangent bundle TM and T � M, as well as the space of its sections, is denoted as EA and �Ea, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Symmetric and skew-symmetric tensors, as well as the corresponding operations, are denoted by round and square brackets, respectively, around indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' For instance, α(AB) ∈ E(AB) denotes the symmetrization of a bilinear form αAB ∈ EAB on M, βab = β[ab] ∈ �E[ab] denotes a bivector on � M etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' A projective structure on M is given by an equivalence class of torsion-free affine con- nections, where two connections DA and �DA are projectively equivalent if there is a 1-form ΥA ∈ EA such that �DAξB = DAξB + δB AΥCξC + ξCΥA, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1) for all ξA ∈ EA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Assuming the manifold M is oriented and a volume form on M is fixed, there is a unique connection in the projective class for which the volume form is parallel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' any such connection is called special.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Special connections are characterized by the fact that their Ricci (equivalently, Schouten) tensor is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Adapted to the projective structure on M, we MODIFIED CONFORMAL EXTENSIONS 5 use appropriate parametrizations of density bundles as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' For any w ∈ R, the density bundle of projective weight w is defined as E(w) := (∧nTM)− w n+1 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2) where dim M = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Weighted tensor bundles are denoted accordingly, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=', EA(3) stays for EA ⊗ E(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' A conformal structure on � M is given by an equivalence class of (pseudo-)Riemannian metrics, where two metrics gab and �gab are conformally equivalent if there is a function f ∈ �E such that �gab = e2fgab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Similarly to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2), the density bundle of conformal weight w on � M is defined as �E[w] := � ∧2n T � M �− w 2n , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3) where dim � M = 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The conformal structure can be seen as a section of �E(ab)[2] so that the individual metrics from the conformal class correspond to so-called scales, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' everywhere positive sections of �E[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The standard PW construction works as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Let DA be a torsion-free affine connection on M and let π : T ∗M → M be the cotangent bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Let TT ∗M = V ⊕ H be the decom- position so that V ∼= T ∗M is the vertical distribution of the projection π and H ∼= TM is the horizontal distribution determined by DA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The (standard) PW metric associated to DA is the split-signature metric �gab on � M := T ∗M given by the natural pairing between V and H and requiring that both V and H are isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Since the distribution V is parallel with respect to the corresponding Levi-Civita connection, the PW metric is a Walker metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' For a non-trivial symmetric bilinear form ΦAB ∈ E(AB) on M, let Φab ∈ �E(ab) be its pull-back to � M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The PW metric can be modified so that gab := �gab + Φab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='4) The above mentioned properties change only so that the distribution H is no more isotropic with respect to gab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The metric (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='4) is called the modified PW metric associated to the affine connection DA and the symmetric tensor ΦAB on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Note that other names for gab are used in the literature, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=', it is called the Riemann extension metric in [25] and [14], and the deformed Riemannian extension metric in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Both the standard and the modified PW metric provide an identification of the tangent and cotangent bundle, �Ea ∼= �Ea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Per default, the indices are raised and lowered with respect to the standard PW metric �gab, the use of any other metric will always be mentioned explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In particular, the inverse metric of gab has the form (g−1)ab = �gab − Φab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' A coordinate expression of a modified PW metric is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' For local coordinates (xA) on M, let Γ C A B be the Christoffel symbols of DA and let (pA) be the canonical fibre coordinates on T ∗M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Then the metric (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='4) is expressed as g = 2 dxA ⊙ dpA − 2 � Γ C A B pC + ΦAB � dxA ⊙ dxB .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='5) There is a projective-to-conformal variant of the standard PW construction provided that we pass to an appropriately weighted cotangent bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' To do so, we assume that the man- ifold M is oriented and the affine connection DA on M is special, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' preserves a volume form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' This provides a trivialization of E(w), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' an identification T ∗M(w) ∼= T ∗M, for any 6 HAMMERL, SAGERSCHNIG, ˇSILHAN, ˇZ´ADN´IK w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' By [23, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1], projectively equivalent torsion-free special connections on M in- duce conformally equivalent PW metrics on T ∗M(w) if and only if w = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Thus, adjusting accordingly the previous setup, we have the following projectively invariant definition: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The modified conformal extension or the modified conformal PW met- ric associated to an oriented projective structure on M and the symmetric weighted tensor ΦAB ∈ E(AB)(2) is the split-signature conformal structure on � M := T ∗M(2) represented by the modified PW metric (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='4) of a special torsion-free affine connection from the projective class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' To investigate various quantities associated to the standard and the modified PW metric, we primarily need the relation between corresponding covariant derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' By �Da and Da we denote the Levi-Civita connection of the standard and the modified PW metric, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' They are related as Da = �Da + Fa •, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='6) where the difference tensor Facd ∈ �Eacd is seen as a 1-form with values in the endomorphisms of T � M and • denotes the algebraic action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' This tensor can be specified as follows: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Let the standard and the modified PW metric be related by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='4) and let the corresponding covariant derivatives be related by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Then Facd = �D(aΦd) c − 1 2 �DcΦad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='7) In particular, Facd ∈ �Eacd is strictly horizontal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' it is a section of V ⊗ V ⊗ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='7) follows from the characterizing properties of the Levi-Civita con- nection, namely, from Dagbc = 0 and the torsion-freeness, by a direct calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The rest follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='7), the strict horizontality of Φab and the parallelity of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' □ For instance, for any va ∈ �Ea and αa ∈ �Ea, we have Davb = �Davb + vr �D(aΦr) b − 1 2vr �DbΦar, Daαb = �Daαb − αr �D(aΦb) r + 1 2αr �DrΦab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='8) Recall that the indices are raised with respect to the standard PW metric �gab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Curvature relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' There is a tight relation between the curvature tensors of the affine connection DA and the induced PW metric �gab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' This, for example, dictates that �gab is flat, Ricci-flat and conformally flat if and only if DA is flat, Ricci-flat and projectively flat, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' These facts easily follow from explicit relations between the initial and the induced curvatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' A particularly simple relation, which is often used below, is that the Ricci, respectively Schouten, tensor of �gab is just the pull-back of the Ricci, respectively Schouten, tensor of DA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' See [23, Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='5] and the surrounding formulas for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' We are going to relate the respective tensors associated to the standard and the modified PW metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Some of these relations can also be found in the existing literature, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' [25, Section 8] or [1, Theorem 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Nevertheless, we offer a self-contained derivation of these results and, notably, a conceptual interpretation of a condition characterizing the flatness of modified conformal extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The objects associated to �gab and gab are generally adorned by tilde and bar, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' MODIFIED CONFORMAL EXTENSIONS 7 Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Let the standard and the modified PW metric be related by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Then the corresponding Riemann, Ricci and Schouten tensors are related by Rabrd¯grc = �Rabcd − �Dc �D[aΦb]d + �Dd �D[aΦb]c − �Rabr [cΦd]r, Ricab = � Ricab, Pab = �Pab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='9) In particular, the modified PW metric is Ricci-flat if and only if the original affine connection is Ricci-flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' With the relation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='6), it follows that Rabcd = �Rabcd + 2 �D[aFb]cd since F[a • Fb]• = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Substituting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='7) and contracting with gab, the relation of Riemann tensors follows after some computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The Ricci and the Schouten relations follow easily, taking into account that Φab is strictly horizontal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The last statement follows from Ricab = � Ricab and the characterization of the Ricci-flatness of �gab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' □ The basic curvature quantity in conformal geometry is the Weyl tensor, the conformally invariant part of the Riemann tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' For the metric gab, it can be expressed as W abrd¯grc = Rabrdgrc − 4 Proj[ab][cd] � gacPbd � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='10) where Proj[· ·][· ·] denotes skew-symmetrization over the embraced indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Substituting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='4) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='9) allows to express the right-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='10) in terms of Φab and tensors associated to �gab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' This can further be rearranged using the second BGG operator in the sequence �Ea[2] � B1 −→ �E(ab)0[2] � B2 −→ �E[ab][cd]0[2] −→ · · · The first operator in this sequence is the conformal Killing operator, �B1(v)ab = �D(avb)0, which is implicitly present in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The second operator is �B2(Φ)abcd = Proj⊞0 � �Da �DcΦbd + �PacΦbd � = = Proj[ab][cd]0 � �Da �DcΦbd + �PacΦbd + 1 4 � WabrcΦdr − 1 4 � WcdraΦbr � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='11) where Proj⊞0 denotes the trace-free part of the ‘window’ symmetry corresponding to the indicated Young tableau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' A short general introduction to the BGG theory and details for relevant projective BGG operators is given in appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The role of the second BGG operator in measuring the change of the harmonic curvature (which is the Weyl curvature in conformal geometry) under deformations of parabolic geometries (which are represented by Φab in our situation) is in general described in [10, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Let the standard and the modified PW metric be related by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Then the corresponding Weyl tensors are related by W abrdgcr = � Wabcd − 2 �B2(Φ)abcd − 1 2 �� Wabr [cΦd]r + � Wcdr [aΦb]r � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='12) where �B2 is the second BGG operator (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In particular, the condition W abcd vawd = 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='13) holds for all vertical vectors va, wd ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Substituting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='4) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='9) into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='10) yields W abrd¯grc = � Wabcd − 2 Proj[ab][cd] � �Dc �D[aΦb]d � − �Rabr [cΦd]r − 4 Proj[ab][cd] � Φac�Pbd � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' 8 HAMMERL, SAGERSCHNIG, ˇSILHAN, ˇZ´ADN´IK The tensor Φab is strictly horizontal, hence it is trace-free, Φab ∈ �E(ab)0[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='11), the formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='12) follows after some computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='13) follows immediately from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='12) and � Wabcd vawd = 0, which is one of the characteristic conditions of the standard PW metric, see [23, Theorem 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' □ The conformal BGG operator (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='11) is visibly related to its projective counterpart in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Since Φab and �Pab is the pullback of ΦAB and PAB, respectively, also the tensor �B2(Φ)abcd is the pullback of B2(Φ)ABCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Altogether, we conclude with Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Let the modified PW metric gab be induced by a special affine connection DA and a modification tensor ΦAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Then gab is conformally flat if and only if DA is projectively flat and B2(Φ) = 0, where B2 : E(AB)(2) → E[AB][CD](2) is the second BGG operator (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' For the standard PW metric, the relation between the projective and the conformal Weyl tensor, W and � W, is described in [23, equation (32)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In particular, � W = 0 if and only if W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' From that relation and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='12) we see that difference tensor W − � W is strictly horizontal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Hence, vanishing of W is equivalent to vanishing of both � W and �B2(Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Since �B2(Φ) is just the pullback of B2(Φ), the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Our next aim is to provide a local characterization of split-signature conformal structures that arise as modified conformal extensions of projective structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' We have already observed that for such conformal structures the integrability condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='13) on the Weyl tensor holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Another ingredient needed for the characterization is a pure twistor spinor that emerges as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The vertical distribution V ⊂ T � M of the projection � M → M can always be represented by a pure spinor field χ such that V = ker χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In the case of standard conformal PW metrics, the spinor field can be chosen so that it satisfies the twistor spinor equation, which is a rather restrictive condition, see [23, Theorem 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In fact, for a pure twistor spinor χ with an integrable kernel, there are metrics in the conformal class for which χ is parallel and any two such metrics are related by a conformal factor constant along the leaves of ker χ, see [23, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' We will use this fact repeatedly in the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Investigating the influence of the modification gives the following Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' A modified conformal PW metric admits a pure twistor spinor χ such that ker χ = V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Let χ be the twistor spinor of the standard conformal PW metric such that ker χ = V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' As mentioned just before the Lemma, there is a scale for which χ is parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2, it follows that χ is parallel also with respect to the modified PW metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In particular, it satisfies the corresponding twistor spinor equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' □ We are now ready to locally characterize modified conformal extensions of projective struc- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The proof is based on an analogous result for modified pseudo-Riemannian extensions of affine structures due to [1] and [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Locally, a conformal structure of split signature is a modified conformal ex- tension of a projective structure if and only if the following properties are satisfied: (a) there is a pure twistor spinor χ with integrable kernel ker χ, MODIFIED CONFORMAL EXTENSIONS 9 (b) the following integrability condition holds Wabcd vawd = 0, for all va, wd ∈ ker χ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='14) where Wabcd is the conformal Weyl curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='4 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='6, a conformal spin structure that arises as a modified conformal extension satisfies conditions (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' For the converse, let a split-signature conformal spin structure on � M satisfy (a) and (b) and let M be the local leaf space of the integrable distribution ker χ ⊂ T � M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Here and below, all accounts are local and we implicitly assume that any neighborhood shrinks as needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' By [23, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2], there is a metric gab in the conformal class such that Daχ = 0, where Da is the Levi-Civita connection of gab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Consequently, the Schouten tensor Pab of gab is strictly horizontal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' From the relation among the Weyl, Riemann and Schouten tensors, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='10), it follows that the integrability condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='14) is equivalent to Rabcd vawd = 0, for all va, wd ∈ ker χ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='15) where Rabcd is the Riemann curvature of gab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The latter condition is the standard obstruction for the connection Da to descend to an affine connection DA on the local leaf space M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' It follows from [14, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='5] that a section of the projection � M → M (interpreted as a zero section) provides a local diffeomorphism � M ∼= T ∗M under which the metric gab corresponds to a modified PW metric of the affine connection DA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' From [23, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2] we also know that any two metrics leaving χ parallel are related by a conformal factor constant along the leaves of ker χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The transformation of the Levi- Civita connections under such rescaling then shows that the descended affine connections on M are projectively related, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' the proof of [23, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Any descended connection is necessarily special, since the Ricci tensor is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The corresponding volume form provides a trivialization of any density bundle so, in particular, an identification T ∗M ∼= T ∗M(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Altogether, the conformal spin structure on � M satisfying (a) and (b) determines a projective structure on the local leaf space M and, under the local identification � M ∼= T ∗M(2), its modified conformal extension corresponds to the initial conformal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Comparing the current characterization with the one for standard conformal PW metrics in [23, Theorem 1], we see that the latter are distinguished by the presence of a conformal Killing field k ∈ ker χ satisfying Lkχ = − 1 2(n + 1)χ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='16) where n is the dimension of the underlying projective manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' After a modification, the vector field k still satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='16) but need not be a conformal Killing field;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' the conformal Killing operator is related to the modification term Φ as follows: D(akb)0 = −Φab, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='17) where Da is the Levi-Civita connection of the modified PW metric gab = �gab + Φab and the trace-free part is taken with respect to gab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' These facts are related to the property (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='26) recalled below and the transformation formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The field k ∈ ker χ can be changed so that the conditions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='16) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='17) still hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Indeed, putting k′ a := ka + αa, Φ′ AB := ΦAB − D(AαB), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='18) 10 HAMMERL, SAGERSCHNIG, ˇSILHAN, ˇZ´ADN´IK where αa ∈ �Ea[2] is the pull-back of αA ∈ EA(2) with respect to the projection � M → M, the Lie derivative is Lk′χ = − 1 2(n + 1)χ, since �Daαb is strictly horizontal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Further, one easily verifies that D(ak′ b)0 = −Φ′ ab, where Φ′ ab ∈ �E(ab)[2] is the pull-back of Φ′ AB ∈ E(AB)(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The freedom in the choice of αA ∈ EA(2) corresponds to the freedom in the choice of local sections of the projection � M → M as in the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Altogether, two modified conformal PW metrics of the same projective structure whose modification tensors are related by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='18) are basically comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Note that the difference ΦAB − Φ′ AB = D(AαB) is the image of the first BGG operator (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3) on αA ∈ EA(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In particular, the current observations matches nicely with Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Further conventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In the rest of this section, we collect necessary spinor-related notions that are repeatedly used below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' This is a digest of [23, section 2], which is based on a general calculus developed in [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Let DA be a special torsion-free affine connection on M, let �gab be the induced PW metric on � M and let �S+ and �S− be the irreducible spinor bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' To distinguish, we decorate the respective sections with primed and unprimed capital indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In particular, the related gamma matrices are denoted as γ B′ a A ∈ �Ea ⊗ �S+ ⊗ �S∗ − and γ B a A′ ∈ �Ea ⊗ �S− ⊗ �S∗ +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The above mentioned pure spinor field annihilating V ⊂ T � M is written as χA′ ∈ �S+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Similarly, there is a pure spinor field ˇηA′ ∈ �S∗ + annihilating H ⊂ T � M, the horizontal distribution given by DA, such that χA′ ˇηA′ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' To the spinors χA′ and ˇηA′, we associate the sections χA a := γ A a B′ χB′ ∈ �Ea ⊗ �S−, ˇηaA := ˇηB′ γ B′ a A ∈ �Ea ⊗ �S∗ −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Besides V ∼= ker χA a and H ∼= ker ˇηaA, we can identify V ∼= im ˇηaA and H ∼= im χA a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The freedom in the choice of χA′ and ˇηA′ can be fixed by χaA �Da = ∂ ∂pA , ˇηa A �Da = ∂ ∂xA + Γ C A B pC ∂ ∂pB , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='19) where �Da is the Levi-Civita connection of �gab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The previous setup allows the following useful interpretations: the field χA a , respectively ˇηa A, is seen as the pull-back of 1-forms, respectively the horizontal lift of vector fields, from M to � M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In particular, the PW metric can be written as �gab = 2χA (aˇηb)A (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='20) and its defining properties are reflected in χA a χaB = 0 , ˇηa AˇηaB = 0 , χA a ˇηa B = δB A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='21) For later use, note also that the pull-back of the modification term ΦAB ∈ E(AB)(2) is Φab = ΦABχA a χB b (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='22) and the horizontal–vertical decomposition of a vector field va ∈ �Ea looks like va = υAˇηa A + βAχaA, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='23) where the coefficientts are interpreted as υA ∈ EA and βA ∈ EA(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' For vector fields (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='23) with coefficients υA and βA depending only on xA, the covariant derivative has the form �Davb = � DAυB� χA a ˇηb B + � DAβB − υCR D CB ApD � χA a χbB .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='24) MODIFIED CONFORMAL EXTENSIONS 11 Concerning the distinguished vertical vector field from Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='8, its standard coordinate expression is k = 2pA ∂ ∂pA .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' We will also need the following relations ka = 2pAχaA, pA = 1 2krˇηrA, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='25) where, compared with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='23), pA is interpreted as a section of EA(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In fact, ka is a conformal Killing field of satisfying �Dakb = µab + �gab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='26) where µab = �D[akb].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In particular, ka is a homothety of ˜gab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Finally, we note that the endomor- phism µab acts as plus and minus the identity on the horizintal and the vertical distribution, respectively, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=', µarχrA = −χaA, µarˇηrA = ˇηaA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Einstein metrics The aim of this section is to characterize Einstein representative metrics of modified con- formal extensions in terms of underlying projective data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' This section culminates in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Contrary to the non-modified situation, Einstein metrics in the conformal class are not necessarily Ricci-flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' An almost Einstein scale of a conformal structure is a scale σ ∈ �E[1] such that the corre- sponding metric in the conformal class is Einstein off the zero set of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' For conformal class represented by a metric gab, the scale σ is almost Einstein if and only if the trace-free part of (DaDb +Pab)σ with respect to gab vanishes, where Da and Pab are the Levi-Civita connection and the Schouten tensor of gab, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' If we consider a modified conformal extension represented by gab = �gab +Φab, this condition can be also written in terms of the non-modified PW metric �gab as � �Da �Db + �Pab � σ = ψ(�gab + Φab), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1) for some ψ ∈ �E[−1] that need not be specified for now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In order to characterize solutions of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1) in underlying projective terms, we employ the first projective BGG operators from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Solutions of the former BGG equation are the scales τ ∈ E(1) that determine Ricci-flat affine connections from the projective class, solutions of the latter BGG equation are the weighted vector fields ξA ∈ EA(−1) satisfying DAξB − 1 nδABDRξR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2) As a consequent condition we get WABCRξR = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3) cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Note that the prolongation of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2) gives D(ADB)ξR + δR (A PB)S ξS = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='4) cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' [23, eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (54)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Moreover, we shall need the bilinear differential operator F : EA(−1) × E(AB)(2) → E(AB)(1) given by F(ξ, Φ)AB = ξR� D(AΦB)R − 1 2DRΦAB � + 1 n(DRξR)ΦAB, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='5) which, indeed, is projectively invariant too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' 12 HAMMERL, SAGERSCHNIG, ˇSILHAN, ˇZ´ADN´IK Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' A section σ ∈ �E[1] is an almost Einstein scale of the modified conformal extension represented by gab = �gab + Φab if and only if σ is linear in pA, σ = ξRpR + τ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='6) and the underlying sections τ ∈ E(1) and ξA ∈ EA(−1) satisfy the integrability condition ξRWRBCD = 0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='7) and the differential conditions � DAξB� 0 = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='8) (DADB + PAB) τ = F(ξ, Φ)AB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='9) Note that, for ΦAB = 0, the right-hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='9) vanishes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=', both τ and ξA are solutions of the corresponding BGG equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Let σ ∈ �E[1] be an almost Einstein scale, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=', the equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1) be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Both gab = �gab +Φab and �Pab vanish when contracting with two vertical vectors, hence the previous assumption implies χaAχbB �Da �Dbσ = ∂2 ∂pA∂pB σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Thus, σ is a linear polynomial in pA, for which we fix the notation as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' To express the Einstein scale equation in terms of τ ∈ E(1) and ξA ∈ EA(−1), one has to substitute (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='6) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1) and expand according to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='24), taking into account (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Considering the decomposition of the left-hand side as � �Da �Db + �Pab � σ = Θ′ A Bχ(a Aˇηb)B + Θ′′ ABχ(a Aχb) B, direct computation reveals that Θ′ A B = 2DAξB, Θ′′ AB = � D(ADB)ξR + δR (A PB)S ξS − ξSWS(A R B) � pR + + (DADB + PAB)τ − ξR(D(AΦB)R − 1 2DRΦAB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Recasting the right-hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1) according to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='20) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='22), the Einstein scale equa- tion is written as � Θ′ A B − 2ψδAB� χ(a Aˇηb)B + � Θ′′ AB − ψΦAB � χ(a Aχb) B = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='10) where ψ = 1 nDRξR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The two summands in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='10) must vanish separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Taking into account the fact that all expressions are polynomial in pA, the first condition Θ′ A B − 2ψδAB = 0 is equivalent to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='8) and the second condition Θ′′ AB − ψΦAB = 0 is equivalent to the pair D(ADB)ξR + δR (A PB)S ξS − ξSWS(A R B) = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='11) (DADB + PAB)τ − ξR(D(AΦB)R − 1 2DRΦAB) − 1 n(DRξR)ΦAB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='12) From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='11) and the condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='4), which is a consequence of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='8), we conclude that ξSWS(ARB) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' From the symmetries of Weyl tensor and the condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3), which is another consequence of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='8), we conclude that ξSWS[ARB] = − 1 2WABRSξS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Thus, the integrability condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='7) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Conversely, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='7) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='8) clearly imply (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' With the notation from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='5), the equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='12) is just (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' □ MODIFIED CONFORMAL EXTENSIONS 13 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1 establishes a bijective correspondence between almost Einstein scales and pairs τ ∈ E(1) and ξA ∈ EA(−1) satisfying certain projectively invariant conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The following theorem shows that the components can also be identified in conformal terms, and we have full control over the Ricci-flatness of rescaled metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' There is a bijective correspondence between almost Einstein scales of the modified conformal extension and pairs τ ∈ E(1) and ξA ∈ EA(−1) satisfying conditions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='7)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' More precisely, an almost Einstein scale σ ∈ �E[1] can be uniquely decomposed as σ = σ+ + σ−, where Lkσ+ = σ+ and Lkσ− = −σ−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' These components correspond to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='6) as ξA = χaA �Daσ+, τ = σ−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='13) Moreover, the scalar curvature of the rescaled metric corresponding to σ is 2n(2n − 1)ΦRS ξRξS, off its zero set, where n is the dimension of the underlying projective manifold and ΦAB ∈ EAB(2) is the modification tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' For the decomposition of an almost Einstein scale as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='6), let σ+ := ξRpR and σ− := τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The properties Lkσ+ = σ+ and Lkσ− = −σ− follow by the same argument as in [23, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2] (the degree of homogeneity with respect to pA plays a key role there).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The expressions in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='13) are clear, recalling that χaA �Da = ∂ ∂pA .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The remaining part is based on a similar reasoning as in [23, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3] with regard to our current setting as in the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The scalar curvature of a metric is proportional to the trace of its Schouten tensor with the constant factor 2(2n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' This follows from definitions which, together with transformation formulas reflecting a change of scale, can be found e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Let gab = �gab + Φab be the modified PW metric and P := (g−1)rsPrs be the trace of the corresponding Schouten tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Analogous quantities corresponding to an Einstein scale σ are denoted as �gab and �P, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The key relation is �P = P − (g−1)rs(DrΥs + (n − 1)ΥrΥs), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='14) where Υa = −σ−1Daσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Since Pab is strictly horizontal, P vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Similarly, for the decom- position σ = σ+ + σ− as above, the component σ− does not contribute to the trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Hence we may consider just σ = σ+ = ξRpR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' To compute the differential, we use (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='25), expand according to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='24) and substitute (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2), which yields Daσ+ = 1 2n(DRξR)ka + ξRˇηaR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The trace of the derivative of Υa simplifies as (g−1)rsDrΥs = (g−1)rsΥrΥs − 2σ−1 + DRξR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Putting things together, one verifies that the divergence terms vanish and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='14) reduces to �P = n ΦRS ξRξS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Hence the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' □ We conclude this section with an additional relation that will be helpful in the next section where we discuss some special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' It is a consequence of the conditions from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1: 14 HAMMERL, SAGERSCHNIG, ˇSILHAN, ˇZ´ADN´IK Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' If τ ∈ E(1), ξA ∈ EA(−1) and ΦAB ∈ E(AB)(2) satisfy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='7)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='9) then � WABRCDR − YCAB � τ = ξR � 3 4WABSCΦRS − 2B2(Φ)ABCR � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='15) where B2 : E(AB)(2) → E[AB][CD](2) is the second BGG operator (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Applying the second BGG operator (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1) to the left-hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='9) yields − 1 2 � WABRCDR − YCAB � τ, where YCAB = 2D[A PB]C is the Cotton tensor, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Applying the same operator to the right-hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='9), a tedious computation leads to ξR� B2(Φ)ABCR − 3 8WABSCΦRS + 1 4WCRS [AΦB]S � , where B2 is the BGG operator (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3), more explicitly described in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Here one has to take into account that ξA satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='8) and, consequently, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The previous two displays together with the integrability condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='7) give the stated result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Einstein metrics: special cases The conditions from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1 significantly simplify in certain special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' We dis- cuss the case of modified conformal extensions of projectively flat structures, the case when the modification term ΦAB is in the image of the first BGG operator and the lowest dimen- sional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In all these cases, we are able to untangle the interrelations between the source sections τ and ξA, relative to the modification term ΦAB, and specify the dimension of the space of almost Einstein scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Projectively flat case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In this case, many curvature related objects disappear which leads to significant simplifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Primarily, both the Weyl and the Cotton tensor vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In particular, the condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='7) is satisfied trivially and there is only one term in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='15) that survives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Also, in the flat case any BGG sequence is a complex and, working locally, it is actually exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' These facts lead to the following splitting of the characterizing conditions: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Consider a modified PW metric associated to a projectively flat affine connec- tion DA and a modification tensor ΦAB ∈ E(AB)(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Locally, there is a bijective correspondence between almost Einstein scales of the conformal class and pairs τ ∈ E(1) and ξA ∈ EA(−1) satisfying (DADB + PAB)τ = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1) (DAξB)0 = 0, ξR B2(Φ)ABCR = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2) where B2 : E(AB)(2) → E[AB][CD](2) is the second BGG operator (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The dimension of the space of almost Einstein scales equals to d + n + 1, where d is the dimension of the space of solutions to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2) and n is the dimension of the underlying projective manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Only the conditions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='8) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='9) from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1 are relevant in our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Applying the second BGG operator to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='9) yields 0 = ξR B2(Φ)ABCR, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Locally, by the exactness of the BGG sequence, the right-hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='9) is in the image of the first BGG operator on a section of E(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' This guarantees the existence of τ ∈ E(1) satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='9) and all such sections are parametrized by solutions to the equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' However, in the flat case, solutions to the later equation allow a coordinate expression τ = cAxA + c0, where c0, c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' , cn are arbitrary constants, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' □ MODIFIED CONFORMAL EXTENSIONS 15 If the tensor field B2(Φ) is generic, the dimension of the space of almost Einstein scales equals to n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Here the genericity means that the field, interpreted as a bundle map EA → E[AB]C, is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Special modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Here we assume that ΦAB = D(AϕB), for some ϕA ∈ EA(2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' ΦAB is in the image of the first BGG operator (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' With regard to Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='8, the characterization of almost Einstein scales has to correspond to the one for standard conformal extensions as in [23, Theorem 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' To keep the presentation self-contained, we derive the characterization directly from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The effect of ΦAB is so that the corresponding system of equations, which is homogeneous in the standard case, becomes non-homogeneous and one seeks for a particular solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In particular, the space of almost Einstein scales of a modified conformal extension of the current type is an affine space over the vector space of almost Einstein scales of its non-modified companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Let the standard PW metric be modified by the term of the form ΦAB = D(AϕB), for some ϕA ∈ EA(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' There is a bijective correspondence between almost Einstein scales of the conformal class and pairs τ ∈ E(1) and ξA ∈ EA(−1) satisfying (DADB + PAB)τ = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3) ξRWRACB = 0, (DAξB)0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='4) The dimension of the space of almost Einstein scales equals to d1 +d2, where d1 and d2 is the dimension of the space of solutions to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='4), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' A straightforward computation reveals that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='5) has the form F(ξ, Φ)AB = 1 2(DADB + PAB)(ξRϕR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='5) Thus, τ = 1 2ξRϕR is a particular solution to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='9) and all such solutions are parametrized by solutions to the equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='4) are just the remaining conditions from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' □ Note that, compared with the standard extension, the modification tensor enters the game so that the rescaled metrics need not be Ricci-flat, see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Dimension four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Here we examine 4-dimensional modified conformal extensions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' those of 2-dimensional projective structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In this case, the Weyl tensor WABCD vanishes automatically and the key curvature invariant is the Cotton tensor YCAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' This simplifies the condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='15), namely, YCAB τ = 2ξR B2(Φ)ABCR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='6) Also, B2(Φ) is a section of the bundle E[AB][CD](2) which is—in this dimension—a density bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Using the projective volume form ǫAB ∈ E[AB](3), respectively its inverse ǫAB ∈ E[AB](−3), it is identified with E(−4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In particular, equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='6) can be rewritten as (⋆Y )A τ = 2ξA(⋆B2(Φ)), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='7) where (⋆Y )A := YCDEǫACǫDE ∈ EA(−6) and ⋆B2(Φ) := B2(Φ)ABCDǫABǫCD ∈ E(−4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' For modified conformal extensions of 2-dimensional projective structures, the dimension of the space of almost Einstein scales is 0, 1, 3 or 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' 16 HAMMERL, SAGERSCHNIG, ˇSILHAN, ˇZ´ADN´IK Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Let the dimension of the space of almost Einstein scales be denoted by d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Whenever we consider sections that do not vanish identically, we restrict off their zero sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' For non-flat projective structures, the solution space to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='8) is at most 1-dimensional;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' this follows from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='5) and the neighbouring discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Indeed, we have EA(−1) ∼= EA(2) thus the equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='8) corresponds to the first BGG equation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3), for which (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='5) gives the related integrability condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Any solution ξA ∈ EA(−1) determines τ ∈ E(1) uniquely via the condition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Such a pair defines an almost Einstein scale if and only if it satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Altogether, we conclude that d ≤ 1 in such cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' For flat projective structures and conformal extensions with B2(Φ) ̸= 0, it follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='7) that ξA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The conditions from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1 are reduced to the single equation (DADB + PAB)τ = 0, whose solution space is 3-dimensional, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Thus, d = 3 in such cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' For flat projective structures and conformal extensions with B2(Φ) = 0, the induced con- formal structure is flat, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In such cases, the dimension d is maximal possible, which is well known to be d = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' □ All values listed in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3 are realizable: (0) For conformal extensions of generic projective structures there are no almost Einstein scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (1) For the projective structure given by the Levi-Civita connection of a generic surface of revolution, let us consider its standard (non-modified) conformal extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The Killing field generating the rotation gives rise to the vector field satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='8), see [23, section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1], and this is the only source for almost Einstein scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (3) A generic modified conformal extension of flat projective structure has 3-dimensional space of almost Einstein scales;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' a particular example of this type is discussed in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (6) The standard conformal extension of the flat projective structure has 6-dimensional space of almost Einstein scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Symmetries In this section, we characterize infinitesimal conformal symmetries of a modified PW metric in terms of underlying projective data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Before we come to the main statement in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3, we need several preparatory observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' As the first step, we express the conformal Killing equation and its prolongation for the modified PW metric in terms of the original (non-modified) one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' For a conformal Killing field va of the modified PW metric gab = �gab + Φab, the trace-free part of D(avb) with respect to gab vanishes, where Da is the Levi-Civita connection of gab and vb = vb + vrΦrb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Compactly written, D(avb) − ψgab = 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1) where ψ is the divergence that need not be specified at the moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' An expansion of this condition in the sense of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2 gives the needed expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In comparison with the non-modified case, an extra term (ωab) and its further derivatives (ω′ abc and ω′′ ab) are contained in the final formulas: MODIFIED CONFORMAL EXTENSIONS 17 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Let va ∈ �Ea be a conformal Killing field of the modified PW metric gab = �gab + Φab and let us decompose �Davb = µab + ψ�gab + ωab, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2) where µab = �D[avb], ψ = 1 2n �Drvr and ωab = �D(avb)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Then: (i) The symmetric trace-free part of �Davb is ωab = − 1 2(LvΦ)ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3) In particular, it is strictly horizontal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=', χaAωab = χbBωab = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (ii) Differential consequences of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2) are �Daψ = �Parvr − βa, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='4) �Daµbc = −2�ga[bβc] − 2�Pa[bvc] − � Wbcarvr + ω′ abc, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='5) �Daβb = −�Yabrvr − ψ�Pab + �Parµrb + ω′′ ab, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='6) for some βa ∈ �Ea, ω′ abc ∈ �Ea[bc][2] and ω′′ ab ∈ �Eab satisfying χaAω′ abc = 0 and χbBω′′ ab = 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (i) On the one hand, it follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='8) that, for vb = vb + vrΦrb, Davb = �Davb − vr �D(aΦb) r + 1 2vr �DrΦab + ( �Davr)Φbr + vr �DaΦbr = = �Davb + 1 2vr �DrΦab + ( �Davr)Φbr + vr �D[aΦb] r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The conformal Killing equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1) then reads as �D(avb) + 1 2vr �DrΦab + ( �D(avr)Φb)r − ψ(�gab + Φab) = 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='7) where ψ = 1 2n �Drvr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' On the other hand, the Lie derivative of Φab ∈ �Eab[2] in the direction of va is (LvΦ)ab = vr �DrΦab + 2( �D(avr)Φb)r − 1 n( �Drvr)Φab (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='8) (where the last coefficient reflects the conventions from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Thus, the equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='7) can be written as �D(avb) + 1 2(LvΦ)ab − ψ�gab = 0, which gives (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Since Φab is strictly horizontal and the flow of va preserves the vertical distribution, the rest follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (ii) To derive the differential consequences of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2), we shall mimic the computation of standard prolonged systems, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='4) is read as the definition of βa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Applying �Dc to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2), commuting covariant derivatives on the left-hand side and skewing over b and c, the equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='5) follows after some manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' From the computation it further follows that ω′ abc = 2 �D[bωc]a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Since ωab is strictly horizontal, the property χaAω′ abc = 0 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Further, applying �gab to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='5) yields �Dr�µra = −(2n − 1)�βa + �Parvr + �Drωra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='9) Finally, applying �Dc to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='5), commuting covariant derivatives and using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='9), the equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='6) follows after a tedious but straightforward computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' From the computation it further 18 HAMMERL, SAGERSCHNIG, ˇSILHAN, ˇZ´ADN´IK follows that ω′′ ab is a linear combination of �Dr �Drωab and �Da �Drωbr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In particular, the property χbBω′′ ab = 0 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' □ As the next step, to express infinitesimal conformal symmetries in underlying projective terms, we shall need several projectively invariant operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Primarily, we employ the first BGG operators from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='8) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3) and the operator (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The corresponding equations, respectively their solutions, are expounded as follows: The equation associated to (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='8) is equivalent to the system DAwBC − 2δA[BνC] = 0, where νC = 1 n−1DRwRC, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='10) DAνB + PAR wRB + 1 2(n−2)wRSWRSBA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='11) Solutions associated to (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='11) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3) are the infinitesimal projective symmetries and the so-called projective Killing forms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Moreover, we shall need the bilinear differential operator F : E[CD](−2) × E(AB)(2) → EC(AB) given by F(w, Φ)C AB = wRC� D(AΦB)R − 1 2DRΦAB � + νCΦAB, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='12) where νC is as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' One can check it is projectively invariant too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Analogously to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='8), we also have the Lie derivative on sections of E(AB)(2), (LvΦ)AB = vRDRΦAB + 2(D(AvR)ΦB)R − 2 n+1(DRvR)ΦAB, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='13) where vR ∈ ER (the last coefficient reflects the conventions from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' We are ready to characterize infinitesimal conformal symmetries of a modified PW metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The description is based on the horizontal–vertical decomposition of conformal vector fields whose components are identified with projectively invariant objects: Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Let va = υAˇηa A+βAχaA be the decomposition of a vector field with respect to �gab as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Then va is a conformal Killing field of the modified PW metric gab = �gab+Φab if and only if υA and βA are polynomials in pA, υA = wABpB + vA, βA = ψABCpBpC + ϕABpB + αA, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='14) and the underlying sections wAB ∈ EAB(−2), vA ∈ EA, ψABC ∈ EA(BC)(−2), ϕAB ∈ EAB, αA ∈ EA(2) satisfy the algebraic condition w(AB) = 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='15) the integrability condition wR(CWR(A D) B) = 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='16) the differential conditions � DAwBC� 0 = 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='17) � DADBvC + PAB vC + vRWR(A C B) � 0 = − � F(w, Φ)C AB � 0 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='18) D(AαB) = − 1 2(LvΦ)AB + 1 2 ˚ ψ ΦAB, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='19) for some ˚ ψ ∈ E, such that DA˚ ψ = 2 n+1 F(w, Φ)RAR, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='20) MODIFIED CONFORMAL EXTENSIONS 19 where νC = 1 n−1DRwRC as in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='10), and the remaining coefficients are given by ψABC = δA(BνC), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='21) ϕAB = − � DAvB + wSBΦSA � 0 + � n−1 n(n+1)(DSvS) + ˚ ψ � δAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='22) Note that, for ΦAB = 0, the right-hand sides of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='18)–(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='20) vanish, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=', each of wAB, vA, αA is a solution to the corresponding BGG equation and ˚ ψ is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Further comments relating the previous description to the non-modified case are at the end of section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Let va = υAˇηa A + βAχaA be a vector field of the form (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' We start by expressing the conformal Killing equation of gab = �gab + Φab in terms of the underlying objects wAB, ψABC, ϕAB and αA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The proper weights declared in the statement follow from the discussion around (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='25) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='23) and need not be emphasized below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' From the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1 we know that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1) is equivalent to D(avb) = �D(avb) + 1 2(LvΦ)ab + ψΦab, where vb = vb +vrΦrb and ψ = 1 2n �Drvr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Now, one has to substitute (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='14), respectively (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='22), and expand according to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='24), taking into account (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Considering the decomposition of the form D(a¯vb) = ΘAB ˇη(a|A|ˇηb)B + Θ′ A B χ(a Aˇηb)B + Θ′′ AB χ(a Aχb) B, direct, albeit rather lengthy, computation reveals that ΘAB = w(AB), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='23) Θ′ A B = � DAwBR + 2ψABR� pR + � DAvB + wSBΦSA + ϕAB� , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='24) Θ′′ AB = � D(AψB) RS − wT(RWT(A S) B) + wT(R PT(A δS) B) � pRpS+ + � D(AϕB) R + D(A(wSRΦB)S) − vSWS(A R B) − vR PAB + + vS PS(A δR B) − wSRD(AΦB)S + 1 2wSRDSΦAB � pR + + � D(AαB) + (D(AvR)ΦB)R + 1 2vSDSΦAB � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='25) With the reference to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='20) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='22), the conformal Killing equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1) reads as ΘAB ˇη(a|A|ˇηb)B + � Θ′ A B − 2ψδB A � χ(a Aˇηb)B + � Θ′′ AB − ψΦAB � χ(a Aχb) B = 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='26) where ψ = 1 2n � DSwSR + 2ψSSR� pR + 1 2n � DSvS + ϕSS� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='27) The three summands in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='26) must vanish separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Taking into account the fact that the expressions (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='23)–(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='25) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='27) are polynomial in pA, the first condition ΘAB = 0 is w(AB) = 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='28) the second condition Θ′ A B − 2ψδB A = 0 is equivalent to the pair DAwBR + 2ψABR − 1 nδB A � DSwSR + 2ψSSR� = 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='29) � DAvB + wSBΦSA + ϕAB� 0 = 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='30) 20 HAMMERL, SAGERSCHNIG, ˇSILHAN, ˇZ´ADN´IK and the third condition Θ′′ AB − ψΦAB = 0 is equivalent to the triple D(AψB) RS − wT(RWT(A S) B) + wT(R PT(A δS) B) = 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='31) D(AϕB) R + D(A(wSRΦB)S) − vSWS(A R B) − vR PAB + + vS PS(A δR B) − wSRD(AΦB)S + 1 2wSRDSΦAB − 1 2n � DSwSR + 2ψSSR� = 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='32) D(AαB) + (D(AvR)ΦB)R + 1 2vSDSΦAB − 1 2n � DSvS + ϕSS� ΦAB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='33) The system (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='28)–(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='33) provides the desired characterization of conformal Killing fields of the form (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='14) in purely underlying terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Now we analyze individual conditions in detail: The condition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='28) is just (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='15), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=', wAB is a bivector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Skew-symmetrization of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='29) over B and R gives (DAwBR)0 = 0, which is just the condition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Symmetrization of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='29) over B and R gives (ψABR)0 = 0, which means that ψABC = δA(BνC) for some νC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Using this notation and taking the trace of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='29) give DRwRA = (n − 1)νA, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='34) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=', νC is as stated and we get the condition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='30) dictates the trace-free part of ϕAB, while its trace is undetermined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' It will be convenient to express it as ϕRR = n−1 n+1DRvR + n˚ ψ, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='35) for some function ˚ ψ ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Thus, we have the equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' With (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='21), the equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='31) is rewritten as δ(A (RDB)νS) − wT(RWT(A S) B) + wT(R PT(A δS) B) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='36) In particular, the trace-free part of wT(RWT(AS)B) vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Taking the full trace of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='36) and comparing with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='11), which is a consequence of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='17), gives wTRWTRBA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='37) Thus, wT(RWT(AS)B) = 0, which is just the integrability condition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' With (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='21) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='34), the equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='32) is rewritten as D(AϕB) R + D(A(wSRΦB)S) − vSWS(A R B) − vR PAB + + vS PS(A δR B) − wSRD(AΦB)S + 1 2wSRDSΦAB − νRΦAB = 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='38) The trace-free part of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='38) is built from operators (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='11) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='12) as stated in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='18), whereas the trace of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='38) leads to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' For the later claim, one has to use the following consequence of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='22), respectively (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='35), D(AϕR) R = 1 2(n + 1)DA˚ ψ − 1 2(n − 1) PAR vR − 1 2DR(wSRΦSR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' With (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='34) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='35), the equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='33) is rewritten as D(AαB) + (D(AvR)ΦB)R + 1 2vSDSΦAB − 1 n+1(DSvS)ΦAB − 1 2 ˚ ψΦAB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='39) Referring to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='13), this is just the equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Altogether, we have derived all the conditions (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='15)–(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='22) in the statement from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='28)– (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Retracing, respectively adapting, the previous account, it is easy to proceed in the opposite direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Hence the two systems are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' MODIFIED CONFORMAL EXTENSIONS 21 To finish the proof, we have to show that any conformal Killing field of the modified PW metric has the form (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' We shall mimic part of the computation from the proof of [23, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='5], taking into account the current modifications gathered in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Let the scale be chosen so that the twistor spinor χ is parallel with respect to �D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Differentiating (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2) and substituting (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='4) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='5) give �Da �Dbvc = −2�ga[bβc] − 2�Pa[bvc] − � Wbcarvr + ω′ abc + �Parvr�gbc − βa�gbc + �Daωbc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' After vertical contractions we get χaAχbB �Da �Dbvc = 2χaAχbB�gc[aβb].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='40) In particular, another contraction gives χaAχbBχcC �Da �Dbvc = ∂2 ∂pA∂pB υC = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=', υC is a linear polynomial in pA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Further differentiation of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='40), substitution of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='6) and the vertical contraction yield χaAχbBχcC �Da �Db �Dcvd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In particular, we have χaAχbBχcC ˇηd D �Da �Db �Dcvd = ∂3 ∂pA∂pB∂pC βD = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=', βD is a polynomial of second order in pA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' □ The vector field va = υAˇηa A + βAχaA of the form (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='14) can alternatively be written as va = va + + va 0 + va −, where va + = wABpB ˇηa A + ψABCpBpCχaA , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='41) va 0 = vAˇηa A + ϕABpBχaA , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='42) va − = αAχaA .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='43) The key coefficients are wAB, vA, αA and ˚ ψ, whereas the remaining two, ψABC and ϕAB, are determined by the others via (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='21) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Though the two prescriptions are not pro- jectively invariant, formulas (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='42) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='41) provide well-defined lifts of underlying objects, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=', the vector fields are indeed independent of the choice of affine connection in the projective class (the invariance of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='43) is clear).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' For the non-modified case, the independence is shown in [23, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1], respectively [23, Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The modification ΦAB enters only the component va 0, according to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='22), which clearly does not spoil the projective invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' We are now prepared to state the main theorem of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' There is a bijective correspondence between conformal Killing fields of the modified PW metric and quadruples wAB ∈ E[AB](−2), vA ∈ EA, αA ∈ EA(2), ˚ ψ ∈ E (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='44) satisfying the conditions (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='16)–(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Locally, the last condition is equivalent to wRS� D[AD|RΦS|B] + P[A|R ΦS|B] � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='45) More precisely, a conformal Killing field va ∈ �Ea can be uniquely decomposed as va = va + + va 0 + va −, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='46) 22 HAMMERL, SAGERSCHNIG, ˇSILHAN, ˇZ´ADN´IK where Lkva + = 2va +, Lkva − = −2va − and Lkva 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The components can be expressed as the lifts (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='41)–(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='43), where the essential coefficients are wAB = χaAχB b �Davb +, vA = χA b vb 0, αA = ˇηaAva −, ˚ ψ = 1 n+1 � 1 n �Dbvb 0 − µab �Davb 0 � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='47) the remaining ones being given by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='21) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The first part of the statement is clear from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The condition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='20) means that the 1-form F(w, Φ)RAR = νRΦAR − wRSDRΦSA (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='48) is exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Locally, it is equivalent to its closedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Putting the differential of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='48) equal to zero and applying (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='10), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='11), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='37) lead to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Given the decomposition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='46) along (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='41)–(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='43) of a vector field va = υAˇηa A + βAχaA of the form (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='14), the properties Lkva + = 2va +, Lkva − = −2va − and Lkva 0 = 0 follow from [23, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (Note that just the degree of homogeneity with respect to pA plays a role here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=') The expressions of vA, αA and wAB in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='47) are clear, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' the omnipresent relations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='21) and χaA �Da = ∂ ∂pA .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The expression for ˚ ψ can be verified by a direct computation using the form of va 0 from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Specifically, one computes that �Dbvb 0 = � 1 + n−1 n+1 � DRvR + n˚ ψ and µab �Davb 0 = DRvR − ϕRR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Hence the formula for ˚ ψ follows using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' □ For ΦAB = 0, the function ˚ ψ has to be constant and we can single out the distinguished vector field ka from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' This—as well as each of the other components—is a conformal Killing field and we recover [23, Theorem 3], respectively [23, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In particular, each component is given solely by one of the sections from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' For general modifications, none of the components of the decomposition above has to be a conformal Killing field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Also, considering the sources (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='44) individually, the corresponding lifts may, but need not, be conformal Killing fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' On the one hand, for a projective Killing form αA, respectively an infinitesimal projective symmetry vA satisfying LvΦ = 0, the lift has the form va = va −, respectively va = va 0, and is an infinitesimal conformal symmetry, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' conditions (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='18) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' On the other hand, a bivector wAB such that the right-hand side of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='18), respectively (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='20), does not vanish cannot lift to an infinitesimal symmetry without a counterbalancing influence of vA, respectively ˚ ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Symmetries: special cases The conditions from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2 significantly simplify in some special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' As in section 4, we focus on the case of modified conformal extensions of projectively flat structures, the case when the modification term ΦAB is in the image of the first BGG operator and the lowest dimensional case, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In all these cases, we attempt to specify the dimension of the algebra of infinitesimal symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Contrary to section 4, the details are getting more technical, which is why we combine several specifications in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In any case, we are still able to obtain interesting partial results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In this context, we also discuss an example with submaximal algebra of conformal Killing fields in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' MODIFIED CONFORMAL EXTENSIONS 23 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Projectively flat case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' As in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1, we explore some consequences of projective flatness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The simplifications are obtained by applying second BGG operators to particular conditions in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2 and local exactness of BGG complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' However, we have to manage more involved conditions than in the case of almost Einstein scales, which leads us to impose an additional genericity assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Consider a modified PW metric associated to a projectively flat affine con- nection DA and a modification ΦAB such that B2(Φ) is generic, where B2 : E(AB)(2) → E[AB][CD](2) is the second BGG operator (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Locally, there is a bijective correspondence between its conformal Killing fields and triples vA ∈ EA, αA ∈ EA(2) and ˚ ψ ∈ R satisfying � DADBvC + PAB vC� 0 = 0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1) Lv(B2(Φ)) = ˚ ψB2(Φ), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2) D(AαB) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3) The dimension of the space of conformal Killing fields equals to d + 1 2n(n + 1), where d is the dimension of the space of solutions to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2) and n is the dimension of the underlying projective manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The genericity of B2(Φ) means that this field, interpreted as a bundle map EA → E[AB]C, is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Recall the condition (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1) indicates that vA is an infinitesimal projective symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The key conditions to control are (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='17)–(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Applying the second BGG operator (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='10) to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='18) yields 0 = � wCR B2(Φ)ABRD � 0, where B2 is as in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Here one has to take into account that wAB satisfies (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='17) or, equivalently, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='10)–(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In the flat case, the condition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='45), which is a consequence of the key ones, means wRSB2(Φ)ARSD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Altogether, we have wCR B2(Φ)ABRD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' For generic B2(Φ), this implies wAB = 0, hence the conditions (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='18) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='20) reduces to � DADBvC + PAB vC� 0 = 0, DA˚ ψ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Applying the second BGG operator (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3) to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='19) yields 0 = −Lv(B2(Φ)) + ˚ ψB2(Φ), where we have used that B2 commutes with Lv and ˚ ψ is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Locally, by the exactness of the BGG sequence, the right-hand side of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='19) is the image of the first BGG operator on a section of EA(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' This guarantees the existence of αA ∈ EA(2) satisfying (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='19) and all such sections are parametrized by solutions to the equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' However, in the flat case, the solution space has dimension 1 2n(n + 1), see section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Special modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Here we assume that ΦAB = D(AϕB), for some ϕA ∈ EA(2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' ΦAB is in the image of the first BGG operator (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' As discussed at the beginning of section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2, this case reduces to the standard conformal extension whose infinitesimal symmetries are characterized in [23, Theorem 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Again, we derive the characterization directly from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2, observe the effect of ΦAB and note that the space of conformal Killing fields of a modified conformal extension of the current type is an affine space over the vector space in the non-modified situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' 24 HAMMERL, SAGERSCHNIG, ˇSILHAN, ˇZ´ADN´IK Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Let the standard PW metric be modified by the term of the form ΦAB = D(AϕB), for some ϕA ∈ EA(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' There is a bijective correspondence between its conformal Killing fields and triples wAB ∈ E[AB](−2), vA ∈ EA and αA ∈ EA(2) satisfying � DCwAB� 0 = 0, wR(CWR(A D) B) = 0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='4) � D(ADB)vC + PAB vC + vSWS(A C B) � 0 = 0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='5) D(AαB) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='6) The dimension of the space of conformal Killing fields equals to d1 + d2 + d3 + 1, where d1, d2 and d3 is the dimension of the space of solutions to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='4), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='5) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='6), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' A direct computation shows that the bilinear operator from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='12) has the form F(w, Φ)CAB = 1 2wRC� D(ADB)ϕR + PAB ϕR − PR(A ϕB) + WR(A S B)ϕS � + νCD(AϕB), where νA = 1 n−1DRwRA as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Its trace-free and trace part, which appear on the right- hand side of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='18) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='20), respectively, are � F(w, Φ)C AB � 0 = 1 2 � D(ADB)(wRCϕR) + PAB(wRCϕR) + (wRSϕR)WS(A C B) � 0, F(w, Φ)RAR = 2DA � wRSDRϕS − 2νRϕR � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Thus, vA = − 1 2wRAϕR is a particular solution to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='18) and all such solutions are parametrized by solutions to the equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Similarly, ˚ ψ = 4 n+1(wRSDRϕS − 2νRϕR) is a particular solution to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='20) and all such solutions differ by an additive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Using the previous displays, the right-hand side of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='19) can be arranged as − 1 2D(A(Lvϕ)B) − 1 2 � F(w, Φ)RAB � ϕR + 1 2ϕ(ADB)˚ ψ + 1 2 ˚ ψD(AϕB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The problematic term � F(w, Φ)RAB � ϕR actually equals to D(AγB), where γB = 1 2(wRSDBϕR+ νSϕB)ϕS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Thus, αB = − 1 2(Lvϕ)B − 1 2γB + 1 2 ˚ ψϕB is a particular solution to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='19) and all such solutions are parametrized by solutions to the equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Equations in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='4) are just the remaining conditions from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Dimension four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Analogously to section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3, we inspect specific features of the lowest dimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In particular, the projective flatness is controlled by the vanishing of Cotton tensor YCAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Indices in computations below are lowered and raised via the projective volume form ǫAB ∈ E[AB](3) and its inverse ǫAB ∈ E[AB](−3), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' As the first result we show that, for non-flat extensions, one of the four building blocks necessarily vanishes: Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' For a non-flat modified conformal extension of a 2-dimensional projective structure, there is a bijective correspondence between its conformal Killing fields and triples vA ∈ EA, αA ∈ EA(2) and ˚ ψ ∈ R satisfying � DADBvC + PAB vC� 0 = 0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='7) D(AαB) = − 1 2(LvΦ)AB + 1 2 ˚ ψ ΦAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='8) Recall that the condition (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='7) indicates that vA is an infinitesimal projective symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' MODIFIED CONFORMAL EXTENSIONS 25 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Sections wAB ∈ E[AB](−2) satisfying (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='17) are identified with densities wABǫAB ∈ E(1) satisfying the first BGG equation corresponding to (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Its non-trivial solutions has to satisfy the condition (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2), which implies projective flatness in dimension 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In such case, the condition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='45) reads as wRSB2(Φ)ARSD = 0, which implies B2(Φ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='5, the conformal extension would be flat, which contradicts our assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Thus, wAB has to vanish identically and the key conditions from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2 simplify to those displayed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' □ As above, the composition of an appropriate relation with the corresponding BGG operator may provide extra information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Let us continue in the setting of the previous Proposition and apply the second BGG operator (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3) to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' This yields α[CYD]AB = Lv(B2(Φ))ABCD − ˚ ψ(B2(Φ))ABCD, where we have used (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='5), the fact that B2 commutes with Lv and that ˚ ψ is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' With the associated quantities (⋆Y )A = YCDEǫACǫDE ∈ EA(−6) and ⋆B2(Φ) = B2(Φ)ABCDǫABǫCD ∈ E(−4), the previous display is rewritten as αR(⋆Y )R = Lv(⋆B2(Φ)) − ˚ ψ(⋆B2(Φ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='9) Off its zero, the density ⋆B2(Φ) ∈ E(−4) plays a role of projective scale, hence fixes a unique affine connection from the projective class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Compared to the context of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3, there are currently more terms entering the game and a complete discussion seems to be rather complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Restricting to the case of non-flat extensions of flat projective structures, we conclude with the following specification of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1: Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' For a non-flat modified conformal extension of a flat 2-dimensional projective structure, let DA be the affine connection corresponding to the scale ⋆B2(Φ) and RABCD be its curvature tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Locally, off the zero set of ⋆B2(Φ), there is a bijective correspondence between the conformal Killing fields and pairs vA ∈ EA and αA ∈ EA(2) satisfying DADBvC + vSRSACB = 0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='10) D(AαB) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='11) The dimension of the space of conformal Killing fields equals to d+3, where d is the dimension of the space of solutions to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Recall that the flatness of induced conformal structure is controlled by the vanishing of ⋆B2(Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The condition (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='10) indicates that vA is an infinitesimal affine symmetry of DA, see appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The scale ⋆B2(Φ) is parallel with respect to DA, hence Lv(⋆B2(Φ)) = 4 3(DRvR)(⋆B2(Φ)), where the (otherwise unimportant) coefficient on the right-hand side reflects the conventions from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' With this observation we easily compare the actual conditions with those of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1: Let vA ∈ EA be an infinitesimal projective symmetry satisfying (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2) for some ˚ ψ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' With the current reformulations, that condition means DRvR = 3 4 ˚ ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In particular, DRvR is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Hence, by Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1, vA is an infinitesimal affine symmetry of DA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Conversely, let vA ∈ EA be an infinitesimal affine symmetry of DA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Then vA is an infinitesimal projective 26 HAMMERL, SAGERSCHNIG, ˇSILHAN, ˇZ´ADN´IK symmetry and, by Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1, DRvR is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Hence, for ˚ ψ = 4 3DRvR, the condition (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The rest is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' □ By the local exactness of BGG sequences in the flat case, every projective scale is locally of the form ⋆B2(Φ), for some Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' It then follows from Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='4 and Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2 that the corresponding modified conformal extension of a flat projective structure has the dimension of the symmetry algebra limited as follows: Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' For non-flat modified conformal extensions of flat 2-dimensional projective structures, the Lie algebra of infinitesimal conformal symmetries can be of any dimension from 3 to 9, except for 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The flat 4-dimensional conformal structure, which is the standard conformal extension of flat projective structure, has maximal symmetry algebra, whose dimension is 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' It turns out, that the submaximal dimension of the symmetry algebra of 4-dimensional conformal structures of split signature is 9, see [24, section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' This case is discussed in more detail below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Submaximal example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In this section, we illustrate some of the previously obtained general results in a very concrete setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' For this purpose, we take a non-flat modified con- formal extension with non-trivial algebra of infinitesimal symmetries and show how this (po- tentially complicated) structure can be assorted in simpler underlying projective terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Inter- estingly, examples of metrics of split signature whose algebra of conformal Killing fields has submaximal dimension are modified PW metrics, see [24, section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The richest structure appears in dimension four, which is the case we discuss here in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Let us take the metric g = dx1 ⊙ dp1 + dx2 ⊙ dp2 + (x2)2 dx1 ⊙ dx1, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='12) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' the modification with Φ = (x2)2 dx1⊙dx1 of the flat PW metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' According to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='4, the Weyl curvature of g is W = −2B2(Φ) = −4 dx1 ∧ dx2 ⊙ dx1 ∧ dx2, hence the conformal extension is non-flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' According to Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='4, conformal Killing fields of g are parametrized by infinitesimal affine symmetries of the flat affine connection and projective Killing forms, which form spaces of dimension 6 and 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Altogether, the space of conformal Killing fields has dimension 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' To be more precise, we describe both the underlying sources and their corresponding lifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The infinitesimal affine symmetries and the projective Killing forms have the form, respec- tively, v = (c1 + c3x1 + c5x2)∂x1 + (c2 + c4x1 + c6x2)∂x2, αk = (c7 + c9x2) dx1 + (c8 − c9x1) dx2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' where c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' , c9 are arbitrary constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' A particular solution to the key equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='19) is αp = � −c2x1x2 − 1 2c4(x1)2x2 − 1 3c5(x2)3� dx1 + � 1 2c2(x1)2 + 1 6c4(x1)3� dx2, for the constant ˚ ψ = 4 3(c3 + c6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Thus, conformal Killing fields of g are uniquely given by the quadruples w, v, α and ˚ ψ, where w = 0 and α = αp + αk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' According to Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3, they MODIFIED CONFORMAL EXTENSIONS 27 are described by the following lifts: v0 = � c1 + c3x1 + c5x2� ∂x1 + � c2 + c4x1 + c6x2� ∂x2+ + ((c3 + 2c6)p1 − c4p2) ∂p1 + (−c5p1 + (2c3 + c6)p2) ∂p2, v− = � c7 + c9x2 − c2x1x2 − 1 2c4(x1)2x2 − 1 3c5(x2)3� ∂p1+ + � c8 − c9x1 + 1 2c2(x1)2 + 1 6c4(x1)3� ∂p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' All in all, conformal Killing fields of the modified PW metric (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='12) have the form v = v0 +v− with the summands as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Also, the structure of the Lie algebra of conformal Killing fields is given by the underlying data, typically in a rather intricate way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' To decipher the current example, let us denote the algebra of infinitesimal affine and conformal symmetries by g and g, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Let us further denote by ei and ei the generator of g and g corresponding to the coefficient ci, where i runs from 1 to 6 and 9, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Of course, g = g0 ⊕ g1, where the reductive and the nilpotent subalgebra is, respectively, g0 = ⟨e4, e3 − e6, e5⟩ ⊕ ⟨e3 + e6⟩, g1 = ⟨e1, e2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The indicated decomposition of g0 exhibits its simple part, which is sl(2, R), and the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' It turns out that g is a graded Lie algebra, g = g0 ⊕ g1 ⊕ g2 ⊕ g3, where g0 = ⟨e4, e3 − e6, e5⟩ ⊕ ⟨e3 + e6⟩, g1 = ⟨e1, e2⟩, g2 = ⟨e9⟩, g3 = ⟨e7, e8⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The subalgebra g0 is isomorphic to g0, the effect of the modification is visible in the nilpotent part of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Nonetheless, the action of g0 on g1, respectively on g2 ⊕ g3, corresponds to the action of g0 on g1, respectively on projective Killing forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The resulting refined structure on the conformal side is a priori not obvious and emerges as a consequence of the very specific modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Moreover, note that g is isomorphic to a parabolic subalgebra of the split real form of 14-dimensional exceptional simple Lie algebra g2, namely, to the one corresponding to the short root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Remarks 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The appearance of a parabolic subalgebra of g2 as the Lie algebra of conformal infinitesimal symmetries of the metric (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='12) has the following geometric explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' There is a natural (twistor) distribution of rank 2 on the bundle of self-dual null-planes of a 4- dimensional split-signature conformal manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Locally, this is a (2, 3, 5) distribution if and only if the self-dual part of the Weyl tensor is non-trivial, see [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In this correspondence, infinitesimal symmetries of the conformal structure lift to infinitesimal symmetries of the twistor distribution, and thus the symmetry algebra of the former structure is identified with a subalgebra of the symmetry algebra of the latter one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' It is a classical result by Cartan, that the maximal symmetry algebra of (2, 3, 5) distributions is the exceptional simple Lie algebra g2 and the submaximal one has dimension 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Now, the self-dual part of the Weyl tensor of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='12) is non-trivial and its algebra of conformal symmetries has dimension 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' It follows that the Lie algebra of infinitesimal conformal symmetries of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='12) has to be a subalgebra of g2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Finally, note that there are just two 9-dimensional subalgebras in g2, namely, the two maximal parabolic ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The modified PW metric (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='12) is also a pp-wave, symmetric and Einstein metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' All Einstein metrics in the conformal class of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='12) correspond to the scales of the form σ = 28 HAMMERL, SAGERSCHNIG, ˇSILHAN, ˇZ´ADN´IK c1x1 +c2x2 +c3, for c1, c2, c3 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' They form a space of dimension 3 and provide a realization of one of the possibilities listed in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Ambient metric and Q-curvature There is a natural geometric construction of the Fefferman–Graham ambient metric for modified conformal PW metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Compared to the non-modified situation studied in [22], the modification tensor causes no serious complication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In this context, partial observations with more general modifications were done, which we comment at the end of section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The general setting of the Fefferman–Graham ambient construction is as follows, see [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Let a conformal class on a smooth manifold � M be represented by a metric g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The ambient manifold M is locally described by two more coordinates, say t ∈ R+, which parametrizes metrics in the conformal class, and ρ ∈ R, which represents the essentially new dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The normal form for the ambient metric on M is g = 2ρ dt ⊙ dt + 2t dt ⊙ dρ + t2g(ρ), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1) where g(ρ) is a smooth 1-parameter family of metrics on � M with g(0) = g being a repre- sentative metric in the conformal class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' On of the key features of such metrics is that they are homogeneous of degree 2 with respect to t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The Fefferman–Graham ambient metric is a metric of the form (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1) that is Ricci-flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The existence of such metrics is a subtle question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' For a given g, the metric (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1) is typically constructed iteratively by the Taylor expansion of g(ρ) in ρ so that the Ricci-flatness condition is controlled asymptotically for ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In even dimension, the construction is obstructed in finite order by the so-called Fefferman–Graham tensor, a conformally invariant tensor, which is the Bach tensor in dimension 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' For modified conformal extensions of projective structures, the ambient metric exists and has the simplest conceivable form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Namely, the term g(ρ) is linear in ρ and g is Ricci-flat globally, not only asymptotically: Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Let g be a modified PW metric with the Schouten tensor P and let t, ρ be the additional coordinates on the ambient manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Then g = 2ρ dt ⊙ dt + 2t dt ⊙ dρ + t2� g + 2ρ P � (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2) is a globally Ricci-flat Fefferman–Graham ambient metric of the conformal class of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The Ricci, respectively Schouten, tensor of a modified PW metric is the pullback of the Ricci, respectively Schouten, tensor of the underlying affine connection, independently of the modification tensor Φ, see section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Hence, in local coordinates (xA, pA) as above, the Schouten tensor of g has the form P = 1 n−1 RicAB dxA ⊙dxB, where RicAB is the Ricci tensor of the underlying affine connection and n is the dimension of the corresponding manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The normal form of the metric (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2) is obvious, thus it is enough to check the Ricci- flatness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' This can be done directly, using [18, formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='17)] and the just mentioned properties of the Ricci tensor of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Alternatively, the metric g can be obtained as the modified PW metric of the Thomas ambient connection associated to the initial projective structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The Ricci- flatness of g then follows from the Ricci-flatness of the Thomas ambient connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Details in this picture are as follows: Let C and ∇ be the Thomas ambient cone and connection, respectively, associated to the underlying projective structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The modification tensor Φ ∈ E(AB)(2) lifts to a symmetric MODIFIED CONFORMAL EXTENSIONS 29 2-tensor �Φ on C which is homogeneous of degree 2 with respect to the standard R+-action on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Finally, the ˆΦ-modified PW metric of ∇ is identified with (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2) by the same coordinate transformation as in the proof of [22, Theorem 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' □ Another related notion is the Q-curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Although it is associated to a particular metric rather than to the conformal class, it is an important quantity in conformal geometry, see [6], [13], [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The explicit and simple form of the Fefferman–Graham ambient metric allows the Q-curvature to be computed, which is generally a rather difficult task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' To do so, one analyzes ∆n log(t), where ∆ is the ambient Laplacian associated to (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2) and t as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In the non-modified case, it easily follows that ∆ log(t) = 0, hence the Q- curvature of any PW metric vanishes, see [22, Theorem 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Checking details in that reference, it follows that the modification tensor Φ does not play any role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Hence we have the following generalization: Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The Q-curvature of any modified PW metric vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Remarks 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Modified PW metrics fit into the class of so-called null Ricci Walker metrics discussed in [3], for which the existence of explicit ambient metrics has been shown in some interesting cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Modified PW metrics also form a subclass of more general modifications studied in [9], which include, in particular, self-dual Walker metrics in dimension 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The Bach tensor of a self-dual conformal metric, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' the Fefferman–Graham obstruction tensor, vanishes identically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' While no geometric construction of the associated ambient metric is known for this class in general, an explicit formula for specific cases can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' For the sake of illustration, we allow extra modifications parametrized by smooth functions α ∈ E on the underlying projective manifold and determined by the distinguished vertical field ka ∈ �Ea from Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='8, namely, the homothety of the standard PW metric �gab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Thus, we allow the generalization of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='4) so that gab := �gab + Φab + α k(akb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3) Particular modifications of the form (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3) appear also in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In the case when �g is the flat PW metric, Φ = 0 and α = 1 we have the para Fubini–Study metric, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' [5], [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Let g be the extra modified PW metric (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3), for some α ∈ E, on a 4- dimensional manifold with the Schouten tensor P and let t, ρ be the additional coordinates on the ambient manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Then g = 2ρ dt ⊙ dt + 2t dt ⊙ dρ + t2 � g + 2ρ P + ρ2� α2g + 2α (dα) ⊙ k �� (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='4) is a globally Ricci-flat Fefferman–Graham ambient metric of the conformal class of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' More- over, the Q-curvature of the metric g is Q = −56α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Again, the metric (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='4) is in the normal form, so it suffices to check its Ricci-flatness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' This was done directly using the computational systems Mathematica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' It follows that the expression (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='4) is not unique: adding an arbitrary constant multiple of (�∆α) g − 2α (dα) ⊙ k, where �∆ is the Laplacian of �g, into the inner parentheses provides also a Ricci-flat solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Projective BGG operators The BGG theory is a conceptual framework comprehending many problems concerning in- variant operators, respectively equations, in parabolic geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' For a general introduction, we refer to seminal papers [12] and [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' BGG operators appear in sequences, in which the 30 HAMMERL, SAGERSCHNIG, ˇSILHAN, ˇZ´ADN´IK first operator is the most frequent in applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Solutions to the corresponding equations often have significant geometrical meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Prominent instances in conformal geometry are related to almost Einstein scales and infinitesimal symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' These problems are studied in sections 3–6, where the corresponding overdetermined equations are reduced to systems of projectively invariant equations among which projective first BGG operators play an essential role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Each BGG sequence is determined by a concrete tractor bundle and the related exterior covariant differential, from which an explicit description of the operators can be deduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' On the underlying level, the type of sequence is encoded in the source bundle of the first operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In the locally flat case, BGG sequences form complexes, which are locally exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Moreover, solutions to the first BGG equation form a vector space whose dimension equals to the rank of the background tractor bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' For purposes of this article, we need first two operators of several projective BGG sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' We denote these operators as B1 and B2, keeping in mind they are always related to the actual type of the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Each sequence corresponds to certain tensor product of the standard tractor bundle T and its dual T ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The standard tractor bundle is associated to the standard action of the group SL(n + 1, R), the Lie group of projective symmetries of the (oriented) projective sphere of dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In particular, the rank of T is n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The common pattern of the following subsections is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Firstly, we specify the type of sequence and explicit formulas for first two operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Secondly, composing these two operators, we get an integrability condition for the existence of solutions to the first BGG equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Thirdly, we count the dimension of the solution space to that equation in the locally flat case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Some operators depend on the dimension n of the projective manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' This is caused by natural isomorphisms in low dimensions that are, on the underlying level, provided by the projective volume form ǫA1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='An ∈ E[A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='An](n + 1), respectively its dual ǫA1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='An ∈ E[A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='An](−n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Sequence for T ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The corresponding BGG sequence starts as follows: E(1) B1 −→ E(AB)(1) B2 −→ E[CA]B(1) −→ · · · where B1(τ)AB = (DADB + PAB)τ, B2(T)CAB = D[CTA]B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1) The composition yields (B2 ◦ B1)(τ)CAB = − 1 2 � WABRCDRτ − YCABτ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2) For n = 2, the vanishing of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2) means YCAB = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' the local flatness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In locally flat case, the dimension of the solution space of B1 equals to rank T ∗ = n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Sequence for ∧2T ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The corresponding BGG sequence starts as follows: EB(2) B1 −→ E(AB)(2) B2 −→ E[AB][CD](2) −→ · · · where B1(ϕ)AB = D(AϕB), B2(Φ)ABCD = Proj⊞ � DADCΦBD + PAC ΦBD � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3) MODIFIED CONFORMAL EXTENSIONS 31 Here Proj⊞ denotes the projection to the subspace of the ‘window’ symmetry correspond- ing to the indicated Young tableau, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' the subspace of E[AB][CD](2) such that the skew- symmetrization over any triple of indices vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Otherwise put, a short computation reveals that B2(Φ)ABCD = Proj[AB][CD] � DADCΦBD + PAC ΦBD + + 1 4WABRCΦDR − 1 4WCDRAΦBR � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='4) where Proj[· ·][· ·] denotes the skew-symmetrization over the embraced indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The composition yields (B2 ◦ B1)(ϕ)ABCD = Proj[AB][CD] � − 1 2WABRCD[RϕD] − 1 2WCDRAD[RϕB] + + 1 4WABRCD(DϕR) − 1 4WCDRAD(BϕR) − − 1 2(DAWCDRB)ϕR − 1 2ϕCYDAB � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='5) For n = 2, the vanishing of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='5) means ϕ[CYD]AB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' This condition can be expressed so that ϕA = f(⋆Y )A, where (⋆Y )A := YABCǫBC, for some f ∈ E(5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In locally flat case, the dimension of the solution space of B1 equals to rank ∧2T ∗ = 1 2n(n + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Sequence for T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The corresponding BGG sequence starts with EA(−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' For n = 2, we have the identification EA(−1) ∼= EA(2), hence the sequence coincides with the one in section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' For n ≥ 3, the sequence is as follows: EB(−1) B1 −→ � EAB(−1) � 0 B2 −→ � E[CA] B(−1)) � 0 −→ · · · where B1(ξ)AB = � DAξB� 0, B2(Ξ)CAB = � D[CΞA] B� 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='6) The composition yields (B2 ◦ B1)(ξ)CAB = 1 2WCABRξR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='7) In locally flat case, the dimension of the solution space of B1 equals to rank T = n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Sequence for ∧2T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The corresponding BGG sequence starts with E[AB](−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' For n = 2, we have the identification E[AB](−2) ∼= E(1), hence the sequence coincides with the one in section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' For n = 3, we have the identification E[AB](−2) ∼= EA(2), hence the sequence coincides with the one in section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' For n ≥ 4, the sequence is as follows: E[AB](−2) B1 −→ � ED[AB](−2) � 0 B2 −→ � E[CD] [AB](−2) � 0 −→ · · · where B1(w)DAB = � DDwAB� 0, B2(V )CDAB = � D[CVD] AB� 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='8) The composition yields (B2 ◦ B1)(w)CDAB = − � WCD[ARwB]R� 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='9) 32 HAMMERL, SAGERSCHNIG, ˇSILHAN, ˇZ´ADN´IK Note that, for n = 3, the first operator actually coincide with the one in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='8) (whereas the second does not) and the expression (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='9) vanishes identically for any wAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In locally flat case, the dimension of the solution space of B1 equals to rank ∧2T = 1 2n(n + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Sequence for (T ∗ ⊗ T )0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The corresponding BGG sequence starts as follows: EC B1 −→ � E(AB) C� 0 B2 −→ � E[DA]B C� 0 −→ · · · where B1(v)(AB) C = � DADBvC + PAB vC� 0, B2(V )DABC = � D[DVA]B C� 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='10) In the context of projective infinitesimal symmetries, we refer also to the following modifica- tion of the first operator: B# 1 (v)ABC = � DADBvC + PAB vC + vRWR(A C B) � 0, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='11) One can verify that both B1(v)ABC and B# 1 (v)ABC are, indeed, symmetric in lower indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In fact, the target space of B2 is the subspace of � E[DA]BC� 0 of such tensor fields that vanish upon skew-symmetrization over lower indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The composition yields (B2 ◦ B# 1 )(v)DABC = 1 2(LvW)DACB, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='12) where Lv denotes the Lie derivative in the direction of the vector field vA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In locally flat case, the dimension of the solution space of B1 equals to rank(T ∗ ⊗ T )0 = n(n + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Projective and affine infinitesimal symmetries Both parts of this section concern affine, respectively projective, infinitesimal symmetries of an affine connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The following results are relevant primarily for section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Affine symmetries among projective ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Let DA be a special torsion-free affine connection and RABCD be its curvature tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Let PAB and WABCD be the projective Schouten and Weyl tensor of DA, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' These tensors are related by W C AB D = R C AB D + PAD δC B − PBD δC A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1) It is a common knowledge that a vector field vA ∈ EA is an affine infinitesimal symmetry of the affine connection DA if and only if it satisfies DADBvC + vSRSACB = 0 (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2) and it is a projective infinitesimal symmetry of the projective structure [DA] if and only if it satisfies � DADBvC + PAB vC + vRWR(A C B) � 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3) Clearly, any affine infinitesimal symmetry is also the projective one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The opposite considera- tion is specified as follows: Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Let vA ∈ EA be a projective infinitesimal symmetry of a projective struc- ture [DA].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Then vA is an affine infinitesimal symmetry of a representative special affine connection DA if and only if the divergence DRvR is constant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=', DADRvR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' MODIFIED CONFORMAL EXTENSIONS 33 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The ‘only if’ direction of the equivalence is read directly from (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' For the ‘if’ direc- tion, we assume that vA satisfies (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3) and ψ := 1 nDRvR is a constant function and want to show that (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Let us decompose the left-hand side of (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2) into the skew-symmetric, the symmetric trace-free and the trace part, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The skew-symmetric part vanishes trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The condition (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3) implies that the symmetric trace-free part also vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' It re- mains to show that traces vanish as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' An easy computation shows that (any) trace is a nonzero constant multiple of PAR vR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=', (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2) is satisfied if and only if PAR vR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3) is overdetermined and its prolongation was computed in [23, section 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Besides vA and ψ, two additional variables are needed to form a closed system, namely, φB A := DAvB − δB Aψ, βA := − 1 n+1DADRvR − PAR vR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='4) Now, solutions to (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='3) are exactly the solutions to the prolonged system, which is displayed in [23, formulas (71)] and which we do not reproduce here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' By the assumption that ψ is constant, the second quantity in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='4) reads as βA = − PAR vR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Substituting everything to the third equation of the prolonged system, one obtains PAR vR = 0 after some computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' □ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Affine symmetries in dimension 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The specification of dimensions of algebras of affine infinitesimal symmetries is a classical subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' It is known that, for affine connections on 2-dimensional manifolds, all dimension less or equal to 6, except for 5, are possible, see [17] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' In fact, all these dimensions can be realized as follows: Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The Lie algebra of affine infinitesimal symmetries of an affine connection on a 2-dimensional manifold can be of any dimension less or equal to 6, except for 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Each of these cases can be realized by a special projectively flat affine connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Let (x1, x2) be a local coordinates and let us consider the family of affine connections obtained from the flat affine connection by the projective change (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='1), where Υ = (a2(x1)2 + a1x1 + a0) dx1 + (b1x2 + b0) dx2, where a0, a1, a2, b0, b1 are real parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=', we consider the family of connections whose Christoffel symbols are 1 2Γ111 = Γ122 = Γ221 = a2(x1)2 + a1x1 + a0, 1 2Γ222 = Γ112 = Γ211 = b1x2 + b0, Γ212 = Γ121 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' For particular values of parameters, we achieve all the possibilities as follows: (0) Generic values of parameters provide no affine symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (1) For a2 = 0, the symmetry algebra has dimension 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (2) For a2 = b1 = 0, the symmetry algebra has dimension 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (3) For a2 = b1 = b0 = 0, the symmetry algebra has dimension 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (4) For a2 = b1 = b0 = a1 = 0, the symmetry algebra has dimension 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' (6) For all parameters vanishing, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' for the flat connection, the symmetry algebra has di- mension 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' The symmetry of the Ricci tensor was checked and the affine symmetry algebra was generated using the computational system Maple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' □ 34 HAMMERL, SAGERSCHNIG, ˇSILHAN, ˇZ´ADN´IK References [1] Z.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=': Center for Theoretical Physics PAS, Al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' Lotnik´ow 32/46, 02-668 Warszawa, Poland J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' ˇS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' : Masaryk University, Faculty of Science, Kotl´aˇrsk´a 2, 61137 Brno, Czech Republic V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content=' ˇZ: Masaryk University, Faculty of Education, Poˇr´ıˇc´ı 31, 60300 Brno, Czech Republic Email address: matthiasrh@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='com, kat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='sagerschnig@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='com, silhan@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='muni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='cz, zadnik@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='muni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} +page_content='cz' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E2T4oBgHgl3EQf-Qn4/content/2301.04238v1.pdf'} diff --git a/FNAyT4oBgHgl3EQfrPlH/content/tmp_files/2301.00556v1.pdf.txt b/FNAyT4oBgHgl3EQfrPlH/content/tmp_files/2301.00556v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3d6177ca64edfab125f4a4caf5c2fea3c5207a49 --- /dev/null +++ b/FNAyT4oBgHgl3EQfrPlH/content/tmp_files/2301.00556v1.pdf.txt @@ -0,0 +1,1198 @@ +arXiv:2301.00556v1 [q-bio.PE] 2 Jan 2023 +Competition of alliances in a cyclically dominant eight-species population +Junpyo Parka, Xiaojie Chenb, and Attila Szolnokic,∗∗ +aDepartment of Applied Mathematics, Kyung Hee University, Yongin 17104, Republic of Korea +bSchool of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China +cInstitute of Technical Physics and Materials Science, Centre for Energy Research, P.O. Box 49, H-1525 Budapest, Hungary +Abstract +In a diverse population, where many species are present, competitors can fight for surviving at individual +and collective levels. In particular, species, which would beat each other individually, may form a specific +alliance that ensures them stable coexistence against the invasion of an external species. Our principal goal +is to identify those general features of a formation which determine its vitality. Therefore, we here study a +traditional Lotka-Volterra model of eight-species where two four-species cycles can fight for space. Beside +these formations, there are other solutions which may emerge when invasion rates are varied. The complete +range of parameters is explored and we find that in most of the cases those alliances prevail which are formed +by equally strong members. Interestingly, there are regions where the symmetry is broken and the system +is dominated by a solution formed by seven species. Our work also highlights that serious finite-size effects +may emerge which prevent observing the valid solution in a small system. +Keywords: +cyclic dominance, alliances, competition +1. Introduction +If a species is stronger and beats the other one +then how can the “weaker” species survive? This +is a fundamental problem of ecology to explain the +amazing biodiversity of life [1, 2]. A possible expla- +nation could be the so-called intransitive or cycli- +cally dominated relation among competing species +where a third (or more) species beat the preda- +tor species, hence establishing a delicate balance +among all competitors. In the simplest case, these +relations remind the well-known rock-scissors-paper +game and this type of interaction can really be +observed among animals, plants, or even among +bacterias [3, 4, 5, 6, 7, 8, 9]. Evidently, a three- +member loop can be easily enlarged for more par- +ticipants, as it was made in the extended Lotka- +Volterra model [10, 11, 12, 13]. Motivated by the +complexity of such interactions, researchers have +studied related models actively over the last decade +and several interesting observations have been made +[14, 15, 16, 17, 18]. +∗Email +addresses: +junpyopark@khu.ac.kr; +xiao- +jiechen@uestc.edu.cn; szolnoki.attila@ek-cer.hu +∗∗Corresponding author +It would be almost impossible to sum all impor- +tant observations, such as reported in Refs. [19, 20, +21, 22, 23, 24, 25]. Instead, the reader is referred +to specific review papers about the milestones along +this research path [26, 27, 28, 29]. Nevertheless, it is +also worth noting here that the mentioned cyclic or +intransitive relation among participants should not +necessarily be defined by a Lotka-Volterra-type mi- +croscopic rule, but could be the result of collective +behavior among competing strategies in evolution- +ary game models [30, 31, 32, 33, 34, 35, 36, 37, 38] +The most exciting question is whether we can +predict the direction of evolution based on the food- +web that defines the microscopic dynamics? Can +we identify general principles which may inform us +about the vitality of an alliance? An early observa- +tion was, for instance, that when two three-member +cycles fight then the one, where the inner invasion +is faster, is more viable [39]. +The picture, how- +ever, is more subtle, because a faster general rota- +tion does not necessarily provide an advantage if +the members of the trio are unequal. In the lat- +ter case, the triplet becomes vulnerable no matter +the average of inner invasion rates is higher than in +the rival triplet which is formed by less aggressive, +Preprint submitted to Chaos Solitons and Fractals +January 3, 2023 + +but equally strong partners [40]. Notably, hetero- +geneous invasion rates may emerge temporarily or +locally [41, 42]. Or, the extension of the original +food-web by adding a reverse link can also change +a stable solution [43]. Furthermore, the number of +members could also be a decisive factor because a +smaller loop is generally stronger than a large one +[44]. +In this work we introduce a minimal eight-species +model where two four-member loops fight for space. +The interaction between these quartets practically +establishes a traditional Lotka-Volterra circle of +eight species, which can be described by an inva- +sion rate. Not to break the balance between the +quartets, we assume that the inner invasion is uni- +form and equally strong in both formations. The +only difference between them is we introduce two +short-cuts in one of the loops which generates extra +interactions within the related alliance. Our major +question is how the stability of alliances change due +to this extension and can we identify new solutions +which would be hidden otherwise? +It could also +be interesting whether the previously established +principles, observed for simpler systems formed by +trials and duets, remain valid when we apply them +to alliances formed by larger groups. +2. Alliances formed at different levels +To make our observations comparable with pre- +vious works [45, 46, 47, 48, 49, 50], we assume +that species are distributed on an L × L square +lattice where periodic boundary conditions are ap- +plied. Each node is occupied by one of the species +which are denoted by Xi where indexes are from +i = 0 to 7. During an elementary step, we select +a neighboring pair nodes. If they are occupied by +the same species or represent species which are neu- +tral then nothing happens. Otherwise, an invasion +happens with a specific probability and predator +species invades prey species. The microscopic rules +are defined in the following way: +XiXi+1 +γ−→ XiXi +(1) +XiXi+2 +α−→ XiXi +(2) +X2X6 +β−→ X2X2, X4X0 +β−→ X4X4, +(3) +where i + 1 and i + 2 are calculated in cyclic man- +ner. The parameters α, β and γ define the proba- +bilities of successful invasions between the involved +predator-prey neighbors. +γ +α +β +1 +2 +4 +0 +3 +5 +6 +7 +Figure 1: Food-web of competing species. In the basic model +eight species are invade cyclically each other with probabil- +ity γ, similar to the extended Lotka-Volterra model. +Ad- +ditionally, we introduce a cyclic inner invasion among odd +and separately among even species with probability α. To +break the symmetry among the quartets, we also introduce +an inner invasion between species “2” and “6” and between +species “4” and “0” with probability β. Importantly, we use +the color-code of species in later figures where spatial distri- +butions are presented. +For a deeper insight about the model, in Fig. 1 we +show the food-web of competing species. Our first +note is about the biggest loop where all species are +involved, according to the extended Lotka-Volterra +type system. Here every species is simultaneously +a predator and a prey of another one, hence es- +tablishing a possible solution. +The invasion rate +is γ for all interactions here. Another solution is +formed by “1”, “3”, “5”, and “7” species who in- +vade each other cyclically with probability α. For +later reference, we denote this alliance as A4 +1,3,5,7. +A similar A4 +0,2,4,6 quartet is formed among species +“0”, “2”, “4” and “6”. Additionally, we introduce +an extra chance of invasions among group members +here. In particular, species “2” invades species “6” +and species “4” invades species “0” with probabil- +ity β. +In this way, there are no neutral pairs in +the mentioned quartet anymore. As a consequence, +the average inner invasion is augmented among the +four species, which could be a support to their via- +bility. Interestingly, however, there are many other +possible solutions in this system. For example, the +new invasions among even-numbered species offer +2 + +the chance for trials to emerge: A3 +0,2,4 or A3 +0,2,6 can +work as a rock-scissors-paper-type solution. Beside +the mentioned formations, there are A5 (five-), A6 +(six-), or A7 seven-member solutions, as well. Per- +haps, it is not necessary to specify them because +the reader can easily construct examples, based on +the food-web, shown in Fig. 1. +As we noted, there are three parameters in our +model and in the following we systematically scan +the whole parameter field to identify the dominant +solution for each combination of parameters. For +this reason, we executed Monte Carlo simulations +where a Monte Carlo step (MCS) means that on +average every node has a chance to update its state. +The linear system size varied between L = 100 to +L = 3200 and the necessary relaxation steps were +between 103 to 3 × 105 MCSs. According to the +standard protocol, simulations were launched from +an initial state where species are distributed ran- +domly on the lattice and monitor the ρi concentra- +tion of Xi for all species. We, however, would like +to stress that this initial state does not always could +be a good choice to find the solution which is valid +in the large size limit. It is because a small sys- +tem size does not necessarily “offer” equal chance +for all possible solutions to emerge. Some solution, +which involves large typical lengths, would emerge +just later, but to that stage of the evolutionary pro- +cess other solutions may invade the whole available +space. Therefore, a more complex solution practi- +cally has no chance to emerge at a small system size. +As a consequence, other solutions may win and an +independent run can easily terminates onto a differ- +ent solution, no matter we use the same set of model +parameters. To overcome this uncertainty, we not +simply enlarged the system size, but also used alter- +native initial states where larger and homogeneous +domains of competing species were distributed ran- +domly on the lattice. Similar approach was used +previously in Refs.[51, 52], for instance. In this way, +we can create all types of interfaces which could +be the foundation to build more sophisticated solu- +tions. To make our analysis more complete, at the +critical values of control parameters we also gener- +ated sub-solutions independently in restricted areas +of space and after we let them fight directly along +the separating interface [53, 54, 55]. This technique +can identify the dominant solution unambiguously. +The details of the above mentioned protocols will +be specified in the next section. +3. Results +3.1. Phase diagrams +We first summarize our main findings and later +we give further details about the microscopic mech- +anisms which are responsible how dominant solu- +tions emerge. +According to our simulations, we +can distinguish three main cases which determine +the system behavior. The decisive condition is the +intensity of interaction between the quartets men- +tioned in Sec. 2. If this interaction is weak, means +the value of γ is small, then the A4 +{1,3,5,7} solution +dominates the whole β − α parameter plane. +In +other words, only the quartet of “1+3+5+7” sur- +vive independently of the α and β values. +For intermediate values of γ, when the invasion +flow in the large cycle becomes substantial, a con- +ceptually new behavior emerges. We represent this +phenomenon by showing the phase diagram ob- +tained at γ = 0.5. +Figure 2 shows that beside +the mentioned A4 +{1,3,5,7} quartet, other solutions be- +come dominant at certain values of β − α pairs. +When both α and β are small then the “large cir- +cle” is the winner, hence all eight species coexist. +For intermediate β values this solution is replaced +by A7 +{0,1,2,3,4,6,7} where only species “5” is missing. + 0 + 0.2 + 0.4 + 0.6 + 0.8 + 1 + 0 + 0.2 + 0.4 + 0.6 + 0.8 + 1 +A4 +{1,3,5,7} +A7 +{0,1,2,3,4,6,7} +all +β +α +Figure 2: Phase diagram on β − α parameter plane obtained +at γ = 0.5. If α is large enough, the quartet of “1+3+5+7” +species always win. If both β and α are small, all species +survive. Interestingly, for low α and intermediate β a seven- +member solution dominates where species “5” is missing. +Dashed blue line denotes discontinuous, while solid red line +marks the positions of continuous phase transition points. +Qualitatively similar system behavior can be ob- +served when γ is large, hence the flow in the ex- +3 + +ternal loop is intensive. In other words, the inter- +action between the quartets becomes large. This +is illustrated in Fig. 3 where we present the com- +plete phase diagram on the β − α plane. The only +difference between the diagrams is the area, where +complete eight-species solution dominates, is larger +and the seven-member solution is shifted toward +larger β values. + 0 + 0.2 + 0.4 + 0.6 + 0.8 + 1 + 0 + 0.2 + 0.4 + 0.6 + 0.8 + 1 +A4 +{1,3,5,7} +A7 +{0,1,2,3,4,6,7} +all +β +α +Figure 3: Phase diagram on β − α parameter plane obtained +at γ = 1. The diagram is similar to the one shown in Fig. 2. +The only difference is the eight-member full solution occu- +pies larger area in the parameter field, while seven-member +solution is shifted toward higher β values. +We must stress that the presented phase dia- +grams are valid in the large system size limit. If +the system size is small then we may observe that +the system can easily terminate onto different so- +lutions. To illustrate it, in Fig. 4 we present the +survival probabilities at each (α, β) parameter pairs +for γ = 1, when the linear system size was L = 100. +More precisely, we launched the evolution from a +random initial state and monitored the fractions of +species until N = L×L MCSs. After, we recorded +the number of surviving species and repeated the +whole process 100 times. In this way we can esti- +mate the probability that S species survive for long +time. +The panels of Fig. 4 show these surviving +probabilities for all possible S values at x different +(α, β) pairs on the parameter plane. +These pan- +els indicate that different solutions may emerge no +matter we use the same values of (α, β) parameters. +Therefore, the destination is rarely unambiguous, +hence the surviving probability equals to 1 only for +S = 4 when α is high and β is low. Nevertheless, +0 +0.5 +1 +0 +0.5 +1 +S=1 +0 +0.5 +1 +0 +0.5 +1 +S=2 +0 +0.5 +1 +0 +0.5 +1 +S=3 +0 +0.5 +1 +0 +0.5 +1 +S=4 +0 +0.5 +1 +0 +0.5 +1 +S=5 +0 +0.5 +1 +0 +0.5 +1 +S=6 +0 +0.5 +1 +0 +0.5 +1 +S=7 +0 +0.5 +1 +0 +0.5 +1 +S=8 +0 +0.2 +0.4 +0.6 +0.8 +1 +Figure 4: Heat maps of the survival probability on β − α +parameter plane obtained for γ = 1 by using 100×100 system +size. Each panel shows the probability to reach a state which +contains S species after N = L2 MCSs. The value of S is +indicated on each panel. The results are averaged over 100 +independent runs. +the contour of areas around “maximum” values are +roughly agree with the diagram shown in Fig. 3. +But, as Fig. 4 illustrated, this system size is insuffi- +cient to make reliable conclusions about the proper +system behavior. +3.2. Detecting phase transitions +Therefore, to detect the phase transition points +more precisely, we need to apply systematic finite- +size analysis. An example is given in Fig. 5 where +we present the “lack” of species “5” in dependence +of α at fixed β = 0.75 and γ = 1 values. When the +value of 1 − ρ5 reaches 1 then the system enters +onto the mentioned A7 +{0,1,2,3,4,6,7} solution. +Evi- +dently, we also checked the portions of other species, +because ρ5 = 0 is fulfilled in other solutions, too. +Here the symbols are the average of many indepen- +dent runs. +As an example, for L = 100 we exe- +cuted 2000 times. Importantly, the average hides +the proper system behavior for small system size, +because it mixes different destinations. For exam- +ple, at L = 100, α = 0.15 we never measured +ρ5 = 0.063. +Instead, the majority of indepen- +dent runs terminated onto a state where ρ5 = 0 +or ρ5 = 0.25. This ambiguity disappears for large +system sizes. The plot also highlights that the tran- +sition from the eight-member to seven-member so- +lution is continuous because species “5” vanishes +gradually as we increase α. The transition, how- +ever, between A7 +{0,1,2,3,4,6,7} and A4 +{1,3,5,7} phases is +discontinuous. +To illustrate another type of phase transition, in +Fig. 6 we show an alternative order parameter in +dependence of β at γ = 0.5 and α = 0.01. Here, +we calculate the difference between the portions of +4 + + + 0.8 + + 0.9 + + 1 + 0 + + 0.1 + + 0.2 + + 0.3 + + 0.4 +all +A7 +{0,1,2,3,4,6,7} +A4 +{1,3,5,7} +1 - ρ5 +α +100 +200 +400 +800 +1600 +Figure 5: The absence of species “5”, as an order param- +eter, in dependence of α at γ = 1, β = 0.75 for different +system sizes. The linear sizes are indicated in the legend. +The dominant solutions are marked on the top. The plots +are the average of 100-2000 runs depending on the system +size. Lines are just to guide the eye. +quartets formed by odd- and even-indexed species, +respectively. When β is small, all available species +coexist, the system is in the “all” phase, hence the +mentioned difference is small. When this parame- +ter becomes 1, then species with even indexes van- +ish and the quartet of “1+3+5+7” species becomes +dominant. +Interestingly, this formation is viable +even if the inner invasion, due to the tiny α = 0.01, +is extremely slow. The explanation of this surpris- +ing behavior is given in the next subsection. +Similarly to the previously discussed Fig. 5 case, +the average of the order parameter could be mis- +leading when the system size is small. This can be +clearly seen for L = 100 and L = 200, where the av- +erage is larger than the value for larger system sizes. +It is because the system can easily be trapped in the +A4 +{1,3,5,7} state already at small β values. The pos- +sible destinations, however, are less ambiguous for +larger sizes, but the jump in the order parameter is +valid signaling a discontinuous phase transition at +β = 0.66. +3.3. Battles of solutions +In the following, we analyze the possible mech- +anisms which explain the system behavior summa- +rized in Fig. 2 and in Fig. 3. To get a deeper insight +about the dominant processes, which drive the pat- +ter formation, it is instructive to use a specific ini- +tial state where we divide the available space into +two halves and both regions are occupied by one of +the main quartets formed by odd- or even-indexed + 0 + 0.2 + 0.4 + 0.6 + 0.8 + 1 + 0 + 0.1 + 0.2 + 0.3 + 0.4 + 0.5 + 0.6 + 0.7 + 0.8 +all +A4 +{1,3,5,7} +(ρ1+ρ3+ρ5+ρ7) - (ρ0+ρ2+ρ4+ρ6) +β +100 +200 +400 +800 +1600 +Figure 6: The difference between symmetric quartets in de- +pendence of β at γ = 0.5, α = 0.1. When this order param- +eter becomes 1, the system enters to a four-species state. +The dominant solutions are marked on the top. The plots +are the average over 20-2000 runs, depending on the system +size. The linear size of squares are indicated. +species. In this way, we can follow how these solu- +tions compete, or how new possibilities may emerge +due to their interaction. +The β − α parameter plane can be divided into +four main sectors which fundamentally determine +the relation of different solutions. +We first con- +sider the low α – low β region. +It is impor- +tant to note that the A4 +{0,2,4,6} quartet formed +by even-indexed species is not a proper solution +here. +Instead, we can see the battle of A3 +{0,2,4} +and A3 +{0,2,6} triplets when even-indexed species are +present. Since species “4” beats species “6”, this +fight ends up with the victory of the former solu- +tion. +This domain is marked by “A” on the left +panel of Fig. 7. When γ is small then there is just +a weak interaction between odd- and even-indexed +species. Therefore, the mentioned domains remain +compact and they fight along the separating inter- +face. Finally, A4 +{1,3,5,7} quartet win this battle and +the whole space will be occupied by the domain, +marked by “B” on the left panel. +If γ is high enough then the evolution changes +drastically. This is illustrated on the right panel +of Fig. 7. Because of high γ, species “7”, who has +no predator in the “0+2+4” triplet, can easily en- +ter into the A3 +{0,2,4} domain. The raid of species +“7” results in a neutral duo. +This A2{2, 7} so- +lution is marked by “C” on this panel. Interest- +ingly, however, this solution has limited life, be- +cause the intensive interactions of the original quar- +tets at the border offer a chance for the complete +5 + +A +B +C +D +Figure 7: Pattern formation in the low α – low β region when +we launch the evolution from a prepared state where left +(right) side of the space is occupied by even-indexed (odd- +indexed) species. In the former case only the A3 +{0,2,4}triplet +survive, shown by “A” on the left panel. For small γ, when +the interaction is weak around the main loop, the A4 +{1,3,5,7} +quartet, marked by “B”, can beat the other solution and +gradually invades the whole space. +If γ is high enough, +species “7” can easily invade the triplet leaving a neutral +A2 +{2,7} pair behind. But this solution, marked by “C”, can- +not survive because the interface between the original quar- +tets is a birthplace of the complete eight-species solution. +This is marked by “D”. Since γ is high enough, this solution +is viable and prevails. Color codes agree with those we intro- +duced in Fig. 1. Parameters are: α = 0.1, β = 0.1, γ = 0.05 +(left panel), and γ = 1 (right panel). +eight-member solution to emerge. +This solution +is marked by “D” on this panel. Because of the +high γ value, the general invasion flow is intensive +in the largest loop, which makes it strong against +A4 +{1,3,5,7}. Notably, the latter quartet is weak due to +small α. The above described mechanism explains +the left-down corners of phase diagrams shown in +Fig. 2 and in Fig. 3. +We can face a conceptually different situation +in the large α – small β corner of the parame- +ter plane, because it provides supporting conditions +both for A4 +{0,2,4,6} and A4 +{1,3,5,7} quartets. There- +fore, both solutions would be viable in the absence +of the other. Importantly, the high α value makes +the difference from the previously discussed cases, +which generates a fast rotation withing both quar- +tets. There is, however, a crucial difference between +these solutions: while A4 +{1,3,5,7} is formed by equal +partners, the extra inner invasions described by β +probability break this delicate balance for the ben- +efit of species “0” at the expense of species “6”. In +this way the pattern formed by even-indexed species +becomes heterogeneous, including larger domains of +“0” species. This makes the whole alliance vulner- +able against the attack of the rival well-balanced +loop. +This process is illustrated in Fig. 8, where we + 0 + + 0.1 + + 0.2 + + 0.3 + 0 + 20000 + 40000 + 60000 + 80000 100000 120000 +0 +6 +2 +4 +1 3 5 7 +ρi +time [MCS] +Figure 8: The time evolution of species when quartets are +fighting at high α, low β values. +Initially both A4 +{0,2,4,6} +and A4 +{1,3,5,7} solutions evolve independently in separated +areas of available space. When separating borders opened, +marked by an arrow, symmetric quartet gradually crowd out +the rival gang. Color codes are the usual, except white line +is replaced by grey one. Parameters are: α = 0.5, β = 0.1, +γ = 0.1, L = 1600. +monitor the battle of these quartets. Initially, we +allowed both solutions to emerge peacefully in re- +stricted areas and the stationary portions of species +“0” and “6” change, as we described previously. Let +us stress that in the initial state all species repre- +sented equally, which is practically invisible because +the unequal stationary distributions of “0+2+4+6” +species evolve very fast. After 20000 MCSs, when +the separating borders opened, the symmetric so- +lution gradually invades the whole space. Notably, +the presented simulation was recorded at γ = 0.1, +where the interaction between the quartets is mod- +erate. Still, this light communication is capable to +reveal the advantage of A4 +{1,3,5,7} solution. +If we +apply larger γ values then nothing changes concep- +tually, but the victory of A4 +{1,3,5,7} becomes faster. +Hence, in agreement with the phase diagrams, the +symmetric four-member solutions is always the win- +ner in the mentioned corner of the β − α parameter +plane. +The lastly discussed observation also answers one +of our original questions. Namely, one may argue +that the introduction of an additional inner inva- +sion within A4 +{0,2,4,6} solution results in a faster ro- +tation among these species, which could be useful +them [39]. But the weakening consequence of sym- +metry breaking is stronger, as it was also the case +for three-member loops [40]. +Next, we discuss the small α – large β corner of +the parameter plane. Here, the situation is partly +6 + +similar to the first discussed case. More precisely, +A4 +{0,2,4,6} solution is unstable and replaced very +soon by A3 +{0,2,4}. But this triplet is vulnerable, be- +cause the large heterogeneity of inner flow, (α ≪ β), +results in huge homogeneous domains, as it was al- +ready reported in [56] for traditional rock-scissors- +paper game. +Such a large homogeneous domain +is also an easy prey of the rival A4 +{1,3,5,7} quartet. +This picture is valid for small and medium γ values, +where the interaction between the even- and odd- +indexed species is not too strong. For large γ val- +ues, however, the evolution could be more complex, +which can be observed more easily from a “random- +patch” initial state. This starting pattern is illus- +trated in Fig. 9. As we already mentioned in Sec. 2, +this prepared initial state can offer all possible in- +terfaces to be present at the very beginning, which +is extremely useful when there are large difference +among the invasion rates. In this way, we can ob- +serve the valid solution already at smaller system +size. We stress, however, that the mentioned state +should be reached from all kind of initial states if +all available species are present and the system size +is large enough. +(a) +(b) +(c) +Figure 9: Alternative initial states from where evolution is +launched when L = 200. +Panel (a) shows the traditional +starting state where every node is uploaded by a randomly +chosen species. Panel (b) shows a state where two competing +quartets are generated first. Panel (c) shows a state where +larger patches of species are distributed randomly. The so- +lution, which is valid in the large size limit, can be reached +from all starting points, but the necessary system size could +be largely different. +Turning back to the large γ, large β, small α re- +gion, the above specified prepared initial state can +help us to identify the valid A7 +{0,1,2,3,4,6,7} solution +already at relatively small system sizes. +At first +sight, this solution may seem weird or exotic, but its +emergence can be understood if we follow how the +portion of species change by increasing β. This phe- +nomenon is illustrated in Fig. 10 where we present +the stationary fractions of all species when we in- +crease the intensity of additional invasions at fixed +γ = 1 and α = 0.2. At β = 0 we have a completely +symmetric food-web where there are equally strong + 0 + + 0.1 + + 0.2 + 0 + 0.1 + 0.2 + 0.3 + 0.4 + 0.5 + 0.6 + 0.7 +all +A7 +{0,1,2,3,4,6,7} +0 +1 +2 +3 +4 +5 +6 +7 +ρi +β +Figure 10: Stationary fractions of species in dependence of +β at γ = 1, α = 0.2. The results are obtained at L = 3200 +system size. Species are denoted to every lines. For clarity +we only show the lines without symbols here. +The stable +phases are shown on the top. +quartets. Consequently, all species are present at +the same level here. When we introduce a non-zero +β then not only the fractions of species “0” and +“6” start decaying, but simultaneously, the portions +of their principal preys, species “1” and “7” start +growing. This increment involves the decay of their +preys, which are species “2” and “0”. This dou- +ble stress explains why species “0” suffers the most +at small β values. Naturally, the decay of species +“6” affects the population of its principal predator, +hence the portion of species “5” also a decreasing +function. One may argue that the decay of species +“0” should affect its main predator species “7”. But +the predator of the latter, which is species “6”, can- +not grow here, hence in sum species “7” should not +necessarily decrease. This difference between the +status of species “5” and “7” explains why there are +diverse consequences when both of their principal +preys, species “6” and “0”, are attacked directly via +an inner invasion described by parameter β. Impor- +tantly, the direct support of species “4” via the in- +tensive “4”→“2” helps the spreading of species “4”. +This process is also dangerous for species “5”. On +the other hand, the intensive “2”→“6” process will +not simply strengthen species “2”, but also weaken +its prey species “3”, which consequence is also en- +joyed by species “4”. When the latter species die +out, the remaining seven species form a heteroge- +neous seven-member Lotka-Volterra loop where the +“weakest” species (species “4”, who beats species +“6” with probability α) occupies the largest frac- +tion of space. This effect is an example for the phe- +7 + +C +B +F +H +G +A +D +E +I +Figure 11: Intermediate stage of the evolution taken from +the battlefield when different solutions emerge and fight for +space. The domains mark the following solutions: A4 +{0,1,3,4} +(A), +A5 +{1,3,4,5,7} +(B), +A4 +{1,3,5,7} +(C), +A5 +{0,2,3,5,7} +(D), +A5 +{0,1,3,4,6} (E), A3 +{0,2,4} (F ), A6 +{1,2,3,4,5,7} (G), A5 +{1,2,4,5,7} +(H), and A5 +{1,2,3,5,7} (I). Finally “C” domain wins. Param- +eters are α = 0.9, β = 0.9, γ = 1, and L = 400. +nomenon observed first by Tainaka in the simplest +three-member loop [57]. +Last, we discuss the remaining large α – large β +corner of the parameter plane. Here the A4 +{0,2,4,6} +solution is not stable again, hence the remaining +A3 +{0,2,4} solution fights against A4 +{1,3,5,7} quartet. +When γ is low and the interaction is weak between +these groups then the latter formation wins. This +evolution is similar to the case we reported in the +left panel of Fig. 7. Practically, the same happens +when γ is high, but in this case a large set of solu- +tions may emerge temporarily. Despite of this di- +versity, however, the winning alliance remains the +symmetric quartet. +To give an impression about +the variety of different candidates, we present an in- +termediate snapshot about the “battlefield” taken +at α = 0.9, β = 0.9, and γ = 1. Figure 11 shows +that in the intermediate stage of the evolution at +least nine(!) solutions emerge locally and fight for +space. But eventually the symmetric A4 +{1,3,5,7} so- +lution crowds out all other competitors. +4. Conclusions +Which are the most important features of a cycli- +cally dominant alliance that determine its viabil- +ity against alternative formations? +Motivated by +this question, we introduced an eight-species model +where there are two four-member quartets who in- +teract each other, hence forming a complete eight- +member loop, too. +The original model is com- +pletely symmetric where the inner invasion within +the quartets are equally strong. Additionally, we +break the symmetry and introduced an extra inner +invasion within one of the loops. +Based on pre- +vious observations about three-member loops, one +may argue that this faster rotation of alliance mem- +bers could be positive to make them stronger. From +the other viewpoint, the broken symmetry is always +detrimental, hence the new opportunity would just +weaken the involved quartet. It is also worth not- +ing that the slight alteration of the original “sym- +metric” food-web offers the chance several potential +solutions to emerge. Indeed, an armada of candi- +dates can be observed at intermediate stage of the +evolution, but it is believed that the final pattern +is determined by some basic concepts. +According to our findings, keeping the symme- +try is vital and the solutions, which maintain a +balance among their members, are fitter. +In the +complete parameter space we studied, it can hap- +pen that there are more than one solution which +possess this attractive character. In this case the +general speed of inner invasion could be a decisive +factor. If the rotations are equally strong then we +could give examples when a short or a larger loop is +the winner. For example, a quartet can beat a trio, +but a quartet can also beat an octet. Therefore, it +cannot be made a simple conclusion that a smaller +or larger alliance is better. Probably, there are two +competing effects whose relation determines the ac- +tual fitness of a loop. On one hand, a short loop +may involve relatively large homogeneous domains, +which could always be dangerous. +On the other +hand, too large loop also means that the reaction of +the alliance could be delayed, because several inner +invasions should happen to produce the predator of +the external intruder. Therefore, the sum of these +adverse impacts could be case-sensitive. +Naturally, our present study is a sterile theoreti- +cal model, but we strongly believe that the observa- +tions we made could be useful when real systems are +studied. Our another important message is the im- +portance of appropriately chosen system size, which +8 + +is always a central issue when cyclic dominance is +present. Otherwise, the observations could be di- +verse without deeper understanding. This is why +we should always consider the actual size when we +want to give predictions about the pattern forma- +tion in a finite system driven by intransitive inter- +actions. +This work was supported by the National Re- +search Foundation of Korea (NRF) grant funded +by the Korea government (MSIT) (No. +NRF- +2021R1A4A1032924). J.P. was also supported by +a grant from Kyung Hee University in 2022 (KHU- +20220901). +X.C. was supported by the National +Natural Science Foundation of China (Grants Nos. +61976048 and 62036002) and the Fundamental Re- +search Funds of the Central Universities of China. +A.S. was supported by the National Research, De- +velopment and Innovation Office (NKFIH) under +Grant No. K142948. +References +[1] K. Sigmund, Games of Life: Exploration in Ecology, +Evolution and Behavior, Oxford University Press, Ox- +ford, UK, 1993. +[2] J. Bascompte, G. Sol´e (Eds.), Modeling Spatiotemporal +Dynamics in Ecology, Springer, New York, 1998. +[3] B. Sinervo, C. M. 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A 176 (1993) 303–306. +10 + diff --git a/FNAyT4oBgHgl3EQfrPlH/content/tmp_files/load_file.txt b/FNAyT4oBgHgl3EQfrPlH/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..483a2556490d6854a57c368cef3003b2bf88e96a --- /dev/null +++ b/FNAyT4oBgHgl3EQfrPlH/content/tmp_files/load_file.txt @@ -0,0 +1,772 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf,len=771 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='00556v1 [q-bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='PE] 2 Jan 2023 Competition of alliances in a cyclically dominant eight-species population Junpyo Parka, Xiaojie Chenb, and Attila Szolnokic,∗∗ aDepartment of Applied Mathematics, Kyung Hee University, Yongin 17104, Republic of Korea bSchool of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China cInstitute of Technical Physics and Materials Science, Centre for Energy Research, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Box 49, H-1525 Budapest, Hungary Abstract In a diverse population, where many species are present, competitors can fight for surviving at individual and collective levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' In particular, species, which would beat each other individually, may form a specific alliance that ensures them stable coexistence against the invasion of an external species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Our principal goal is to identify those general features of a formation which determine its vitality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Therefore, we here study a traditional Lotka-Volterra model of eight-species where two four-species cycles can fight for space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Beside these formations, there are other solutions which may emerge when invasion rates are varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The complete range of parameters is explored and we find that in most of the cases those alliances prevail which are formed by equally strong members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Interestingly, there are regions where the symmetry is broken and the system is dominated by a solution formed by seven species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Our work also highlights that serious finite-size effects may emerge which prevent observing the valid solution in a small system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Keywords: cyclic dominance, alliances, competition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Introduction If a species is stronger and beats the other one then how can the “weaker” species survive?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' This is a fundamental problem of ecology to explain the amazing biodiversity of life [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' A possible expla- nation could be the so-called intransitive or cycli- cally dominated relation among competing species where a third (or more) species beat the preda- tor species, hence establishing a delicate balance among all competitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' In the simplest case, these relations remind the well-known rock-scissors-paper game and this type of interaction can really be observed among animals, plants, or even among bacterias [3, 4, 5, 6, 7, 8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Evidently, a three- member loop can be easily enlarged for more par- ticipants, as it was made in the extended Lotka- Volterra model [10, 11, 12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Motivated by the complexity of such interactions, researchers have studied related models actively over the last decade and several interesting observations have been made [14, 15, 16, 17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' ∗Email addresses: junpyopark@khu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='kr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' xiao- jiechen@uestc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' szolnoki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='attila@ek-cer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='hu ∗∗Corresponding author It would be almost impossible to sum all impor- tant observations, such as reported in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [19, 20, 21, 22, 23, 24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Instead, the reader is referred to specific review papers about the milestones along this research path [26, 27, 28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Nevertheless, it is also worth noting here that the mentioned cyclic or intransitive relation among participants should not necessarily be defined by a Lotka-Volterra-type mi- croscopic rule, but could be the result of collective behavior among competing strategies in evolution- ary game models [30, 31, 32, 33, 34, 35, 36, 37, 38] The most exciting question is whether we can predict the direction of evolution based on the food- web that defines the microscopic dynamics?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Can we identify general principles which may inform us about the vitality of an alliance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' An early observa- tion was, for instance, that when two three-member cycles fight then the one, where the inner invasion is faster, is more viable [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The picture, how- ever, is more subtle, because a faster general rota- tion does not necessarily provide an advantage if the members of the trio are unequal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' In the lat- ter case, the triplet becomes vulnerable no matter the average of inner invasion rates is higher than in the rival triplet which is formed by less aggressive, Preprint submitted to Chaos Solitons and Fractals January 3, 2023 but equally strong partners [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Notably, hetero- geneous invasion rates may emerge temporarily or locally [41, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Or, the extension of the original food-web by adding a reverse link can also change a stable solution [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Furthermore, the number of members could also be a decisive factor because a smaller loop is generally stronger than a large one [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' In this work we introduce a minimal eight-species model where two four-member loops fight for space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The interaction between these quartets practically establishes a traditional Lotka-Volterra circle of eight species, which can be described by an inva- sion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Not to break the balance between the quartets, we assume that the inner invasion is uni- form and equally strong in both formations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The only difference between them is we introduce two short-cuts in one of the loops which generates extra interactions within the related alliance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Our major question is how the stability of alliances change due to this extension and can we identify new solutions which would be hidden otherwise?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' It could also be interesting whether the previously established principles, observed for simpler systems formed by trials and duets, remain valid when we apply them to alliances formed by larger groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Alliances formed at different levels To make our observations comparable with pre- vious works [45, 46, 47, 48, 49, 50], we assume that species are distributed on an L × L square lattice where periodic boundary conditions are ap- plied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Each node is occupied by one of the species which are denoted by Xi where indexes are from i = 0 to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' During an elementary step, we select a neighboring pair nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' If they are occupied by the same species or represent species which are neu- tral then nothing happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Otherwise, an invasion happens with a specific probability and predator species invades prey species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The microscopic rules are defined in the following way: XiXi+1 γ−→ XiXi (1) XiXi+2 α−→ XiXi (2) X2X6 β−→ X2X2, X4X0 β−→ X4X4, (3) where i + 1 and i + 2 are calculated in cyclic man- ner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The parameters α, β and γ define the proba- bilities of successful invasions between the involved predator-prey neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' γ α β 1 2 4 0 3 5 6 7 Figure 1: Food-web of competing species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' In the basic model eight species are invade cyclically each other with probabil- ity γ, similar to the extended Lotka-Volterra model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Ad- ditionally, we introduce a cyclic inner invasion among odd and separately among even species with probability α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' To break the symmetry among the quartets, we also introduce an inner invasion between species “2” and “6” and between species “4” and “0” with probability β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Importantly, we use the color-code of species in later figures where spatial distri- butions are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' For a deeper insight about the model, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 1 we show the food-web of competing species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Our first note is about the biggest loop where all species are involved, according to the extended Lotka-Volterra type system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Here every species is simultaneously a predator and a prey of another one, hence es- tablishing a possible solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The invasion rate is γ for all interactions here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Another solution is formed by “1”, “3”, “5”, and “7” species who in- vade each other cyclically with probability α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' For later reference, we denote this alliance as A4 1,3,5,7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' A similar A4 0,2,4,6 quartet is formed among species “0”, “2”, “4” and “6”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Additionally, we introduce an extra chance of invasions among group members here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' In particular, species “2” invades species “6” and species “4” invades species “0” with probabil- ity β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' In this way, there are no neutral pairs in the mentioned quartet anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' As a consequence, the average inner invasion is augmented among the four species, which could be a support to their via- bility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Interestingly, however, there are many other possible solutions in this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' For example, the new invasions among even-numbered species offer 2 the chance for trials to emerge: A3 0,2,4 or A3 0,2,6 can work as a rock-scissors-paper-type solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Beside the mentioned formations, there are A5 (five-), A6 (six-), or A7 seven-member solutions, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Per- haps, it is not necessary to specify them because the reader can easily construct examples, based on the food-web, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' As we noted, there are three parameters in our model and in the following we systematically scan the whole parameter field to identify the dominant solution for each combination of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' For this reason, we executed Monte Carlo simulations where a Monte Carlo step (MCS) means that on average every node has a chance to update its state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The linear system size varied between L = 100 to L = 3200 and the necessary relaxation steps were between 103 to 3 × 105 MCSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' According to the standard protocol, simulations were launched from an initial state where species are distributed ran- domly on the lattice and monitor the ρi concentra- tion of Xi for all species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' We, however, would like to stress that this initial state does not always could be a good choice to find the solution which is valid in the large size limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' It is because a small sys- tem size does not necessarily “offer” equal chance for all possible solutions to emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Some solution, which involves large typical lengths, would emerge just later, but to that stage of the evolutionary pro- cess other solutions may invade the whole available space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Therefore, a more complex solution practi- cally has no chance to emerge at a small system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' As a consequence, other solutions may win and an independent run can easily terminates onto a differ- ent solution, no matter we use the same set of model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' To overcome this uncertainty, we not simply enlarged the system size, but also used alter- native initial states where larger and homogeneous domains of competing species were distributed ran- domly on the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Similar approach was used previously in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [51, 52], for instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' In this way, we can create all types of interfaces which could be the foundation to build more sophisticated solu- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' To make our analysis more complete, at the critical values of control parameters we also gener- ated sub-solutions independently in restricted areas of space and after we let them fight directly along the separating interface [53, 54, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' This technique can identify the dominant solution unambiguously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The details of the above mentioned protocols will be specified in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Results 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Phase diagrams We first summarize our main findings and later we give further details about the microscopic mech- anisms which are responsible how dominant solu- tions emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' According to our simulations, we can distinguish three main cases which determine the system behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The decisive condition is the intensity of interaction between the quartets men- tioned in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' If this interaction is weak, means the value of γ is small, then the A4 {1,3,5,7} solution dominates the whole β − α parameter plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' In other words, only the quartet of “1+3+5+7” sur- vive independently of the α and β values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' For intermediate values of γ, when the invasion flow in the large cycle becomes substantial, a con- ceptually new behavior emerges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' We represent this phenomenon by showing the phase diagram ob- tained at γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Figure 2 shows that beside the mentioned A4 {1,3,5,7} quartet, other solutions be- come dominant at certain values of β − α pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' When both α and β are small then the “large cir- cle” is the winner, hence all eight species coexist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' For intermediate β values this solution is replaced by A7 {0,1,2,3,4,6,7} where only species “5” is missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='8 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='8 1 A4 {1,3,5,7} A7 {0,1,2,3,4,6,7} all β α Figure 2: Phase diagram on β − α parameter plane obtained at γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' If α is large enough, the quartet of “1+3+5+7” species always win.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' If both β and α are small, all species survive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Interestingly, for low α and intermediate β a seven- member solution dominates where species “5” is missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Dashed blue line denotes discontinuous, while solid red line marks the positions of continuous phase transition points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Qualitatively similar system behavior can be ob- served when γ is large, hence the flow in the ex- 3 ternal loop is intensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' In other words, the inter- action between the quartets becomes large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' This is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 3 where we present the com- plete phase diagram on the β − α plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The only difference between the diagrams is the area, where complete eight-species solution dominates, is larger and the seven-member solution is shifted toward larger β values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='8 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='8 1 A4 {1,3,5,7} A7 {0,1,2,3,4,6,7} all β α Figure 3: Phase diagram on β − α parameter plane obtained at γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The diagram is similar to the one shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The only difference is the eight-member full solution occu- pies larger area in the parameter field, while seven-member solution is shifted toward higher β values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' We must stress that the presented phase dia- grams are valid in the large system size limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' If the system size is small then we may observe that the system can easily terminate onto different so- lutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' To illustrate it, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 4 we present the survival probabilities at each (α, β) parameter pairs for γ = 1, when the linear system size was L = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' More precisely, we launched the evolution from a random initial state and monitored the fractions of species until N = L×L MCSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' After, we recorded the number of surviving species and repeated the whole process 100 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' In this way we can esti- mate the probability that S species survive for long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 4 show these surviving probabilities for all possible S values at x different (α, β) pairs on the parameter plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' These pan- els indicate that different solutions may emerge no matter we use the same values of (α, β) parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Therefore, the destination is rarely unambiguous, hence the surviving probability equals to 1 only for S = 4 when α is high and β is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Nevertheless, 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='5 1 S=1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='5 1 S=2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='5 1 S=3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='5 1 S=4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='5 1 S=5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='5 1 S=6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='5 1 S=7 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='5 1 S=8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='8 1 Figure 4: Heat maps of the survival probability on β − α parameter plane obtained for γ = 1 by using 100×100 system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Each panel shows the probability to reach a state which contains S species after N = L2 MCSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The value of S is indicated on each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The results are averaged over 100 independent runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' the contour of areas around “maximum” values are roughly agree with the diagram shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' But, as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 4 illustrated, this system size is insuffi- cient to make reliable conclusions about the proper system behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Detecting phase transitions Therefore, to detect the phase transition points more precisely, we need to apply systematic finite- size analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' An example is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 5 where we present the “lack” of species “5” in dependence of α at fixed β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='75 and γ = 1 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' When the value of 1 − ρ5 reaches 1 then the system enters onto the mentioned A7 {0,1,2,3,4,6,7} solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Evi- dently, we also checked the portions of other species, because ρ5 = 0 is fulfilled in other solutions, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Here the symbols are the average of many indepen- dent runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' As an example, for L = 100 we exe- cuted 2000 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Importantly, the average hides the proper system behavior for small system size, because it mixes different destinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' For exam- ple, at L = 100, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='15 we never measured ρ5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='063.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Instead, the majority of indepen- dent runs terminated onto a state where ρ5 = 0 or ρ5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' This ambiguity disappears for large system sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The plot also highlights that the tran- sition from the eight-member to seven-member so- lution is continuous because species “5” vanishes gradually as we increase α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The transition, how- ever, between A7 {0,1,2,3,4,6,7} and A4 {1,3,5,7} phases is discontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' To illustrate another type of phase transition, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 6 we show an alternative order parameter in dependence of β at γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='5 and α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Here, we calculate the difference between the portions of 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='9 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='4 all A7 {0,1,2,3,4,6,7} A4 {1,3,5,7} 1 - ρ5 α 100 200 400 800 1600 Figure 5: The absence of species “5”, as an order param- eter, in dependence of α at γ = 1, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='75 for different system sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The linear sizes are indicated in the legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The dominant solutions are marked on the top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The plots are the average of 100-2000 runs depending on the system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Lines are just to guide the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' quartets formed by odd- and even-indexed species, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' When β is small, all available species coexist, the system is in the “all” phase, hence the mentioned difference is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' When this parame- ter becomes 1, then species with even indexes van- ish and the quartet of “1+3+5+7” species becomes dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Interestingly, this formation is viable even if the inner invasion, due to the tiny α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='01, is extremely slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The explanation of this surpris- ing behavior is given in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Similarly to the previously discussed Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 5 case, the average of the order parameter could be mis- leading when the system size is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' This can be clearly seen for L = 100 and L = 200, where the av- erage is larger than the value for larger system sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' It is because the system can easily be trapped in the A4 {1,3,5,7} state already at small β values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The pos- sible destinations, however, are less ambiguous for larger sizes, but the jump in the order parameter is valid signaling a discontinuous phase transition at β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Battles of solutions In the following, we analyze the possible mech- anisms which explain the system behavior summa- rized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 2 and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' To get a deeper insight about the dominant processes, which drive the pat- ter formation, it is instructive to use a specific ini- tial state where we divide the available space into two halves and both regions are occupied by one of the main quartets formed by odd- or even-indexed 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='8 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='8 all A4 {1,3,5,7} (ρ1+ρ3+ρ5+ρ7) - (ρ0+ρ2+ρ4+ρ6) β 100 200 400 800 1600 Figure 6: The difference between symmetric quartets in de- pendence of β at γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='5, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' When this order param- eter becomes 1, the system enters to a four-species state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The dominant solutions are marked on the top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The plots are the average over 20-2000 runs, depending on the system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The linear size of squares are indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' In this way, we can follow how these solu- tions compete, or how new possibilities may emerge due to their interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The β − α parameter plane can be divided into four main sectors which fundamentally determine the relation of different solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' We first con- sider the low α – low β region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' It is impor- tant to note that the A4 {0,2,4,6} quartet formed by even-indexed species is not a proper solution here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Instead, we can see the battle of A3 {0,2,4} and A3 {0,2,6} triplets when even-indexed species are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Since species “4” beats species “6”, this fight ends up with the victory of the former solu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' This domain is marked by “A” on the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' When γ is small then there is just a weak interaction between odd- and even-indexed species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Therefore, the mentioned domains remain compact and they fight along the separating inter- face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Finally, A4 {1,3,5,7} quartet win this battle and the whole space will be occupied by the domain, marked by “B” on the left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' If γ is high enough then the evolution changes drastically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' This is illustrated on the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Because of high γ, species “7”, who has no predator in the “0+2+4” triplet, can easily en- ter into the A3 {0,2,4} domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The raid of species “7” results in a neutral duo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' This A2{2, 7} so- lution is marked by “C” on this panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Interest- ingly, however, this solution has limited life, be- cause the intensive interactions of the original quar- tets at the border offer a chance for the complete 5 A B C D Figure 7: Pattern formation in the low α – low β region when we launch the evolution from a prepared state where left (right) side of the space is occupied by even-indexed (odd- indexed) species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' In the former case only the A3 {0,2,4}triplet survive, shown by “A” on the left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' For small γ, when the interaction is weak around the main loop, the A4 {1,3,5,7} quartet, marked by “B”, can beat the other solution and gradually invades the whole space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' If γ is high enough, species “7” can easily invade the triplet leaving a neutral A2 {2,7} pair behind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' But this solution, marked by “C”, can- not survive because the interface between the original quar- tets is a birthplace of the complete eight-species solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' This is marked by “D”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Since γ is high enough, this solution is viable and prevails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Color codes agree with those we intro- duced in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Parameters are: α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='1, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='1, γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='05 (left panel), and γ = 1 (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' eight-member solution to emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' This solution is marked by “D” on this panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Because of the high γ value, the general invasion flow is intensive in the largest loop, which makes it strong against A4 {1,3,5,7}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Notably, the latter quartet is weak due to small α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The above described mechanism explains the left-down corners of phase diagrams shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 2 and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' We can face a conceptually different situation in the large α – small β corner of the parame- ter plane, because it provides supporting conditions both for A4 {0,2,4,6} and A4 {1,3,5,7} quartets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' There- fore, both solutions would be viable in the absence of the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Importantly, the high α value makes the difference from the previously discussed cases, which generates a fast rotation withing both quar- tets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' There is, however, a crucial difference between these solutions: while A4 {1,3,5,7} is formed by equal partners, the extra inner invasions described by β probability break this delicate balance for the ben- efit of species “0” at the expense of species “6”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' In this way the pattern formed by even-indexed species becomes heterogeneous, including larger domains of “0” species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' This makes the whole alliance vulner- able against the attack of the rival well-balanced loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' This process is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 8, where we 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='3 0 20000 40000 60000 80000 100000 120000 0 6 2 4 1 3 5 7 ρi time [MCS] Figure 8: The time evolution of species when quartets are fighting at high α, low β values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Initially both A4 {0,2,4,6} and A4 {1,3,5,7} solutions evolve independently in separated areas of available space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' When separating borders opened, marked by an arrow, symmetric quartet gradually crowd out the rival gang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Color codes are the usual, except white line is replaced by grey one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Parameters are: α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='5, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='1, γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='1, L = 1600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' monitor the battle of these quartets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Initially, we allowed both solutions to emerge peacefully in re- stricted areas and the stationary portions of species “0” and “6” change, as we described previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Let us stress that in the initial state all species repre- sented equally, which is practically invisible because the unequal stationary distributions of “0+2+4+6” species evolve very fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' After 20000 MCSs, when the separating borders opened, the symmetric so- lution gradually invades the whole space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Notably, the presented simulation was recorded at γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='1, where the interaction between the quartets is mod- erate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Still, this light communication is capable to reveal the advantage of A4 {1,3,5,7} solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' If we apply larger γ values then nothing changes concep- tually, but the victory of A4 {1,3,5,7} becomes faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Hence, in agreement with the phase diagrams, the symmetric four-member solutions is always the win- ner in the mentioned corner of the β − α parameter plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The lastly discussed observation also answers one of our original questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Namely, one may argue that the introduction of an additional inner inva- sion within A4 {0,2,4,6} solution results in a faster ro- tation among these species, which could be useful them [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' But the weakening consequence of sym- metry breaking is stronger, as it was also the case for three-member loops [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Next, we discuss the small α – large β corner of the parameter plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Here, the situation is partly 6 similar to the first discussed case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' More precisely, A4 {0,2,4,6} solution is unstable and replaced very soon by A3 {0,2,4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' But this triplet is vulnerable, be- cause the large heterogeneity of inner flow, (α ≪ β), results in huge homogeneous domains, as it was al- ready reported in [56] for traditional rock-scissors- paper game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Such a large homogeneous domain is also an easy prey of the rival A4 {1,3,5,7} quartet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' This picture is valid for small and medium γ values, where the interaction between the even- and odd- indexed species is not too strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' For large γ val- ues, however, the evolution could be more complex, which can be observed more easily from a “random- patch” initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' This starting pattern is illus- trated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' As we already mentioned in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 2, this prepared initial state can offer all possible in- terfaces to be present at the very beginning, which is extremely useful when there are large difference among the invasion rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' In this way, we can ob- serve the valid solution already at smaller system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' We stress, however, that the mentioned state should be reached from all kind of initial states if all available species are present and the system size is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' (a) (b) (c) Figure 9: Alternative initial states from where evolution is launched when L = 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Panel (a) shows the traditional starting state where every node is uploaded by a randomly chosen species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Panel (b) shows a state where two competing quartets are generated first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Panel (c) shows a state where larger patches of species are distributed randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The so- lution, which is valid in the large size limit, can be reached from all starting points, but the necessary system size could be largely different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Turning back to the large γ, large β, small α re- gion, the above specified prepared initial state can help us to identify the valid A7 {0,1,2,3,4,6,7} solution already at relatively small system sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' At first sight, this solution may seem weird or exotic, but its emergence can be understood if we follow how the portion of species change by increasing β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' This phe- nomenon is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 10 where we present the stationary fractions of all species when we in- crease the intensity of additional invasions at fixed γ = 1 and α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' At β = 0 we have a completely symmetric food-web where there are equally strong 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='7 all A7 {0,1,2,3,4,6,7} 0 1 2 3 4 5 6 7 ρi β Figure 10: Stationary fractions of species in dependence of β at γ = 1, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The results are obtained at L = 3200 system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Species are denoted to every lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' For clarity we only show the lines without symbols here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The stable phases are shown on the top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' quartets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Consequently, all species are present at the same level here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' When we introduce a non-zero β then not only the fractions of species “0” and “6” start decaying, but simultaneously, the portions of their principal preys, species “1” and “7” start growing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' This increment involves the decay of their preys, which are species “2” and “0”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' This dou- ble stress explains why species “0” suffers the most at small β values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Naturally, the decay of species “6” affects the population of its principal predator, hence the portion of species “5” also a decreasing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' One may argue that the decay of species “0” should affect its main predator species “7”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' But the predator of the latter, which is species “6”, can- not grow here, hence in sum species “7” should not necessarily decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' This difference between the status of species “5” and “7” explains why there are diverse consequences when both of their principal preys, species “6” and “0”, are attacked directly via an inner invasion described by parameter β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Impor- tantly, the direct support of species “4” via the in- tensive “4”→“2” helps the spreading of species “4”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' This process is also dangerous for species “5”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' On the other hand, the intensive “2”→“6” process will not simply strengthen species “2”, but also weaken its prey species “3”, which consequence is also en- joyed by species “4”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' When the latter species die out, the remaining seven species form a heteroge- neous seven-member Lotka-Volterra loop where the “weakest” species (species “4”, who beats species “6” with probability α) occupies the largest frac- tion of space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' This effect is an example for the phe- 7 C B F H G A D E I Figure 11: Intermediate stage of the evolution taken from the battlefield when different solutions emerge and fight for space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The domains mark the following solutions: A4 {0,1,3,4} (A), A5 {1,3,4,5,7} (B), A4 {1,3,5,7} (C), A5 {0,2,3,5,7} (D), A5 {0,1,3,4,6} (E), A3 {0,2,4} (F ), A6 {1,2,3,4,5,7} (G), A5 {1,2,4,5,7} (H), and A5 {1,2,3,5,7} (I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Finally “C” domain wins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Param- eters are α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='9, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='9, γ = 1, and L = 400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' nomenon observed first by Tainaka in the simplest three-member loop [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Last, we discuss the remaining large α – large β corner of the parameter plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Here the A4 {0,2,4,6} solution is not stable again, hence the remaining A3 {0,2,4} solution fights against A4 {1,3,5,7} quartet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' When γ is low and the interaction is weak between these groups then the latter formation wins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' This evolution is similar to the case we reported in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Practically, the same happens when γ is high, but in this case a large set of solu- tions may emerge temporarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Despite of this di- versity, however, the winning alliance remains the symmetric quartet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' To give an impression about the variety of different candidates, we present an in- termediate snapshot about the “battlefield” taken at α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='9, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='9, and γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Figure 11 shows that in the intermediate stage of the evolution at least nine(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=') solutions emerge locally and fight for space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' But eventually the symmetric A4 {1,3,5,7} so- lution crowds out all other competitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Conclusions Which are the most important features of a cycli- cally dominant alliance that determine its viabil- ity against alternative formations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Motivated by this question, we introduced an eight-species model where there are two four-member quartets who in- teract each other, hence forming a complete eight- member loop, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The original model is com- pletely symmetric where the inner invasion within the quartets are equally strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Additionally, we break the symmetry and introduced an extra inner invasion within one of the loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Based on pre- vious observations about three-member loops, one may argue that this faster rotation of alliance mem- bers could be positive to make them stronger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' From the other viewpoint, the broken symmetry is always detrimental, hence the new opportunity would just weaken the involved quartet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' It is also worth not- ing that the slight alteration of the original “sym- metric” food-web offers the chance several potential solutions to emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Indeed, an armada of candi- dates can be observed at intermediate stage of the evolution, but it is believed that the final pattern is determined by some basic concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' According to our findings, keeping the symme- try is vital and the solutions, which maintain a balance among their members, are fitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' In the complete parameter space we studied, it can hap- pen that there are more than one solution which possess this attractive character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' In this case the general speed of inner invasion could be a decisive factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' If the rotations are equally strong then we could give examples when a short or a larger loop is the winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' For example, a quartet can beat a trio, but a quartet can also beat an octet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Therefore, it cannot be made a simple conclusion that a smaller or larger alliance is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Probably, there are two competing effects whose relation determines the ac- tual fitness of a loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' On one hand, a short loop may involve relatively large homogeneous domains, which could always be dangerous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' On the other hand, too large loop also means that the reaction of the alliance could be delayed, because several inner invasions should happen to produce the predator of the external intruder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Therefore, the sum of these adverse impacts could be case-sensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Naturally, our present study is a sterile theoreti- cal model, but we strongly believe that the observa- tions we made could be useful when real systems are studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Our another important message is the im- portance of appropriately chosen system size, which 8 is always a central issue when cyclic dominance is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Otherwise, the observations could be di- verse without deeper understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' This is why we should always consider the actual size when we want to give predictions about the pattern forma- tion in a finite system driven by intransitive inter- actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' This work was supported by the National Re- search Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' NRF- 2021R1A4A1032924).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' was also supported by a grant from Kyung Hee University in 2022 (KHU- 20220901).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' was supported by the National Natural Science Foundation of China (Grants Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 61976048 and 62036002) and the Fundamental Re- search Funds of the Central Universities of China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='S.' 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(2023) +Preprint 3 January 2023 +Compiled using MNRAS LATEX style file v3.0 +Time-averaging Polarimetric and Spectral Properties of GRBs +Liang Li1,2,3⋆ Soroush Shakeri4,1,5† +1ICRANet, Piazza della Repubblica 10, I-65122 Pescara, Italy +2ICRA and Dipartimento di Fisica, Università di Roma “La Sapienza”, Piazzale Aldo Moro 5, I-00185 Roma, Italy +3INAF – Osservatorio Astronomico d’Abruzzo, Via M. Maggini snc, I-64100, Teramo, Italy +4Department of Physics, Isfahan University of Technology, Isfahan 84156-83111 +5ICRANet-Isfahan, Isfahan University of Technology, Isfahan 84156-83111, Iran +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +One of the most fundamental and yet open issues in gamma-ray burst (GRB) physics, is the comprehension of the nature of +their jet composition. The investigation of joint polarimetric and spectral properties is essential to probe the jet composition and +radiation mechanism of GRBs. Several distinct categories of jet properties—the “Kinetic-energy-dominated" (KED), “Poynting- +flux-dominated" (PFD), and “Hybrid-dominated" (HD) jets—have been observed in the observed GRB spectra, and the emission +dominated by different jet properties is expected to have a different level of polarization (πKED ≲ πHD ≲ πPED). In the present +paper, we collected a GRB sample in which all the bursts detected by the Gamma-ray Burst Monitor (GBM) on board the +NASA Fermi Gamma-ray Space Telescope whose polarization measurements are also reported in the literature and the epochs +of prompt emission are heavily overlapped with their polarization observations, aiming to establish a connection between the +polarization and jet properties of GRBs, and to confirm the validity of this correlation (πKED ≲ πHD ≲ πPED) from observations. +With a detailed spectral analysis, we found that all the bursts are classified as the “Hybrid" jet type, implying that one cannot +rule out that the photosphere emission may also be the possible mechanism powering the high levels of polarization. Moreover, +we also discovered that the polarization degrees π are tightly correlated with the cosmological rest-frame peak energy (Ep,z) of +the νFν prompt emission spectrum, the isotropic-bolometric-equivalent emission energy (Eγ,iso), and the blackbody temperature +(kT). Finally, we present different polarization models in the presence of ordered and random magnetic field configurations +with the properties of corresponding hybrid jets in order to interpret polarization measurements of the prompt emission in our +sample. +Key words: gamma-ray burst: general, radiation mechanisms: non-thermal, radiation mechanisms: thermal +1 INTRODUCTION +Gamma-ray bursts (GRBs) are one of the most explosive, and elec- +tromagnetically the brightest transient phenomena in the Universe, +occurrence at cosmological distances. After decades of investiga- +tion, the origin of the jet composition (a hot baryonic-dominated +fireball or a cold Poynting-flux-dominated outflow), and the radi- +ation mechanism and energy dissipation mechanism (synchrotron, +or Comptonization of quasi-thermal emission from the photosphere) +in gamma-ray burst (GRB) physics are still unclear (e.g., Rees & +Meszaros 1994; Mészáros & Rees 2000; Rees & Mészáros 2005; +Pe’er et al. 2006; Dai et al. 2006; Pe’er 2015; Pe’Er & Ryde 2017; +Zhang 2018; Bégué et al. 2022). +There are two crucial clues that can in principle be helpful in di- +agnosing the jet composition of GRBs, as well as their radiation +mechanism and energy dissipation mechanism. The conventional +approach is to examine the spectral properties of prompt emission. +⋆ E-mail: liang.li@icranet.org (LL) +† E-mail:s.shakeri@iut.ac.ir (SS) +Theoretically, a thermal component originating from photosphere +emission, or a non-thermal component originating from synchrotron +radiation, possibly also from inverse Compton scattering, is often +expected to be present in GRB spectral analysis. Phenomenologi- +cally, GRB spectra in the keV-MeV energy range can be typically +well-delineated by an empirical function, known as the Band func- +tion (Band et al. 1993), which is generally considered to be a non- +thermal spectrum. The Band spectrum features a smoothly broken +power law, with the peak energy Ep ≃ 210 keV (the energy at which +most of the energy is released) in νFν space and the asymptotic +power-law photon indices below (α ≃ −0.8) and above (β ≃ −2.5) +the break energy (e.g., Li et al. 2021). The low-energy spectra dur- +ing GRB prompt emission phase are closely related to the energy +distribution of electrons (e.g., Preece et al. 1998; Lloyd & Petrosian +2000; Geng et al. 2018). This fact can be utilized in order to diag- +nose GRBs radiation mechanism as well as their jet properties. For +instance, synchrotron emission predicts two different α values: α=- +3/2 and α=-2/3 (so-called the line-of-death (LOD) of synchrotron +emission, Preece et al. 1998) correspond to the fast-cooling and +slow-cooling synchrotron emission, respectively. It has been shown +© 2023 The Authors +arXiv:2301.00576v1 [astro-ph.HE] 2 Jan 2023 + +2 +Li & Shakeri +that the synchrotron emission in the presence of a decaying mag- +netic field can reproduce the Band-like spectrum of the GRB prompt +phase (Lan et al. 2021). While photosphere models, on the other +hand, predict much harder values of α; e.g., above α=-2/3. For ex- +ample, a recent study (Acuner et al. 2020) suggests that the spec- +tra that prefer the photospheric model all have low-energy power- +law indices α ∼>-0.5, as long as the data has a high significance. +Years of observations have revealed, however, that GRBs have di- +verse spectral properties, making it difficult for a single spectral +model (such as the Band-alone model) to accurately characterize all +the spectral shapes. For instance, time-resolved and time-integrated +spectral analysis inferred from the broadband Fermi observations +have revealed that GRB prompt emission exhibits remarkably di- +verse spectral properties (e.g., Abdo et al. 2009; Ryde et al. 2010; +Axelsson et al. 2012; Acuner et al. 2019; Li 2019a; Li et al. 2019, +2021, 2022b; Deng et al. 2022). A kinetic-energy-dominated (KED) +jet characterised by a quasi-thermal Planck-like spectrum has been +detected in some bursts (e.g. GRBs 090902B, 220426A, Ryde et al. +2010; Deng et al. 2022; Wang et al. 2022; Song et al. 2022), while +a cold Poynting-flux-dominated (PFD) outflow characterised by a +Band (or cutoff power-law1)-only function (Band et al. 1993) has +been also observed in some other bursts (e.g. GRB 080916C2, GRB +130606B, and many others, Abdo et al. 2009; Li 2022a), even a +hybrid-dominated (HD) relativistic outflow with a hot fireball com- +ponent and a cold Poynting-flux component, characterized by either +a composited spectral scenario, with a non-thermal component and +a thermal component, e.g., GRBs 100724B, 110721A, 150314A, +190114C, and several others, Axelsson et al. 2012; Guiriec et al. +2011; Wang et al. 2019; Li 2022a; Li et al. 2022b), or a transition +from a fireball to a Poynting-flux-dominated outflow within a single +burst (e.g. GRBs 140206B, 160625B, and several others, Li 2019a), +have also been observed. +An alternative approach is to investigate their polarization prop- +erties. Theoretically, photon polarizations play a key role to un- +derstand the jet composition, angular structure, geometric config- +uration, magnetic composition and magnetic field configuration of +GRB jets, and radiation mechanism of GRB jets (Toma et al. 2009a; +Lundman et al. 2013; Zhang 2014; Zhang et al. 2019). Although +magnetic field configurations with relatively large coherence lengths +more than gyroradius of charged particles can generate the same +energy spectrum via synchrotron mechanism, the level of polariza- +tion may significantly different for various magnetic field structures. +Therefore joint spectral and polarization analysis is essential to de- +termine the magnetic field structure in outflow materials of GRBs +(Granot 2003; Lyutikov et al. 2003; Granot & Königl 2003a; Kole +et al. 2020). For instance, the center engine is anticipated to gen- +erate strong magnetic fields (a highly magnetized jet) and launch +them concurrently with the relativistic jets. It is unclear, neverthe- +less, whether the GRB emission is caused by shock dissipation or +magnetic reconnection, and whether the outflow is dominated by the +photosphere or synchrotron emission (Toma et al. 2009a). +In fact, the generation of the polarization signal can be intrinsic +to the emission process or due to the propagation effects Shakeri +1 Recent studies (Li 2022b,a) supported by several pieces of additional ev- +idence (e.g., inconsistent spectral parameter distributions and distinct Amati +and Yonetoku correlations) have shown that Band-like spectra and CPL-like +spectra may originate from distinct radiation processes. +2 It has been demonstrated in recent studies (Guiriec et al. 2015; Vereshcha- +gin et al. 2022) that a thermal component needs to be added during the initial +prompt emission of GRB 080916C to obtain an acceptable fit to the spectral +data. +& Allahyari (2018). Several emission models (induced synchrotron +emission, Rybicki & Lightman 1979; photosphere emission, Lund- +man et al. 2014a; and Compton drag model, Lazzati et al. 2004) +have been proposed to explain the intrinsic polarization properties +of relativistic jets during prompt emissions. (i) Synchrotron emis- +sion model. There are some studies (e.g., Rybicki & Lightman 1979; +Toma et al. 2009b; Lan & Dai 2020) showing that higher values +of linear polarized signal (polarization degree π ranging from 20% +to 70%) is expected to be measured with an ordered magnetic field +from the synchrotron emission from a relativistic jet. While jets with +random magnetic fields produce lower levels of polarization, this is +due to the polarization being canceled out so that the net polariza- +tion degree being close to zero for an on-axis observer. A polariza- +tion detection which is less than 15% is believed to be originated +from a random magnetic field configuration within the jet (Mao & +Wang 2013). For example, if the emission is dominated by the inter- +nal shock (IS) model, π is expected to range from 10% (the maxed +magnetic field configuration) to 70% (the large-scale ordered mag- +netic field configuration). (ii) Dissipative photosphere model. The +dissipative photosphere model predicts a relatively low degree of po- +larization in the γ-ray band. However, a structured jet photosphere +model might also generate polarized photons by Compton scatter- +ing, but the degree of polarization would be energy-dependent from +the synchrotron model in ordered magnetic fields. For instance, it +is demonstrated that if the jet has considerable structure, the model +may create polarizations of up to 40% within δΘ ∼ Γ−1. However, in +the absence of dissipation and below the photosphere the polariza- +tion is rather limited to values below 15%-20% (Gill et al. 2018). To +restrict these models, a high-sensitivity gamma-ray polarimeter with +a broad band-pass to detect energy-dependent polarization signals is +required (Zhang 2014; Ito et al. 2014; Lundman et al. 2014a; Lund- +man et al. 2018a). (iii) Internal-collision-induced magnetic recon- +nection and turbulence (ICMART) model. In the ICMART model +(Zhang & Yan 2011a), π is expected to range from 60 percent at the +beginning of the pulse and down to about 10 percent at the end of +the pulse. A decaying polarization degree is predicted. +It is highly speculated that the prompt emission is likely expected +to be strongly polarized owing to its non-thermal origin (a non- +thermal Band-like spectrum). Observationally, higher levels of lin- +ear polarization measured from prompt γ-ray emission have been +reported by several authors (e.g., Coburn & Boggs 2003; Willis +et al. 2005; McGlynn et al. 2007; Yonetoku et al. 2012a). For in- +stance, a higher polarization degree π = 80%±20% in GRB 021006 +was claimed by Coburn & Boggs (2003) using the RHESSI data. +Later, several other cases were also reported, e.g., GRB 930131 (π > +35%), GRB 960924 (π > 50%), GRB 041219A, GRB 100826A +(π = 27% ± 11%), GRB 110301A (π = 70% ± 22%), and GRB +110721A (π = 84%+16% +−28%). Subsequent observations were also ob- +served in the optical band during the afterglow emission and were of +relatively low polarization. Compared with prompt γ-ray emission, +the levels of linear polarization measured from afterglow emission +are relatively lower. e.g., GRB 060418 (π < 8%), GRB 090102 (π = +10.1% ± 1.3%), GRB 091208B (π = 10.4% ± 2.5%), and 120328A +(π = 28% ± 4%). However, higher degrees of polarization observa- +tions are still expected to be measured from early reverse shocks, up +to ∼ 60%. +Generally speaking, we can study GRB polarization and spec- +tral properties either in a time-integrated (e.g., Li 2022a) or time- +resolved (e.g., Li et al. 2021) manner. The former represents average +polarimetric and spectral properties and is treated as a single-time +event for the entire emission period. The latter treats the entire emis- +sion period as divided into multiple-time events, and polarimetric +MNRAS 000, 1–?? (2023) + +Time-averaging Polarimetric and Spectral Properties of GRBs +3 +and spectral analyses are therefore performed on each event individ- +ually. The time-integrated method depends more heavily on the sta- +tistical results of a large sample in order to produce a more trustwor- +thy result because different bursts have distinct observational prop- +erties (e.g., angular structure). However, this issue does not arise +when a time-resolved technique is used in the same burst since it +ensures that other conditions (e.g., magnetic field configuration) are +essentially the same and, therefore, makes it easier to obtain reliable +results. +Following these lines of argument, the emissions dominated by +different jet properties (KED, PFD, and HD) may have different lev- +els of prompt-GRB polarization measurements3 (Li 2019a). As a +result, a different level of polarization degrees (πKED ≲ πHD ≲ πPED) +is naturally expected due to different jet properties if other condi- +tions are basically the same, where πKED, πHD, and πPED are the po- +larization degree in the KED, HD, and PHD jets, respectively. This +may provide a method to study the correlations between polarization +properties and their spectral properties, as well as their jet properties. +A possible connection between the spectral and polarization proper- +ties has not yet been firmly established, though recent works provide +some statistical results (Chattopadhyay et al. 2019; Kole et al. 2020). +Therefore, we dedicate this work to examining whether any possi- +ble connections exist between polarization and jet properties, and +aim to confirm the validity of this correlation (πKED ≲ πHD ≲ πPED) +from observations. Practically, several important factors need to be +taken into account in our analysis. (i) In order to potentially evaluate +all of the frequently-used GRB spectral models and thus diagnose +the jet properties, our analysis focuses on the bursts detected by the +Gamma-ray Burst Monitor (GBM, 8 KeV-40 MeV, Meegan et al. +2009) onboard the NASA Fermi Gamma-ray Space Telescope. (ii) +To directly compare the spectral and polarization properties and thus +make our results more trustworthy, we select the bursts during which +polarization observations and spectral data are available during the +same epoch. (iii) Realistically, systematic error is frequently at play +in polarization measurements and different polarization instruments +have different systematic errors. Therefore, a high-significance sig- +nal from polarization measurement is required. In this paper, we +collect a sample of the Fermi-GBM detected bursts along with the +polarized measurements reported in the literature using the time- +integrated spectral and polarization analysis approach based on their +statistical results, aiming to establish a connection between the po- +larization and jet properties of GRBs. +The paper is organized as follows. The sample and Methodol- +ogy are presented in Section 2 and Section 3, respectively. Our re- +sults and their physical implication are summarized in Section 4 and +Section 5, respectively. The conclusion is presented in Section 5. +Throughout the paper, the standard Λ-CDM cosmology with the pa- +rameters H0 = 67.4 kms−1 Mpc−1, ΩM = 0.315, and ΩΛ = 0.685 are +adopted (Planck Collaboration et al. 2018). +3 Polarization measurement is a measurement where π ranges from 0% to +100% (0% ≤ π ≤ 100%) and includes the non-detection (0%), so measur- +ing something consistent with zero is a measurement. Polarization detection, +however, indicates a different meaning, where 0% < π ≤ 100% and excludes +0%, since we have excluded part of the parameter space equivalent to detec- +tion measurements due to the fact that a non-detection of flux implies that we +did not measure something. +2 THE SAMPLE +A comprehensive database of GRB polarimetric observations has +been created in a recent work (Li et al. 2022a) by extensively search- +ing for those GRBs in the literature whose polarization measure- +ments have been reported. A total of 73 bursts with polarization +detections were included in the database, covering a broad wave- +length range from radio to optical, X-ray, and γ-ray emission (see +Table 1 in Li et al. 2022a). The prompt emission data of these bursts +were observed by different satellites (Fermi, Swift, BeppoSAX, and +BATSE). On the other hand, the diagnosis of jet composition usually +requires a refined spectral analysis. Among these satellites, Fermi +covers the broadest energy range in the observation. Consequently, +in order to fully evaluate all current spectral models we pay spe- +cial attention to these Fermi-detected bursts. The Gamma-ray Burst +Monitor (GBM, 8 KeV-40 MeV, Meegan et al. 2009) and the Large +Area Telescope (LAT, 20 MeV- 300 GeV, Atwood et al. 2009), on- +board the NASA Fermi Gamma-ray Space Telescope, together pro- +vide unprecedented spectral coverage for seven orders of magnitude +in energy (from ∼8 keV to ∼300 GeV). Our statistical analysis in +the current work includes the prompt γ-ray emission spectral anal- +ysis and the connection between the spectrum and polarization are +thus based on the Fermi-detected bursts. On the other hand, we fo- +cus on the GRBs whose polarization measurements were recorded +during the prompt emission in the γ-ray band in order to compare +the polarization and spectral properties during the same time period. +Following are the specific observational properties of the five fas- +cinating bursts (see Table 1) that made up the smaller sample as a +result of these selection criteria. +• GRB 100826A. On August 26, 2010 at 22:58:22.898 (T0) +UT, GRB 100826A, was detected by the Fermi/GBM. A t90 of +(84.993±0.724) s in the 10-1000 keV was measured using the +GBM data. The GBM lightcurve exhibits a complicated shape with +multiple-peak pulses, and the fluence (flux integrated over the burst +duration) in the energy range of 10-1000 keV during the t90 dura- +tion reported by the GBM team is (1.6388±0.001)×10−4 erg cm−2 +(Figure 2). This burst has no redshift measurement, a value of 2.3 +is estimated using the Yonetoku correlation (Yonetoku et al. 2004). +Therefore, the isotropic-equivalent energy released from gamma- +ray emission in the cosmological frame for this burst can be cal- +culated as, Eγ,iso=(3.39+0.36 +−0.33)×1054 erg. The KED-to-PFD transition +pattern belonging to the “HD” jet type is diagnosed for this burst +by using a low-energy spectrum based on a detailed time-resolved +spectral analysis during prompt emission (see Section 4). Yonetoku +et al. (2011a) reported that a relatively higher polarization degree +(πobs=27±11) with a 2.9σ linear polarization signal significance was +detected during the prompt emission (0-100 seconds after the trig- +ger) of GRB 100826A using a GAmma-ray Polarimeter (GAP) on +board a small Japanese solar-power-sail demonstrator, Interplanetary +Kite-craft Accelerated by Radiation of the Sun (IKAROS). +• GRB 110301A. On March 01, 2011 at 05:08:43.070 (T0) UT, +GRB 110301A, was detected by the Fermi-GBM. A t90 of (5.693± +0.362) s in the 10-1000 keV was measured using the GBM data. The +lightcurve exhibits a complicated shape with multiple-peak pulses, +and the fluence in the energy range of 10-1000 keV from T0+0 s +to T0+5.693 s reported by the GBM team is (3.5891±0.003)×10−5 +erg cm−2 (Figure 3). This burst has no redshift measurement, a +value of 0.36 is estimated by using the Yonetoku relation (Yone- +toku et al. 2004). With this estimated value of redshift, the isotropic- +equivalent energy released from gamma-ray emission in the cosmo- +logical frame for this burst can be calculated as, Eγ,iso=1.4×1052 erg. +A KED-to-PFD jet is diagnosed for this burst by using a low-energy +MNRAS 000, 1–?? (2023) + +4 +Li & Shakeri +spectrum based on a detailed time-resolved spectral analysis during +prompt emission. A linear polarization signal (3.7σ) with a very high +polarization degree (π=70±22) for this burst was claimed by Yone- +toku et al. (2012b) using the IKAROS/GAP data 0-7 seconds after +the trigger (Figure 3). +• GRB 110721A. On July 21, 2011 at 04:47:43.761 (T0) +UT, GRB 110721A, was detected by the Fermi-GBM. A t90 of +(21.822±0.572) s in the 10-1000 keV was measured using the GBM +data. The lightcurve exhibits a single-peak pulse, and the fluence in +the energy range of 10-1000 keV from T0+0 s to T0+21.822 s as re- +ported by the GBM team is (3.5891±0.003)×10−5 erg cm−2 (Figure +4). This burst has a value of 0.382 of redshift. With this redshift, we +calculate the isotropic-equivalent energy released from gamma-ray +emission in the cosmological frame for this burst, Eγ,iso=3.0×1052 +erg. A peak-KED jet is diagnosed for this burst by using a low- +energy spectrum based on a detailed time-resolved spectral analysis +during prompt emission. A linear polarization signal (3.3σ) with a +very high polarization degree (π=85+16 +−22) for this burst was claimed +by Yonetoku et al. (2012b) using the IKAROS/GAP data 0-11 sec- +onds after the trigger (Figure 4). +• GRB 140206A. On 6 February 2014 at 06:36:12.843 UT (T0), +a long burst, GRB 140206B, was detected by Fermi-GBM and sev- +eral other satellites (e.g., INTEGRAL/IBIS). Its GBM lightcurve in +the 10-900 keV exhibits three clearly separated emission episodes +(G1, G2, and G3) as shown in the left panel of Figure 5, with T90 of +146.690±4.419 s was measured by GBM data. GRB 140206B is a +very bright burst, and the fluence (flux integrated over the burst dura- +tion) in the energy range of 10-1000 keV from T0+0 s to T0+146.690 +s reported by the GBM team is (1.23±0.003)×10−4 erg cm−2. This +burst has a value of 2.73 for redshift. With this redshift, we calculate +the isotropic-equivalent energy released from gamma-ray emission +in the cosmological frame for this burst, Eγ,iso=2.3×1054 erg. The +spectral analysis has been performed in great detail in Li (2019a), +and three different spectral components (KED-to-PFD) have been +identified, a short-thermalized precursor early on, followed up with +the main burst with a non-thermal emission later on, and in the last +with a fainter burst with still a non-thermal emission. Interestingly, +immediately following up with the short-thermalized precursor, a +clear polarization signal with 90% confidence was observed 4-26 +seconds after the trigger as reported in Götz et al. (2014) using the +INTEGRAL/IBIS data. This signal is a linear polarization with an +up-limit polarization degree of π > 28 observed in the γ-ray energy +band (Figure 5). +• GRB 160802A. GRB 160802A was detected by GBM on Au- +gust 2, 2016 at UT 06:13:29.63. A T90 of (16.384±0.362) s in the +10-1000 keV was measured using the GBM data. The prompt emis- +sion light curve shows two clearly separated active periods, with a +quiescent time interval between the two periods of about 10 sec- +onds. The earlier period consists of overlapping pulses while the lat- +ter period shows a clear single-peak pulse. The fluence in the energy +range of 10-1000 keV from T0+0 s to T0+21.822 s as reported by +the GBM team is (6.8399±0.0057)×10−5 erg cm−2. This burst has +no redshift measurement, a value of 0.36 is estimated by using the +Yonetoku relation (Yonetoku et al. 2004). Using this estimated value +of redshift, we further calculate the isotropic-equivalent energy re- +leased from gamma-ray emission in the cosmological frame for this +burst, Eγ,iso=2.2×1053 erg. A peak-KED jet for each period is di- +agnosed for this burst by using a low-energy spectrum based on a +detailed time-resolved spectral analysis during prompt emission. A +linear polarization signal (∼3σ) with a very high polarization degree +(π=85±29) for this burst was claimed by Yonetoku et al. (2012b) us- +ing the AstroSat/GZTI data 0-20.34 seconds after the trigger (Figure +6). +3 METHODOLOGY +3.1 Spectral Analysis Techniques +In order to diagnose the jet properties for a given burst, a de- +tailed time-integrated or time-resolved spectral analysis is re- +quired. The spectral analysis is performed by a pure Python pack- +age, namely, the Multi-Mission Maximum Likelihood +Framework (3ML,Vianello et al. 2015). Moreover, a Bayesian +approach and Markov Chain Monte Carlo (MCMC) iterations to +explore the best parameter space was used. Our spectral analy- +sis includes the following main steps. (1). First is to select detec- +tors, sources, and background intervals; (2). Second, we used the +Bayesian block method (BBlocks, Scargle et al. 2013) to bin the +Time-Tagged Events (TTE) lightcurve of the brightest detector (with +the minimum viewing angle), and the significance (S, Vianello 2018) +for each BBlocks time bin was also calculated. (3). Third, we use a +typical GRB spectral model (Band function, Band et al. 1993) to fit +all the spectra selected by the BBlocks method and the best model +parameters are obtained by adopting a fully Bayesian approach. For +a detailed Bayesian spectral analysis and the reduction procedure +applied to a GRB spectrum, we refer to Li (2019a,b, 2020); Li et al. +(2021); Li & Zhang (2021). +3.2 Using the Yonetoku correlation to infer their redshift values +In order to study the intrinsic properties in the cosmological rest +frame, a redshift measurement for each burst is needed. Unfortu- +nately, we currently have 3 bursts (GRB 100826A, GRB 110301A, +and GRB 160802A) without known redshift. Several works to in- +fer redshifts using empirical relations of GRB have been reported in +the literature(Amati et al. 2002; Yonetoku et al. 2004). For example, +(Yonetoku et al. 2004) discovered a new and much tighter relation- +ship between the spectral peak energy (Ep) and the peak luminos- +ity (Lp) using the combined data detected by the BeppoSAX and +BATSE satellites, and claimed that one can use the Ep-Lp relation to +estimating redshift without knowing distances in the BSTASE cata- +log. We, therefore, attempt applying the Yonetoku relation(Yonetoku +et al. 2004) to estimate redshift values for these bursts. +Our procedure to estimate the pseudo-redshift using the Yonetoku +relation includes the following steps. +First, we perform a spectral fit to the peak spectrum (the brightest +time bin was selected by using the Bayesian blocks method(Scargle +et al. 2013) at the highest statistical significance S) of each burst. +The peak flux Fγ and peak energy Ep on the observer’s frame from +the spectral fits are thus obtained, where Fγ is the observed peak flux +integrated between (1-104) keV in units or erg cm−2 s−1. +Second, in order to use the Ep,z-Lp,iso relation, a bolometric lu- +minosity in a common cosmological rest-frame energy band (1-104 +keV) is needed. It can be obtained by using the spectral parameters to +conduct a k-correction extrapolating the observed energy band to 1- +104 keV. For a given burst, the k-correction factor (kc) can be derived +using the following procedure. The observed flux Fobs (erg cm−2 s−1), +in a fixed detector energy bandwidth [e1, e2] (for instance, for the +Fermi-GBM observation, e1=8 keV, e2=40 MeV), can be written as: +Fobs +[e1,e2] = +� e2 +e1 +EN(E)dE, +(1) +MNRAS 000, 1–?? (2023) + +Time-averaging Polarimetric and Spectral Properties of GRBs +5 +where E is in units of keV, and N(E) is a GRB photon number spec- +trum. The total luminosity emitted in the bandwidth [e1,e2], defined +in the cosmological rest-frame, is given by: +L[e1(1+z),e2(1+z)] = 4ˆπD2 +L(z)Fobs +[e1,e2], +(2) +which DL(z) is the luminosity distance. To express the luminosity +L in the cosmological rest-frame energy band, [E1=1 keV, E2 =104 +keV], common to all sources, the Eq.(2) can be rewritten as: +L[E1,E2] = 4ˆπD2 +LFobs +� E1 +1+z , E2 +1+z +� = 4πD2 +Lk[e1,e2,E1,E2,z]Fobs +[e1,e2], +(3) +where the k-correction factor, kc, is therefore defined as: +kc = k[e1,e2,E1,E2,z] = +Fobs +� E1 +1+z ; E2 +1+z +� +Fobs +[e1,e2] += +� E2/(1+z) +E1/(1+z) EN(E)dE +� e2 +e1 EN(E)dE +, +(4) +Last, with the k-correction factors known, the peak luminosity can be +derived from the observed γ-ray flux Fγ according to L = 4ˆπd2 +LFγkc, +where dL is the luminosity distance. +Finally, as long as the prompt emission spectral properties can be +obtained, the Yonetoku relation can be used to infer redshift. With +the above Steps, one can use Equation (2) in Yonetoku et al. (2004) +to estimate their redshift values, +Lp +1052ergs−1 = (2.34+2.29 +−1.76)×105 +�Ep(1+z) +1keV +�2.0±0.2 +. +(5) +Table 2 lists the spectral properties obtained from the peak spec- +tral analysis and the estimated redshift values inferred from the Yo- +netoku relation. +4 RESULTS +4.1 Spectral Properties and their inferred jet properties +Five bursts were claimed to have a high-significance polarization +detection (Table 1) and GBM data (Table 4) taken at their prompt +emission, thereby providing an “ideal” sample to study the possi- +ble connection between jet properties and polarization straightfor- +wardly. +In practice, either time-integrated or time-resolved spectral analy- +sis is frequently used to diagnose their jet properties. Several meth- +ods have been widely used to diagnose the jet properties based on +their spectral analysis. The simplest method is to use the low-energy +spectrum (e.g., Preece et al. 1998) to resolve the jet properties for +a given pulse/burst (Method-I). In this method, a single empirical +model (such as the Band model) based on a time-resolved tech- +nique is typically used. The KED jets, therefore, can be defined +as all α indices, within uncertainties, in a given burst, obtained +from time-resolved spectral analysis, are systematically above the +synchrotron limit throughout the entire burst/pulse duration, being +consistent with a matter-dominated fireball jet. The PFD jets, on +the other hand, are consistent with the scenario that all α indices, +within uncertainties, in a given burst are below the synchrotron limit +throughout the burst/pulse duration, this suggests a Poynting-flux- +dominated jet. The HD jets represent the moderate scenario, and +can be presented that some α indices, obtained from time-resolved +spectral analysis, are above the synchrotron limit (α >2/3) while +some others are below the synchrotron limit (α <2/3) throughout +the burst/pulse duration. The HD jets can be further divided into two +subcategories. (1) the peak-KED pattern since the thermal emission +component (the spectra that violate the LOD line) is only detected +around the peak of the pulse (thermal component dominates the peak +of a pulse/burst); and (2) the KED-to-PFD (thermal to non-thermal +component) transition pattern (Li 2019a), since the thermal emission +component is detected at the beginning of the burst, and followed by +non-thermal emission component. However, it may be difficult to +classify jets as either KED or PFD jets based on the spectral index +alone. In contrast to an optically-thin synchrotron emission, a photo- +spheric quasi-thermal component would, in fact, have a harder low- +energy spectral index, but this does not guarantee that the jet is KED +(Gill et al. 2020). Therefore, a more reliable approach is to find the +best model (Method-II) by comparing various frequently-used spec- +tral models using certain statistical information criteria, such as the +Akaike Information Criteria (AIC; Akaike 1974), Bayesian Infor- +mation Criteria (BIC; Schwarz 1978), and the Deviance Information +Criterion (DIC; Spiegelhalter et al. 2002; Moreno et al. 2013). Af- +ter the identification of the best spectral model, one can assess the jet +properties using the spectral properties inferred from the best model. +Moreover, we may also directly fit the spectral data with a physical +model (such as the synchrotron emission model, Burgess et al. 2020; +Method-III), so as to diagnose the jet properties and any potential +connection with their polarization properties. Our analysis in this +task incorporates both Method-I and Method-II (primary method). +Method III will be used elsewhere. +We first perform time-integrated spectral analysis (treating the +entire epoch of polarization observation as one time bin) by using +various GRB spectral models, including power-law (PL), blackbody +(BB), cutoff power law (CPL), Band function, PL+BB, CPL+BB, +and Band+BB, respectively. We adopted both AIC and BIC to eval- +uate different spectral models and select the preferred one, and the +preferred model is the one that provides the lowest AIC and BIC +scores. As such, we define +• PFD jets: a single non-thermal (Band-like) spectral compo- +nent was found in the time-integrated spectral analysis, like GRB +080916C (Abdo et al. 2009). +• KED jets: a dominate thermal (BB-like) spectral compo- +nent was found in the time-integrated spectral analysis, like GRB +090902B (Ryde et al. 2010) and GRB 220426A (Deng et al. 2022). +• HD jets: a hybrid spectrum of thermal (BB-like) and non- +thermal (Band-like) components was observed in the time-integrated +spectral analysis in a single burst, like GRB 110721A (Axelsson +et al. 2012), and GRB 140206A (e.g., Li 2019a). +Our refined time-integrated spectral analysis suggests that the +Band+BB model can best characterize the spectral shape of all the +five bursts (see Table 3). The global properties of our sample used in +the spectral analysis are reported in Table 4. These include the GRB +name (column 1), observed duration T90 of burst4 (column 2), 10- +1000 KeV fluence (column 3), together with the used detectors (col- +umn 4), the selected source (column 5) and background (column 6) +intervals, and the best model (column 7). The best-fit spectral param- +eters of the sample with the Band+BB model are reported in Table 5, +including GRB name (column 1), source duration (column 2), and +corresponding significance (column 3), and Band component nor- +malization K (column 4), low-energy power-law index α (column +5), peak energy Ep (column 6) of the νFν spectrum, and high-energy +power-law index β (column 7); and BB component normalization K +(column 8), and temperature kT (column 9). +We next perform time-resolved spectral analysis (treating the +entire epoch of polarization observation as divided into multiple- +4 The time interval during which 90% of the total observed counts have been +detected. +MNRAS 000, 1–?? (2023) + +6 +Li & Shakeri +time events) for each individual BBlocks time bin using the Band +model. Temporal evolution of α is presented in Figures 2-6 for GRB +110826A, GRB 110301A, GRB110721A, GRB 140206A, and GRB +160802A, respectively. With the BBlock time bins selected and the +corresponding significance (S) values calculated, the data points are +shown in different S ranges5: S < 10, 10 ≤ S ≤ 20, and S > 20. +For a subset of GRBs, a mixture of thermal (blackbody contribution +from the photosphere emission) and non-thermal (synchrotron emis- +sion from relativistic electrons) components was observed in a sin- +gle burst (e.g., Li 2019a). Previous studies (e.g., Burgess et al. 2014) +showed that the characteristic energy (Ep) of non-thermal emission is +correlated to the characteristic energy (kT) of thermal emission with +a power-law relation in form of Ep ∝ T q, where q ranges from ∼1-2. +Burgess et al. (2014) studied a set of bright Fermi single-pulse GRBs +and claimed that one can use this correlation to identify whether the +jet is dominated by kinetic (q ∼ 1) or magnetic energy (q ∼ 2) de- +pending on the value of the exponent. GRB 110721A was identified +as the baryonic jet type in Burgess et al. (2014) with q = 1.24±0.11, +which is also consistent with our finding in the current analysis. +Our time-integrated spectral analysis indicates that all five bursts +belong to the HD jet type. This finding is quite interesting since only +a subset of GRBs (a fairly low percentage) have an observed ther- +mal component in their spectral analysis, as suggested by several +statistical studies (e.g., Li 2022a). Recently, (Li 2022a) has made a +great effort to collect a complete GRB sample in which all bursts +were detected by Fermi/GBM with known redshift, and created a +spectral parameter catalog based on their model-wise properties. He +discovered that ∼ 5% (7/153) of the analyzed bursts were found to +require a subdominant thermal component in their time-integrated +spectral analysis, including GRB 110721A. Our results imply that +high-degree polarization measurements may also be associated with +a thermal component originating from photosphere emission. Our +time-resolved spectral analysis, on the other hand, further suggests +that two bursts exhibit the peak-KED pattern (GRB 110721A and +GRB 160802A) while the other three bursts (GRB 110301A, GRB +140206A, and GRB 160625B) display the KED-to-PFD transition +pattern across the entire burst durations since all α indices are +above the synchrotron limit early on (α >2/3), and then drop be- +low the synchrotron limit later on (α <2/3), indicating a thermal- +to-nonthermal transition signature. Interestingly, after carrying out +a detailed time-resolved spectral analysis for a sample of the multi- +pulsed bursts, Li (2019a) reported that the jet properties for a good +fraction of the multi-pulsed bursts exhibit a transition from thermal +to non-thermal component among pulses within a single burst, and +claimed that such “transition" jet properties are clearly observed in +four bursts (GRB 140206B, GRB 140329B, GRB 150330A, and +GRB 160625B), and the polarization properties in those transition +bursts would also differ. +4.2 The Observed Parameter Correlations: Polarization +Degrees versus Other Relevant Quantities +The most interesting result that draws our attention is that all five +bursts in our target sample have a relatively high-degree polarization +measurement and are associated with the “HD” jet properties. In +the following discussion, we, therefore, pay special attention to this +interesting observation and its theoretical interpretation. +5 Note that the results obtained for those lower-S spectra may not be robust, +since the spectral fits would not be well determined due to lack of enough +photons. +Much evidence points towards the fact that correlation analysis +plays a crucial role in the understanding of GRB physics as it pro- +vides a crucial clue to revealing its nature (e.g., Amati et al. 2002; +Yonetoku et al. 2004; Liang & Zhang 2005; Dainotti et al. 2008; +Xu & Huang 2012; Zhang et al. 2012; Liang et al. 2015; Dain- +otti & Amati 2018; Li 2022a, +and references therein). Here, an +attempt has been made to explore the correlations between the po- +larization properties and several typical GRB observed quantities. +For instance, the polarization degrees π correlated with (i) the cos- +mological rest-frame peak energy (Ep,z) of the νFν prompt emission +spectrum, (ii) the isotropic-bolometric-equivalent emission energy +Eγ,iso, (iii) the magnetization parameter σ0, (iv) the blackbody tem- +perature kT, (v) the redshift z, and (vi) the corresponding energy +fluence Sγ. Using the same observed epoch during the prompt emis- +sion phase, our target sample allows for a reasonable comparison. +Our analysis includes the following steps. (1) For a given burst in our +target sample, we first select the same time interval for the prompt +emission data as for the polarization measurement. (2) We then at- +tempt to perform a spectral fit to the selected prompt emission data +using PL, BB, CPL, Band, PL+BB, CPL+BB, and Band+BB func- +tions, respectively. The Band+BB model has an AIC/BIC-statistic +improvement of at least 10 with respect to the Band-alone and other +models for these bursts, which suggests Band+BB as the preferred +model that would fit the data and a thermal component existing in +the spectrum. Since the thermal flux ratio (FBB/Ftot) for these bursts +is less than 50% (see Table 5), the thermal components thus are sub- +dominant. Interestingly, Chattopadhyay et al. (2019) analyzed the +prompt emission and polarization data of 11 bright bursts detected +during the first year of operation of CZTI, and reported that of these +bursts, four bursts (GRB 160106A, GRB 160509A, GRB 160802A, +and GRB 160910A) in their spectral analysis showed a deviation +from the Band model and an additional thermal blackbody is needed +in order to model their spectrum more precisely. (3) With the spec- +tral analysis done in Step (2), we are able to obtain the spectral peak +energy (Ep,z) and the blackbody temperature (kT). Eγ,iso can also be +calculated in the cosmological frame with a redshift known (GRB +110721A and GRB 140206B), and with a k-correction applied by +integrating the observed energy spectrum over 1 KeV/(1+z) to 10 +MeV/(1+z). We note here that for the remaining three bursts (GRB +100826A, GRB 110301A, and GRB 160802A) with an unknown +redshift, we use the Yonetoku relation (Yonetoku et al. 2004) to es- +timate their redshift values (see Section 3.2). Using these hybrid- +spectrum observed properties and following the method described +in Gao & Zhang (2015) and Li (2020), we also calculate the mag- +netization parameter σ0 for these bursts. Finally, with these Steps +completed, we therefore present π-Ep,z (Fig.7a), π-Eγ,iso (Fig.7b), +π-kTz (Fig.7c), π-σ0 (Fig.7d), π-z (Fig.7e), and π-Sγ (Fig.7f) plots +in Figure 7. Interestingly, these scatter plots all seem to exhibit a +monotonic power-law decay in their log-log space (except for the +π-σ0 correlation), with a similar decay slope ranging from -0.40 to +-0.20. Our results indicate that a higher Ep,z, Eγ,iso, and kTz tend to +have a lower-degree polarization π. However, we should note that +the sample size is too small to be reliable enough to support the de- +rived conclusion, so the results may not be statistically significant. +5 PHYSICAL IMPLICATION +There are several parameters to impact on the degree of polariza- +tion in GRBs including the geometry of the jet, its angular struc- +ture, the bulk Lorentz factor of outflow material, the magnetic field +configuration and observer’s point of view. Here, we consider an +MNRAS 000, 1–?? (2023) + +Time-averaging Polarimetric and Spectral Properties of GRBs +7 +ultra-relativistic axi-symmetric jet which lunched by a central en- +gine weather a black hole or an rapidly rotating magnetar (e.g., Usov +1992; Thompson 1994; Dai & Lu 1998; Wheeler et al. 2000; Zhang +& Mészáros 2001; Liu et al. 2007; Metzger et al. 2008; Lei et al. +2009; Metzger et al. 2011; Bucciantini et al. 2012; Lü & Zhang +2014; Li et al. 2018). During the prompt emission, we have ultra- +relativistic jet with bulk lorentz factor Γ ≫ 1 leading to strong beam- +ing effect of GRB outflow materials where the Doppler factor can be +approximated as +δD ≈ +2Γ +1+(Γ ˜θ)2 , +(6) +Due to the relativistic beaming effect of GRB jets, the measured ra- +diation energy of the bursts is smaller than its isotropic energy as +Eγ = fbEγ,iso by a factor fb = ∆Ω/4π = (1 − cosθ j) ≈ θ2 +j/2 where +θ j is the half opening angle of the ejecta. In principle, different +GRBs can be viewed from different observing angles θobs with re- +spect to the jet’s central axis. Only those observers whose line-of- +sight (LOS) intersects the surface of the jet can detect the GRBs. In +the ultra-relativistic regime, the observed emission mainly receives +from a region that is limited to a cone with angular size ˜θ ≲ 1/Γ +around LOS. At the early time prompt emission when the LOS in- +tersects the jet surface, if θobs/θ j ≲ 1−(Γθ j)−1 and Γθ j ≳ O(10), the +jet’s edge remains invisible to the observer Gill et al. (2018). +In this case, the emission region can be approximated as an ex- +panding thin spherical shell of width ∆ ≪ R/Γ2 (in the lab frame) +in which particles cool relatively fast compared to the dynamical +time scale of the system. As the GRB jet has slowed down signifi- +cantly when the opening angle θ j ≃ Γ−1 then the jet break happens +and the edge effects become important. The flux density measured +by a distant observer from each fluid element in an infinite thin-shell +approximation for the prompt emission is given by (Granot 2005) +Fν(t) = +(1+z) +16π2d2 +L(z) +� +δ3 +DL′ +ν′(r)d ˜Ω, +(7) +where dL(z) is the luminosity distance of the source, and d ˜Ω = +d ˜φd(cos ˜θ) is the solid angle with ˜θ and ˜φ as the polar angle and +the azimuthal angle measured from the LOS, respectively. The co- +moving spectral luminosity L′ +ν′(r) for the synchrotron emission is +L′ +ν′(r) = L′ +ν′(R) +� +1−(ˆn′ · ˆB′)2� 1+α +2 , +(8) +where α = −dlog(Fν)/dlog(ν) is the spectral index, ˆn is the ob- +server’s LOS in the comoving frame of the GRB jet and ˆB is the +local direction of the magnetic field. The spectral luminosity L′ +ν′(R) +in the comoving frame of the fluid in terms of frequency ν′ and the +peak frequency ν′ +p at which the most of the power is emitted is given +by +L′ +ν′(R) = L′ +ν′p +� ν′ +ν′p +�−α +, +(9) +here we consider a constant luminosity with a radius which the peak +value L′ +ν′p. We assume to have synchrotron emission from accelerat- +ing electrons in the magnetig field with isotropic velocity distribu- +tion and the energy distribution as a power law ne ∝ γ−p. +In our scenario, we assume that each pulse originates from a sin- +gle thin shell and Γ can in principle change for different pulses. The +state of polarization of a radiation field can be expressed in terms of +the Stokes parameters I (intensity), Q and U (linear polarizations), V +(circular polarization). In spite of linear polization, only a few mech- +anisms can generate the high value of circular polarization in the +usual scattering processes in GRBs, and the measured circular polar- +ization has only been reported once in a GRB afterglow Wiersema +et al. (2014). Therefore we will not consider circular polarization +in this paper. Stokes parameters Q and U are differences in flux for +two orthogonal directions on the sky which are coordinate dependent +quantities (Rybicki & Lightman 2008; Westfold 1959), we define the +local degree of linear polarization Π = +� +Q2 +U2/I where +U +I = Πsin2θp , +Q +I = Πcos2θp , +θp = 1 +2 arctan +�U +Q +� +, (10) +and θp is the polarization position angle (PA). The direction of the +polarization vector in the synchrotron emission is orthogonal to the +LOS of the observer ˆn and the local direction of the magnetic field +ˆB in the jet, +ˆΠ = (ˆn× ˆB) +|ˆn× ˆB)| +, +(11) +The polarization measurements can help in order to probe the +magnetic field structure inside the shock wave. Moreover, the de- +gree of the polarization depends on the GRB jet’s angular structure +and the observer’s viewing angle from jet symmetry axis (Lazzati +et al. 2004). The magnetic field structure in KED and PFD flows +has a different origin and can be classified into three categories (Gill +et al. 2018; Gill et al. 2021): (i) a locally ordered magnetic field +(Bord) with angular coherent length θ j > θB ≳ 1/Γ, (ii) a toroidal +magnetic field (Btrod) which has an ordered axisymmetric configura- +tion in the transverse direction with respect to the jet (iii) a tangent +magnetic field which could be in principle parallel (B∥) or perpen- +dicular (B⊥) to the local fluid velocity. In the PFD the magnetic field +is dynamically dominated and usually has a large coherence length +such as Btrod which can be produced by a rotating central engine or +in a high magnetized flow, other locally and globally ordered field +configuration are also possible in this case. On the other hand in +KED we may have a tangled magnetic field structure with B⊥ or/and +B∥ components, however generating such an anisotropic field con- +figurations in shock waves seems to be challenging (Gill & Granot +2020). A globally ordered magnetic field may naturally be advected +from near the central source, while the random magnetic fields gen- +erated in the shock dissipation region (Kumar & Zhang 2015; Geng +et al. 2018; Gill et al. 2018; Fan et al. 2008). The magnetic field +structures that are generated at relativistic collision-less shocks, due +to the two-stream instabilities, are expected to be tangled within the +shock plane (Medvedev & Loeb 1999). +The degree of linear polarization generated in the synchrotron +emission from an isotropic electron distribution with power-law en- +ergy spectrum, and for a given direction of magnetic field is given +by (Rybicki & Lightman 2008; Westfold 1959) +Πlin +max = +α+1 +α+5/3 = +peff +1 +peff +7/3, +(12) +where peff = 2α + 1 is the effective power-law index of the electron +distribution. +peff = +� +� +� +2, +νc < ν < νm, +slow cooling +p, +νm < ν < νc, +fast cooling +p+1, +ν > max(νc,νm), +either fast or slow cooling +(13) +and, therefore +Πlin +max = +� +� +� +� +� +9/13, +νc < ν < νm, +fast cooling +(p+1) +(p+7/3), +νm < ν < νc, +slow cooling +(p+2) +(p+10/3), +ν > max(νc,νm),either fast or slow cooling +(14) +MNRAS 000, 1–?? (2023) + +8 +Li & Shakeri +This polarization may originate from a very small region (point- +like emitter) in which the magnetic field has a specific orientation. +Only in the case of ordered magnetic field with the coherence length +comparable or larger than the visible surface of the emitting region, +the highest value of the polarization Πlin +max in Eq. (19) can be gen- +erated. The photon index in the Synchrotron radiation is limited to +−1/3 ⩽ α ≲ 3/2 which regarding Eq. (19) leads to the maximum +degree of polarization 50% ≲ Π ≲ 75%. In the left panel of Fig- +ure (8), we see predicted polarization values using this theoretical +model with observed data using our target sample. Here α indices +are obtained using the spectral analysis defined in §4. We find that +the observed data are well distributed along the line predicted by this +model. +In general, the measured polarization is obtained by integrating +the local stokes parameters over the flux of the GRB jet as +Π(tf ) = +¯ +Q(tf ) +I(t f ) = +� +dFν cos2θp +� +dFν +, +(15) +assuming to have an axisymmetric flow and taking into account sym- +metry consideration, we see that ¯U = 0 and consequently the instan- +taneous total degree of the linear polarization is ¯Π = | ¯Q|/I. We per- +form an integration over the equal time surface (EATS) for a single +pulse +� +¯ +Q(t)dt/ +� +I(t)dt in order to obtain pulse integrated polariza- +tion of the prompt emission which leads to +Πord +Πmax = +� ymax +0 +dy(1+y)−2−α � +dφΛ(y,φ)cos2θp +� ymax +0 +dy(1+y)−2−α � +dφΛ(y,φ) +, +(16) +The above formula is valid for the prompt emission from an ultra- +relativistic thin-shell for an on-axis observer (θobs = 0) where ymax = +(Γθmax)2 and θmax defined as the maximum angle from LOS Granot +(2003a). The factor Λ(y,φ) is an average over the magnetic field +orientations in the plane of the ejecta as +Λ(y,φ) ≡ ⟨(1−(ˆn′ · ˆB′)2) +1+α +2 ⟩ , +(17) +The polarization angle θp and Λ(y,φ) take different forms regarding +the configuration of the magnetic field in the plane the GRB jet. In +the case of an ordered magnetic field Bord we have : +Λord(y,φ) ≈ +� +(1−y +1+y)2 cos2 φ+sin2 φ +� 1+α +2 +, +(18) +θp = φ+arctan +� +(1−y +1+y)cotφ +� +. +(19) +The time-integrated linear polarization in the presence of an ordered +magnetic field in the plane normal to the jet velocity is plotted as +a function of the spectral index in the right panel of Fig. (8). As it +is seen the polarization degree increases towards higher values of α +and lower values of ymax which can cover the observed polarization +of GRB 110721A, GRB 160802A, and GRB 110301A. Therefore +for a configuration with the globally ordered magnetic field, high +values of linear polarization even larger than 50% are obtainable. +The degree of polarization for a magnetic field with locally tan- +gled or random configuration is obtained by averaging over all direc- +tions of the local magnetic field within the plane of the shock (Granot +& Königl 2003a; Sari 1999; Gruzinov 1999; Nava et al. 2016). The +presence of a random magnetic field leads to negligible values of net +linear polarization measured by an on-axis observer. In the case of a +random field behind the shock wave only if the observer is off-axis +and the circular symmetry is broken, non-zero net polarization is +measurable. The total linear polarization arising from the whole jet +which is subjected to a random field with a direction perpendicular +to the jet velocity is given by Granot (2003a) +Π⊥ +Πmax = +� y2 +y1 dy(1+y)−2−α sin[2Ψ1(y)]G(y,α) +Θ(1−ζ) +� y1 +0 +dy H(y,α) +(1+y)α+2 + +� y2 +y1 dy dy H(y,α) +(1+y)α+2 +� π−Ψ(y) +π +�, +(20) +where Θ(1 − ζ) is the Heaviside step-function with ζ ≡ θobs/θ j as a +parameter to define observer’s point of view, and +G(y,α) = 1 +2π +� π +0 +dφ +�(1−y)2 +(1+y)2 cos2 φ−sin2 φ +�� +1− 4ycos2 φ +(1+y)2 +� α−1 +2 +, (21) +H(y,α) = +� π +0 +dφ +� +1− 4ycos2 φ +(1+y)2 +� 1+α +2 +, +(22) +cosΨ(y) = (1−ζ2)y j −y +2ζ√yy j +. +(23) +In above expressions y1,2 = (1∓ζ)2yj and y j = (Γθ j)2. The variation +of the linear polarization in the presence of a random field configura- +tion measured by an off-axis observer is displayed in Fig. (9). In the +left panel, the spectral indices are selected to be consistent with aver- +age values reported in Table (5) for our target sample and for y j = 10. +In the right panel, yj is changed while α = 1, it is found that the ap- +peared peak has a width in order of 1/√y j. From Fig. (9), we see that +the polarization degree is limited to small values for ζ < 1 while it +is sharply increased for ζ ≈ 1 and finally reaches to an asymptotic +limit at ζ > O(1). It is seen that the Synchrotron radiation with B⊥ +can potentially generate wide range of polarization values from low +levels to moderate values which cover observed values associated to +our sample. In principle, various viewing angles θobs and different +angular structures of the jet affect the measured fluence of GRBs. +Note that the fluence significantly decreases for a top-hat jet view- +ing from outside the jet’s sharp edge, so high levels of polarization +in off-axis jets may only be obtainable in very close bursts. In fact, +the detectibility of GRB polarization needs high-fluence sources and +usually, the fluence rapidly drops below the detector threshold for a +large off-axis observer. +The time-resolved spectral analysis in §4.1 showed thermal to +non-thermal (KED-to-PFD) transitions in our sample where a sub- +dominant component of the thermal emission during bursts is ob- +served. Observing hard values of the spectral indices during the +bursts can be served as hints that LOS is not highly off-axis, since +high latitude emission leads to a softer spectrum Lundman et al. +(2013). +As it was reported in §4.2, a correlation between the polariza- +tion and the isotropic energy π-Eγ,iso (Fig.7b) has been observed +within our sample, it is worth mentioning that higher polariza- +tion values are recorded for closer bursts GRB 110301A (z=0.36), +GRB 110721A (z=0.382), GRB 160802A (z=0.90) and lower val- +ues for farther sources GRB 140206B (z=2.73) and GRB 100826A +(z=2.3) (Fig.7e). The observed fluences of GRB 140206B and GRB +100826A is higher than other sources (see Table. 4 and Fig.7f) and +due to their higher redshifts ζ can not obtain large values, however, +low values of ζ would be enough to reproduce their measured polar- +izations. +The local degree of linear polarization for a tangled or random +field configuration for a thin ultrarelativistic shell modeling of the +prompt emission by assuming α = 1 is obtained by averaging over +all local magnetic field directions as +Πlin +rnd = Πlin +max +(b−1)sin2 θB +2+(b−1)sin2 θB +(24) +MNRAS 000, 1–?? (2023) + +Time-averaging Polarimetric and Spectral Properties of GRBs +9 +where b ≡ 2⟨B2 +∥⟩/⟨B2 +⊥⟩ denotes the anisotropy of the magnetic field +distribution as the ratio of the parallel B∥ to the perpendicular B⊥ +components with respects to the shock direction, and θB is the an- +gle between the LOS from the observer and the direction of the +shock Sari (1999); Gruzinov (1999). In the case of a globally ordered +magnetic field configuration aligned with the jet direction (B → B∥, +b → ∞), Eq. (24) returns back to Eq. (19) and gives the maximum +value of the linear polarization. +The polarized emission may also originate from independent +magnetic patches with various field orientation Li (2022a) where +magnetic patches are locally coherent but distributed randomly in +observed emission region. In this case the measured polarization +from different patches is estimated as Π = Πmax/ +√ +N, where N is the +number of magnetic patches or equivalently multiple pulses where +the coherence length of the magnetic field is as large as the emis- +sion region in a single pulse and observed polarization is an average +over multiple pulses (Gruzinov & Waxman 1999; Granot & Königl +2003b). +The magnetic field which is generated within IS for KED jets has +usually a coherence length much smaller than the angular size of the +emission region which causes negligible net polarization. It has been +shown that even by taking into account the angular structure of the +flow the polarization is limited to Π ≲ 20% for photospheric emis- +sion of a relativistically expanding fireball Ito et al. (2014); Lund- +man et al. (2014a); Parsotan et al. (2020). The observed high val- +ues of the polarization for GRB 110721A and GRB 160802A while +they show the peak-KED pattern cannot be explained simply by the +sub-photospheric dissipation model based on Comptonisation. Be- +cause the multiple scatterings at large optical depths region leads to +wash out the directionality of polarization vectors (Lundman et al. +2018b). To explain the strong polarized signals, models invoking dis- +sipation of ordered magnetic field are favored (Lyutikov et al. 2003; +Zhang & Yan 2011b; McKinney & Uzdensky 2012). A structured jet +photosphere model may also generate polarized photons via Comp- +ton scattering but with a different energy-dependence compare to +the synchrotron model in the ordered magnetic field (Chang et al. +2014b,a, 2013). +The Jitter radiation emitted by ultra-relativistic electrons acceler- +ating in a small-scale random magnetic field (Medvedev 2000), can +also generate a hard energy spectrum with the photon index as high +as α = +0.5. Due to the random distribution of the magnetic field, +jitter radiation is highly symmetric in the electron radiative plane, +leading to the vanishing polarization degree for an on-axis observer +(Mao & Wang 2013, 2017; Mao et al. 2018). The maximum level +of polarization is obtainable when the emitting plane is viewed from +the edge on, it can even reach up to 90% (Prosekin et al. 2016). +However, for smaller off-axis viewing angles which can yield mea- +surable fluences, jitter radiation causes almost negligible polariza- +tion degree. Meanwhile, regardless of the viewing angle the Jitter +radiation cannot produce the observed high degree of polarisation +close to the spectral peak energy of the jet. +To summarize, polarization features can be explained either by the +synchrotron radiation in the ordered/random magnetic field (Granot +2003b; Granot & Königl 2003b; Nakar et al. 2003), the jet structure +(Lazzati & Begelman 2009), or the observer’s viewing angle with re- +spect to the jet (Lazzati et al. 2004), even in the case of thermal radi- +ation from the jet photosphere (Lundman et al. 2014b). For a hybrid +spectrum which include thermal and non-thermal components, we +expect to see relatively high values of the polarization in the prompt +emission which can be produced by synchrotron emission mecha- +nism in the ordered magnetic field of the jet, and for random field +configurations only for off-axis observers (Gill et al. 2021). How- +ever, the spectral properties of our target sample demonstrated that +off-axis observations specially for the large viewing angle is not the +case, and the observed values of the polarization most probably is +a hint of the ordered magnetic field originating from the central en- +gine. Since from PFD jets towards HD and KED jets, polarization +washout effects are increased gradually due to thermal photons, we +would expect that the inequality πKED ≲ πHD ≲ πPED is satisfied if +other conditions are fixed for a given jet. Due to the different de- +grees of polarization predicted by different emission models in var- +ious energy bands, it is essential to have a high-sensitivity gamma- +ray polarimeter with a wide band-pass to detect energy-dependent +polarization signals and constrain different models (Zhang 2014). +However, it should be noted that due to several free parameters in +polarization models, upcoming more precise observations and theo- +retical investigations are needed to discriminate between competing +models in order to explain observed joint polarization and spectral +properties. +6 CONCLUSION +Early polarization observations during the prompt emission phase +play a crucial role in understanding the radiation mechanism and +jet composition of GRBs. Observations over the past few decades +suggest that the jet composition of GRBs may have diverse prop- +erties. If the jet composition is matter-dominated (i.e., a fireball), +the GRB prompt emission spectra would include a bright thermal +component originating from the fireball photosphere. Alternatively, +if the jet composition is Poynting-flux-dominated, the GRB prompt +emission spectra would include a dominant non-thermal compo- +nent originating from the synchrotron radiation. Moreover, if the jet +composition is hybrid-dominated, the GRB prompt emission spec- +tra would include a thermal component originating from the fire- +ball photosphere and a non-thermal component originating from the +synchrotron radiation. It is highly speculated that the prompt emis- +sion is likely expected to be strongly polarized owing to its non- +thermal origin. Consequently, a different level of polarization de- +grees (πKED ≲ πHD ≲ πPED) during the prompt emission phase is nat- +urally expected due to the different types of jet composition. In this +paper, we have collected a GRB sample in which all the bursts de- +tected by Fermi/GBM and whose polarization detection in the emis- +sion region was also reported in the literature, containing five inter- +esting bursts (GRB 100826A, GRB 110301A, GRB 110721A, GRB +140206A, and GRB 160802A). Using the time-averaging polariza- +tion observations and selecting the same epoch for the GBM data +taken during the prompt emission phase, we then attempted to ex- +plore the correlations between jet properties and polarization prop- +erties of GRBs and aimed to confirm the validity of this correlation +(πKED ≲ πHD ≲ πPED) from observations. +We first performed a detailed time-averaged spectral analysis for +each burst in our target sample by using several frequency-used GRB +spectral models and selected the best one by using information crite- +ria (AIC and BIC). The jet properties of GRBs can be classified into +three categories based on their spectral analysis: the “KED”, “PFD”, +and “HD” types. Using the spectral properties we then inferred their +jet properties and discovered that all five bursts belong to the “HD”- +jet type. The lack of the other two types of jets (KED and PED) +prevents us from validating this correlation (πKED ≲ πHD ≲ πPED). +Hopefully, upcoming instruments will provide high-sensitivity po- +larization observations in the future, leading to well-sampled, well- +studied data sets, enabling such statistical analysis. +We next conducted a time-resolved spectral analysis for each in- +MNRAS 000, 1–?? (2023) + +10 +Li & Shakeri +dividual burst by dividing the emission period into multiple-time +slices using the BBlocks method using the Band-alone model. Our +refined time-resolved spectral analysis, on the other hand, further +suggested that the “HD”-type has two subcategories: the peak-KED +pattern and the KED-to-PFD transition pattern. In our attempt to as- +sess the jet properties of GRBs using Band-α evolution, we discov- +ered that two bursts exhibit the peak-KED pattern (GRB 110721A +and GRB 160802A) whereas the other three bursts show the KED- +to-PFD transition pattern (GRB 110301A, GRB 140206A, and GRB +160625B). All five bursts found in the “HD”-type imply that the pho- +tosphere emission may also be a possible mechanism to power the +high-degree polarization observation. +We also made an attempt to explore the correlations between the +polarization properties and several typical GRB observed quantities. +Using the same observed epoch during the prompt emission phase, +our target sample allows for a reasonable comparison. The corre- +lations we attempted to study included the polarization degrees π +correlated with (i) the cosmological rest-frame peak energy (Ep,z) +of the νFν prompt emission spectrum, (ii) the isotropic-bolometric- +equivalent emission energy Eγ,iso, (iii) the magnetization parameter +σ0, (iv) the blackbody temperature kT, (v) the redshift z, and (vi) the +corresponding energy fluence Sγ. As a result, we discovered that a +higher Ep,z, Eγ,iso, and kTz tend to have a lower-degree polarization +π. +Lastly, we discovered that all five bursts in our target sample have +a relatively high-degree polarization detection that seems to corre- +late with the “HD”-jet type. If it is an intrinsic characteristic of +GRBs, this could provide a clue to studying the radiation mechanism +and jet composition of GRBs. We have also discussed some physical +interpretations of this interesting phenomenon. Since the configura- +tion of the magnetic field inside the jet is one of the crucial param- +eters to determine the polarization degree, we discussed two main +configurations (i.e. ordered and random fields), and their connec- +tion to the jet composition is clarified. We considered polarization +patterns as a function of different dynamical parameters associated +to the outflow materials, the spectral indices and the observer’s LOS +with respect to the jet. Combining the spectral analysis and the polar- +ization measurements allowed us to find out the detection of polar- +ization values Π > 50% during prompt emission of GRB 160802A, +GRB 110721A and GRB 110301A is a piece of strong evidence for +the synchrotron emission mechanism in the presence of an ordered +magnetic field which can be advected from the GRB central engine. +Regarding the different properties of our target sample, we conclude +that geometrical effects and large off-axis observations are unlikely +responsible for the measured polarizations assuming random mag- +netic fields within the jets. +Finally, there are some caveats that are worth mentioning when +applying our analysis. (i) Spectrum. We have resolved the jet proper- +ties based on the low-energy spectrum. However, it may be difficult +to classify jets as either KED or PFD jets based on the spectral index +alone. Indeed, a photospheric quasi-thermal component would have +a harder low-energy spectral index as compared to an optically-thin +synchrotron, but that does not guarantee that the jet is KED (an ex- +ample, see Gill et al. 2020). (ii) Polarization. The degree of polariza- +tion ultimately probes the (local) structure of the B-field in the emis- +sion region. An ordered field would necessarily yield high polariza- +tion whereas a tangled field would yield a very small polarization. It +is unclear, however, whether these field configurations are exclusive +to a given jet configuration (or a particular level of magnetization). +In addition, the angular structure of the jet also plays an important +role in governing the observed polarization. Thus, due to the large +range of model parameters, it is difficult to attribute a given level of +polarization to a given jet composition. More discussion is provided +in a recent review article (Gill et al. 2021). (iii) Different instrument +analysis. Currently, it is not clear why different instruments, namely, +POLAR, IKAROS-GAP, and ASTROSAT/CZTI, are finding differ- +ent levels of polarization for a small sample of GRBs (Chattopad- +hyay et al. 2019). There is no consensus. POLAR is finding a rather +low-level polarization, which is consistent with zero within 3σ of +their quoted central values, whereas both IKAROS and AstroSAT +are finding much higher levels. Hard X-ray to soft gamma-ray po- +larization measurements are very tricky and the analysis has to be +carried out very carefully. As such, some of these measurements are +probably not representative of GRBs and need to be further verified +by future more precise instruments. (iv) Time-resolved polarization +analysis. In the current analysis, none of the cases have shown time- +resolved polarization measurements. Even though the GRBs in our +target sample have time-resolved spectral indices, not having corre- +sponding polarization measurements makes it difficult to ascertain +the properties of the B-field and outflows. +Acknowledgements. We thank Ramandeep Gill, Jonathan Gra- +not, Rahim Moradi, Mi-Xiang Lan, Asaf Pe’er, Jin-Jun Geng, +Christoffer Lundman, Remo Ruffini, and ICRANet members for +many discussions on GRBs physics and phenomena. This research +made use of the High Energy Astrophysics Science Archive Re- +search Center (HEASARC) Online Service at the NASA/Goddard +Space Flight Center (GSFC). +Data availability. The data underlying this article will be shared +on reasonable request to the corresponding author. +REFERENCES +Abdo, A. 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F. 2012, A&A , 538, A134 +Yonetoku, D., Murakami, T., Nakamura, T., et al. 2004, ApJ , 609, 935 +Yonetoku, D., Murakami, T., Gunji, S., et al. 2011a, ApJ , 743, L30 +—. 2011b, ApJ , 743, L30 +—. 2012a, ApJ , 758, L1 +—. 2012b, ApJ , 758, L1 +Zhang, B. 2014, International Journal of Modern Physics D, 23, 1430002 +—. 2018, The Physics of Gamma-Ray Bursts, doi:10.1017/9781139226530 +Zhang, B., & Mészáros, P. 2001, ApJ , 552, L35 +Zhang, B., & Yan, H. 2011a, ApJ , 726, 90 +—. 2011b, ApJ , 726, 90 +Zhang, F.-W., Shao, L., Yan, J.-Z., & Wei, D.-M. 2012, ApJ , 750, 88 +Zhang, S.-N., Kole, M., Bao, T.-W., et al. 2019, Nature Astronomy, +arXiv:1901.04207 +MNRAS 000, 1–?? (2023) + +12 +Li & Shakeri +Table 1 A sample of GRB polarimetric observations +GRB +PD +PA +Energy band +Time +Significance +Instrument +Ref. +z +(Fermi ID) +(π%) +(◦) +(t-t0) +(σ) +(For polarization) +100826A(957) +27±11 +159±18, 75±20 +γ-ray +0-100s +2.9σ +IKAROS-GAP +Yonetoku et al. (2011b) +NA +110301A(214) +70±22 +73±11 +γ-ray +0-7s +3.7σ +IKAROS-GAP +Yonetoku et al. (2012a) +NA +110721A(200) +84+16 +−28 +160±11 +γ-ray +0-11s +3.3σ +IKAROS-GAP +Yonetoku et al. (2012a) +0.382 +140206A(275) +> 28 +80±15 +γ-ray +4-26s +90% confidence +INTEGRAL-IBIS +Götz et al. (2014) +2.73 +160802A(259) +85±29 +∼-32 +hard X-rays +0-20.34s +∼3σ +AstroSat-CZTI +Chand et al. (2018) +NA +MNRAS 000, 1–?? (2023) + +Time-averaging Polarimetric and Spectral Properties of GRBs +13 +Table 2 Estimated values of redshift using the Yonetoku relation +GRB +tstart∼tstop +S +Eobs +p +Fobs +p +kc +Lp +z +(s) +(keV) +(erg cm−2 s−1) +(erg s−1) +(Estimated) +100826957 +18.208∼22.288 +107.97 +459±20 +(1.42±0.10)×10−5 +0.85 +(1.27±0.10)×1053 +2.3 +110301214 +3.876∼4.126 +107.60 +126±6 +(1.36±0.16)×10−5 +1.02 +(1.46±0.17)×1053 +0.36 +160802259 +0.962∼1.171 +73.79 +385±39 +(3.04±0.80)×10−5 +0.95 +(3.05±0.80)×1053 +0.90 +MNRAS 000, 1–?? (2023) + +14 +Li & Shakeri +Table 3 Comparison of AIC/BIC between the best model and other models +GRB +t1 ∼ t2 +AIC/BIC(1) +AIC/BIC(2) +AIC/BIC(3) +AIC/BIC(4) +AIC/BIC(5) +AIC/BIC(6) +AIC/BIC(7) +(s) +(PL) +(BB) +(CPL) +(Band) +(PL+BB) +(CPL+BB) +(Band+BB) +100826A(957) +0∼100 +11662/11670 +12225/12232 +6819/6830 +6706/6721 +7819/7834 +6693/6712 +6638/6661 +110301A(214) +0∼7 +10969/10978 +20029/20038 +5342/5354 +5284/5301 +6124/6140 +5302/5323 +5254/5278 +110721A(200) +0∼11 +7319/7327 +14194/14203 +5694/5707 +5541/5557 +7323/7340 +5524/5544 +5504/5528 +140206A(275) +4∼26 +12649/12657 +20413/20421 +7617/7630 +7431/7448 +8549/8566 +7345/7366 +7293/7317 +160802A(259) +0∼20.34 +6840/6847 +7480/7487 +3984/3994 +3898/3912 +4383/4397 +3839/3856 +3840/3861 +MNRAS 000, 1–?? (2023) + +Time-averaging Polarimetric and Spectral Properties of GRBs +15 +Table 4 Global properties of the Sample +GRB +T90 +Fluence +Detectors +∆Tsrc +[∆T(bkg,1),∆T(bkg,2)] +Spectral model +(Fermi ID) +(s) +(erg cm−2) +(s) +(s) +(Preferred) +100826A(957) +84.993±0.724 +(1.64±0.01)×10−4 +n7(n8)b1 +(0 to 100) +(-20 to -10, 200 to 250) +Band+BB +110301A(214) +5.693±0.362 +(3.59±0.01)×10−5 +n7(n8)nbb1 +(0 to 7) +(-20 to -10, 40 to 60) +Band+BB +110721A(200) +21.822±0.572 +(3.70±0.01)×10−5 +(n6)n7n9b1 +(0 to 11) +(-20 to -10, 40 to 60) +Band+BB +140206A(275) +146.690±4.419 +(1.23±0.01)×10−4 +n0(n1)n3b0 +(4 to 26) +(-40 to -20, 70 to 90) +Band+BB +160802A(259) +16.384±0.362 +(6.84±0.01)×10−5 +(n2)b0 +(0 to 20.34) +(-20 to -10, 60 to 80) +Band+BB +MNRAS 000, 1–?? (2023) + +16 +Li & Shakeri +Table 5 Spectral Fit Results of the Sample with the Band+BB Model. +GRB +t1∼t2 +S +K +α +Ep +β +K +kT +Flux +Flux ratio +(s) +(Band) +(Band) +(Band) +(Band) +(BB) +(BB) +(FBB/Ftot) +(s) +(ph.s−1.cm−2.keV−1) +(keV) +(ph.s−1.cm−2.keV−1) +(keV) +(erg.cm−2.s−1) +100826957 +0∼82 +102.7 +(3.04+0.16 +−0.16)×10−2 +-0.84+0.04 +−0.04 +518+46 +−46 +-2.28+0.07 +−0.07 +(5.54+1.97 +−1.93)×10−5 +21+2 +−2 +(3.20+0.34 +−0.31)×10−6 +0.03+0.03 +−0.03 +110301214 +0∼7 +276.8 +(3.73+0.34 +−0.30)×10−1 +-0.72+0.07 +−0.07 +114+2 +−3 +-2.87+0.08 +−0.07 +(1.14+0.57 +−0.49)×10−2 +7+1 +−1 +(5.90+0.73 +−0.69)×10−6 +0.04+0.03 +−0.03 +110721200 +0∼11 +114.5 +(3.01+0.09 +−0.09)×10−2 +-1.20+0.02 +−0.02 +1620+234 +−229 +-2.19+0.10 +−0.10 +(1.73+0.33 +−0.34)×10−5 +33+2 +−2 +(6.94+0.66 +−0.63)×10−6 +0.03+0.01 +−0.01 +140206275 +4∼26 +170.2 +(3.87+0.11 +−0.11)×10−2 +-1.06+0.02 +−0.02 +679+43 +−43 +-2.32+0.08 +−0.08 +(4.78+0.53 +−0.52)×10−5 +27+1 +−1 +(4.57+0.25 +−0.24)×10−6 +0.05+0.01 +−0.01 +160802259 +0∼20.34 +151.7 +(3.89+0.21 +−0.21)×10−2 +-1.00+0.03 +−0.03 +515+44 +−44 +-3.23+0.73 +−0.74 +(9.11+1.22 +−1.21)×10−5 +25+1 +−1 +(3.85+0.55 +−0.41)×10−6 +0.09+0.02 +−0.02 +MNRAS 000, 1–?? (2023) + +Time-averaging Polarimetric and Spectral Properties of GRBs +17 +1 +0 +2 +4 +6 +8 +10 +redshift +10 +1 +100 +101 +102 +103 +GRB 100826A +Estimated value +0 +2 +4 +6 +8 +10 +redshift +10 +1 +100 +101 +102 +103 +GRB 110301A +Estimated value +0 +2 +4 +6 +8 +10 +redshift +10 +1 +100 +101 +102 +103 +GRB 160802A +Estimated value +Figure 1. Estimated redshift using the Yonetoku relation for three bursts (GRB 100826A, GRB 110301A, and GRB 160802A). The yellow and cyan lines +represent the left and right function of Eq.(5), and their intersection point (purple color) is the estimated value of redshift. +MNRAS 000, 1–?? (2023) + +18 +Li & Shakeri +1 +20 +0 +20 +40 +60 +80 +100 +120 +140 +Time (s) +0 +20 +40 +60 +80 +100 +=27±11% +2.0 +1.5 +1.0 +0.5 +0.0 +0.5 + (S > 20) + (10 < S < 20) + (S < 10) +100826A +101 +102 +103 +104 +105 +Photon Energy - keV +100 +101 +102 +103 +104 +keV × [keV +1S +1cm +2] +Band fits +Blackbody +Band+Blackbody +NAI7 +NAI8 +BGO1 +Figure 2. Left panel: prompt emission GBM light curve (overlaid in gray) and polarization observations in γ-ray/hard X-ray energy bands (cyan shaded area), +as well as the temporal evolution of α based on the time-resolved spectral analysis. The horizontal dashed line represents the limiting value of α = −2/3 for +electrons in the slow-cooling regime. Right panel: the spectral data and its best-fit model (Band+BB) during the time epoch (see Table 1 and Table 4) of the +matching polarization observations. +MNRAS 000, 1–?? (2023) + +Time-averaging Polarimetric and Spectral Properties of GRBs +19 +1 +5.0 +2.5 +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +Time (s) +0 +20 +40 +60 +80 +100 +=70±22% +2.0 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 + (S > 20) + (10 < S < 20) + (S < 10) +110301A +101 +102 +103 +104 +105 +Photon Energy - keV +100 +101 +102 +103 +104 +keV × [keV +1S +1cm +2] +Band fits +Blackbody +Band+Blackbody +NAI7 +NAI8 +NAIb +BGO1 +nai +bgo +Figure 3. Same as Figure 2 but for GRB 110301A. +MNRAS 000, 1–?? (2023) + +20 +Li & Shakeri +1 +15 +10 +5 +0 +5 +10 +15 +20 +25 +30 +Time (s) +0 +20 +40 +60 +80 +100 += (84+16 +28)% +2.0 +1.5 +1.0 +0.5 +0.0 +0.5 + (S > 20) + (10 < S < 20) + (S < 10) +110721A +101 +102 +103 +104 +105 +Photon Energy - keV +101 +102 +103 +104 +keV × [keV +1S +1cm +2] +Band fits +Blackbody +Band+Blackbody +NAI6 +NAI7 +NAI9 +BGO1 +nai +bgo +Figure 4. Same as Figure 2 but for GRB 110721A. +MNRAS 000, 1–?? (2023) + +Time-averaging Polarimetric and Spectral Properties of GRBs +21 +1 +20 +0 +20 +40 +60 +80 +100 +120 +140 +Time (s) +0 +20 +40 +60 +80 +100 + (Upper limit) +2.0 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 + (S > 20) + (10 < S < 20) + (S < 10) +140206A +101 +102 +103 +104 +105 +Photon Energy - keV +101 +102 +103 +104 +keV × [keV +1S +1cm +2] +Band fits +Blackbody +Band+Blackbody +NAI0 +NAI1 +NAI3 +BGO0 +nai +bgo +Figure 5. Same as Figure 2 but for GRB 140206A. +MNRAS 000, 1–?? (2023) + +22 +Li & Shakeri +1 +20 +10 +0 +10 +20 +30 +40 +Time (s) +0 +20 +40 +60 +80 +100 += 85 ± 29 +2.0 +1.5 +1.0 +0.5 +0.0 +0.5 + (S > 20) + (10 < S < 20) + (S < 10) +160802A +101 +102 +103 +104 +105 +Photon Energy - keV +101 +102 +103 +104 +keV × [keV +1S +1cm +2] +Band fits +Blackbody +Band+Blackbody +NAI2 +BGO0 +nai +bgo +Figure 6. Same as Figure 2 but for GRB 160802A. +MNRAS 000, 1–?? (2023) + +Time-averaging Polarimetric and Spectral Properties of GRBs +23 +1 +101 +102 +103 +104 +Ep, z[keV] +100 +101 +102 +103 +a +100826A +110301A +110721A +140206B +160802A +100 +101 +102 +103 +Power-law index= -0.41±0.24 +1051 +1052 +1053 +1054 +1055 +E , iso (erg) +100 +101 +102 +103 +b +100826A +110301A +110721A +140206B +160802A +100 +101 +102 +103 +Power-law index= -0.20±0.04 +100 +101 +102 +103 +104 +KT,z[keV] +100 +101 +102 +103 +c +100826A +110301A +110721A +140206B +160802A +100 +101 +102 +103 +Power-law index= -0.32±0.18 +100 +101 +102 +1+ +0 +100 +101 +102 +103 +d +100826A +110301A +110721A +140206B +160802A +100 +101 +102 +103 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +redshift +100 +101 +102 +103 +e +100826A +110301A +110721A +140206A +160802A +10 +6 +10 +5 +10 +4 +10 +3 +10 +2 +Fluence (erg cm +2) +100 +101 +102 +103 +f +100826A +110301A +110721A +140206A +160802A +Figure 7. Scatter plots of polarization degree π versus several other observed quantities: (a) the cosmological rest-frame peak energy (Ep,z) of the νFν prompt +emission spectrum, (b) the isotropic-bolometric-equivalent emission energy Eγ,iso, (c) the magnetization parameter σ0, (d) the blackbody temperature kT, (e) +the redshift z, and (d) the corresponding energy fluence Sγ. Data points with different colors indicate the different bursts in our target sample. The solid lines +(grey) are the best fit using the power-law model with 2σ (95% confidence interval) error region (shadow area). +MNRAS 000, 1–?? (2023) + +24 +Li & Shakeri +1 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +100 +101 +102 +100826A +110301A +110721A +160802A +140206A +100 +101 +102 +lin +max = ++ 1 ++ 5/3 +ymax=1 +ymax=2 +ymax=5 +ymax=10 +ymax=102 +110721A +160802A +110301A +100826A +140206A +0.0 +0.5 +1.0 +1.5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +α +Π/Πmax +Figure 8. Left: The maximum degree of the linear polarization applying synchrotron emission model (Πlin +max Eq. (19)) with observed data using α indices based +on a time-integrated spectral analysis. Right: Time integrated polarization degree in the presence of an ordered magnetic field Bord with in the plane of ejecta +(Eq. (19)) measured by an on-axis observer (θobs = 0), the evolution of the polarization is plotted in terms of α for different values of ymax = (Γθmax)2. +MNRAS 000, 1–?? (2023) + +Time-averaging Polarimetric and Spectral Properties of GRBs +25 +1 +α=0.72 +α=0.84 +α=1 +α=1.2 +Random B⟂ +yj = 10 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +1.8 +2.0 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +ζ=θobs/θj +Π/Πmax +yj=103 +yj=102 +yj=10 +yj=1 +α = 1 +Random B⟂ +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +1.8 +2.0 +0.0 +0.2 +0.4 +0.6 +ζ=θobs/θj +Π/Πmax +Figure 9. The time integrated polarization for a random magnetic field B⊥ which lies entirely in the plane of the shock (Eq. (20)) as a function of the off-axis +parameter ζ = θobs/θ j for different values of spectral index α (left) and yj = (Γθ j)2 (right) as labeled. +MNRAS 000, 1–?? (2023) + diff --git a/KNAyT4oBgHgl3EQfsfkS/content/tmp_files/load_file.txt b/KNAyT4oBgHgl3EQfsfkS/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ec147ba00a3d6ada8936511601fa5ddc558d7f85 --- /dev/null +++ b/KNAyT4oBgHgl3EQfsfkS/content/tmp_files/load_file.txt @@ -0,0 +1,1685 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf,len=1684 +page_content='MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (2023) Preprint 3 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='0 Time-averaging Polarimetric and Spectral Properties of GRBs Liang Li1,2,3⋆ Soroush Shakeri4,1,5† 1ICRANet, Piazza della Repubblica 10, I-65122 Pescara, Italy 2ICRA and Dipartimento di Fisica, Università di Roma “La Sapienza”, Piazzale Aldo Moro 5, I-00185 Roma, Italy 3INAF – Osservatorio Astronomico d’Abruzzo, Via M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Maggini snc, I-64100, Teramo, Italy 4Department of Physics, Isfahan University of Technology, Isfahan 84156-83111 5ICRANet-Isfahan, Isfahan University of Technology, Isfahan 84156-83111, Iran Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' in original form ZZZ ABSTRACT One of the most fundamental and yet open issues in gamma-ray burst (GRB) physics, is the comprehension of the nature of their jet composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The investigation of joint polarimetric and spectral properties is essential to probe the jet composition and radiation mechanism of GRBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Several distinct categories of jet properties—the “Kinetic-energy-dominated" (KED), “Poynting- flux-dominated" (PFD), and “Hybrid-dominated" (HD) jets—have been observed in the observed GRB spectra, and the emission dominated by different jet properties is expected to have a different level of polarization (πKED ≲ πHD ≲ πPED).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' In the present paper,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' we collected a GRB sample in which all the bursts detected by the Gamma-ray Burst Monitor (GBM) on board the NASA Fermi Gamma-ray Space Telescope whose polarization measurements are also reported in the literature and the epochs of prompt emission are heavily overlapped with their polarization observations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' aiming to establish a connection between the polarization and jet properties of GRBs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' and to confirm the validity of this correlation (πKED ≲ πHD ≲ πPED) from observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' With a detailed spectral analysis, we found that all the bursts are classified as the “Hybrid" jet type, implying that one cannot rule out that the photosphere emission may also be the possible mechanism powering the high levels of polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Moreover, we also discovered that the polarization degrees π are tightly correlated with the cosmological rest-frame peak energy (Ep,z) of the νFν prompt emission spectrum, the isotropic-bolometric-equivalent emission energy (Eγ,iso), and the blackbody temperature (kT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Finally, we present different polarization models in the presence of ordered and random magnetic field configurations with the properties of corresponding hybrid jets in order to interpret polarization measurements of the prompt emission in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Key words: gamma-ray burst: general, radiation mechanisms: non-thermal, radiation mechanisms: thermal 1 INTRODUCTION Gamma-ray bursts (GRBs) are one of the most explosive, and elec- tromagnetically the brightest transient phenomena in the Universe, occurrence at cosmological distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' After decades of investiga- tion, the origin of the jet composition (a hot baryonic-dominated fireball or a cold Poynting-flux-dominated outflow), and the radi- ation mechanism and energy dissipation mechanism (synchrotron, or Comptonization of quasi-thermal emission from the photosphere) in gamma-ray burst (GRB) physics are still unclear (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=', Rees & Meszaros 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Mészáros & Rees 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Rees & Mészáros 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Pe’er et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Dai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Pe’er 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Pe’Er & Ryde 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Zhang 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Bégué et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' There are two crucial clues that can in principle be helpful in di- agnosing the jet composition of GRBs, as well as their radiation mechanism and energy dissipation mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The conventional approach is to examine the spectral properties of prompt emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' ⋆ E-mail: liang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='li@icranet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='org (LL) † E-mail:s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='shakeri@iut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='ir (SS) Theoretically, a thermal component originating from photosphere emission, or a non-thermal component originating from synchrotron radiation, possibly also from inverse Compton scattering, is often expected to be present in GRB spectral analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Phenomenologi- cally, GRB spectra in the keV-MeV energy range can be typically well-delineated by an empirical function, known as the Band func- tion (Band et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 1993), which is generally considered to be a non- thermal spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The Band spectrum features a smoothly broken power law, with the peak energy Ep ≃ 210 keV (the energy at which most of the energy is released) in νFν space and the asymptotic power-law photon indices below (α ≃ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='8) and above (β ≃ −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='5) the break energy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=', Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The low-energy spectra dur- ing GRB prompt emission phase are closely related to the energy distribution of electrons (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=', Preece et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Lloyd & Petrosian 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Geng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' This fact can be utilized in order to diag- nose GRBs radiation mechanism as well as their jet properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' For instance, synchrotron emission predicts two different α values: α=- 3/2 and α=-2/3 (so-called the line-of-death (LOD) of synchrotron emission, Preece et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 1998) correspond to the fast-cooling and slow-cooling synchrotron emission, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' It has been shown © 2023 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='00576v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='HE] 2 Jan 2023 2 Li & Shakeri that the synchrotron emission in the presence of a decaying mag- netic field can reproduce the Band-like spectrum of the GRB prompt phase (Lan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' While photosphere models, on the other hand, predict much harder values of α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=', above α=-2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' For ex- ample, a recent study (Acuner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2020) suggests that the spec- tra that prefer the photospheric model all have low-energy power- law indices α ∼>-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='5, as long as the data has a high significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Years of observations have revealed, however, that GRBs have di- verse spectral properties, making it difficult for a single spectral model (such as the Band-alone model) to accurately characterize all the spectral shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' For instance, time-resolved and time-integrated spectral analysis inferred from the broadband Fermi observations have revealed that GRB prompt emission exhibits remarkably di- verse spectral properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=', Abdo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Ryde et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Axelsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Acuner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Li 2019a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2019, 2021, 2022b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' A kinetic-energy-dominated (KED) jet characterised by a quasi-thermal Planck-like spectrum has been detected in some bursts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' GRBs 090902B, 220426A, Ryde et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2022), while a cold Poynting-flux-dominated (PFD) outflow characterised by a Band (or cutoff power-law1)-only function (Band et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 1993) has been also observed in some other bursts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' GRB 080916C2, GRB 130606B, and many others, Abdo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Li 2022a), even a hybrid-dominated (HD) relativistic outflow with a hot fireball com- ponent and a cold Poynting-flux component, characterized by either a composited spectral scenario, with a non-thermal component and a thermal component, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=', GRBs 100724B, 110721A, 150314A, 190114C, and several others, Axelsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Guiriec et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Li 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2022b), or a transition from a fireball to a Poynting-flux-dominated outflow within a single burst (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' GRBs 140206B, 160625B, and several others, Li 2019a), have also been observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' An alternative approach is to investigate their polarization prop- erties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Theoretically, photon polarizations play a key role to un- derstand the jet composition, angular structure, geometric config- uration, magnetic composition and magnetic field configuration of GRB jets, and radiation mechanism of GRB jets (Toma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2009a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Lundman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Zhang 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Although magnetic field configurations with relatively large coherence lengths more than gyroradius of charged particles can generate the same energy spectrum via synchrotron mechanism, the level of polariza- tion may significantly different for various magnetic field structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Therefore joint spectral and polarization analysis is essential to de- termine the magnetic field structure in outflow materials of GRBs (Granot 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Lyutikov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Granot & Königl 2003a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Kole et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' For instance, the center engine is anticipated to gen- erate strong magnetic fields (a highly magnetized jet) and launch them concurrently with the relativistic jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' It is unclear, neverthe- less, whether the GRB emission is caused by shock dissipation or magnetic reconnection, and whether the outflow is dominated by the photosphere or synchrotron emission (Toma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2009a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' In fact, the generation of the polarization signal can be intrinsic to the emission process or due to the propagation effects Shakeri 1 Recent studies (Li 2022b,a) supported by several pieces of additional ev- idence (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=', inconsistent spectral parameter distributions and distinct Amati and Yonetoku correlations) have shown that Band-like spectra and CPL-like spectra may originate from distinct radiation processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2 It has been demonstrated in recent studies (Guiriec et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Vereshcha- gin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2022) that a thermal component needs to be added during the initial prompt emission of GRB 080916C to obtain an acceptable fit to the spectral data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' & Allahyari (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Several emission models (induced synchrotron emission, Rybicki & Lightman 1979;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' photosphere emission, Lund- man et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2014a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' and Compton drag model, Lazzati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2004) have been proposed to explain the intrinsic polarization properties of relativistic jets during prompt emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (i) Synchrotron emis- sion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' There are some studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=', Rybicki & Lightman 1979;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Toma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2009b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Lan & Dai 2020) showing that higher values of linear polarized signal (polarization degree π ranging from 20% to 70%) is expected to be measured with an ordered magnetic field from the synchrotron emission from a relativistic jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' While jets with random magnetic fields produce lower levels of polarization, this is due to the polarization being canceled out so that the net polariza- tion degree being close to zero for an on-axis observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' A polariza- tion detection which is less than 15% is believed to be originated from a random magnetic field configuration within the jet (Mao & Wang 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' For example, if the emission is dominated by the inter- nal shock (IS) model, π is expected to range from 10% (the maxed magnetic field configuration) to 70% (the large-scale ordered mag- netic field configuration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (ii) Dissipative photosphere model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The dissipative photosphere model predicts a relatively low degree of po- larization in the γ-ray band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' However, a structured jet photosphere model might also generate polarized photons by Compton scatter- ing, but the degree of polarization would be energy-dependent from the synchrotron model in ordered magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' For instance, it is demonstrated that if the jet has considerable structure, the model may create polarizations of up to 40% within δΘ ∼ Γ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' However, in the absence of dissipation and below the photosphere the polariza- tion is rather limited to values below 15%-20% (Gill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' To restrict these models, a high-sensitivity gamma-ray polarimeter with a broad band-pass to detect energy-dependent polarization signals is required (Zhang 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Ito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Lundman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2014a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Lund- man et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2018a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (iii) Internal-collision-induced magnetic recon- nection and turbulence (ICMART) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' In the ICMART model (Zhang & Yan 2011a), π is expected to range from 60 percent at the beginning of the pulse and down to about 10 percent at the end of the pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' A decaying polarization degree is predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' It is highly speculated that the prompt emission is likely expected to be strongly polarized owing to its non-thermal origin (a non- thermal Band-like spectrum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Observationally, higher levels of lin- ear polarization measured from prompt γ-ray emission have been reported by several authors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=', Coburn & Boggs 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Willis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' McGlynn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Yonetoku et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2012a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' For in- stance, a higher polarization degree π = 80%±20% in GRB 021006 was claimed by Coburn & Boggs (2003) using the RHESSI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Later, several other cases were also reported, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=', GRB 930131 (π > 35%), GRB 960924 (π > 50%), GRB 041219A, GRB 100826A (π = 27% ± 11%), GRB 110301A (π = 70% ± 22%), and GRB 110721A (π = 84%+16% −28%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Subsequent observations were also ob- served in the optical band during the afterglow emission and were of relatively low polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Compared with prompt γ-ray emission, the levels of linear polarization measured from afterglow emission are relatively lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=', GRB 060418 (π < 8%), GRB 090102 (π = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='1% ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='3%), GRB 091208B (π = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='4% ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='5%), and 120328A (π = 28% ± 4%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' However, higher degrees of polarization observa- tions are still expected to be measured from early reverse shocks, up to ∼ 60%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Generally speaking, we can study GRB polarization and spec- tral properties either in a time-integrated (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=', Li 2022a) or time- resolved (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=', Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2021) manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The former represents average polarimetric and spectral properties and is treated as a single-time event for the entire emission period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The latter treats the entire emis- sion period as divided into multiple-time events, and polarimetric MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (2023) Time-averaging Polarimetric and Spectral Properties of GRBs 3 and spectral analyses are therefore performed on each event individ- ually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The time-integrated method depends more heavily on the sta- tistical results of a large sample in order to produce a more trustwor- thy result because different bursts have distinct observational prop- erties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=', angular structure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' However, this issue does not arise when a time-resolved technique is used in the same burst since it ensures that other conditions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=', magnetic field configuration) are essentially the same and, therefore, makes it easier to obtain reliable results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Following these lines of argument, the emissions dominated by different jet properties (KED, PFD, and HD) may have different lev- els of prompt-GRB polarization measurements3 (Li 2019a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' As a result, a different level of polarization degrees (πKED ≲ πHD ≲ πPED) is naturally expected due to different jet properties if other condi- tions are basically the same, where πKED, πHD, and πPED are the po- larization degree in the KED, HD, and PHD jets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' This may provide a method to study the correlations between polarization properties and their spectral properties, as well as their jet properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' A possible connection between the spectral and polarization proper- ties has not yet been firmly established, though recent works provide some statistical results (Chattopadhyay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Kole et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Therefore, we dedicate this work to examining whether any possi- ble connections exist between polarization and jet properties, and aim to confirm the validity of this correlation (πKED ≲ πHD ≲ πPED) from observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Practically, several important factors need to be taken into account in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (i) In order to potentially evaluate all of the frequently-used GRB spectral models and thus diagnose the jet properties, our analysis focuses on the bursts detected by the Gamma-ray Burst Monitor (GBM, 8 KeV-40 MeV, Meegan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2009) onboard the NASA Fermi Gamma-ray Space Telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (ii) To directly compare the spectral and polarization properties and thus make our results more trustworthy, we select the bursts during which polarization observations and spectral data are available during the same epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (iii) Realistically, systematic error is frequently at play in polarization measurements and different polarization instruments have different systematic errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Therefore, a high-significance sig- nal from polarization measurement is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' In this paper, we collect a sample of the Fermi-GBM detected bursts along with the polarized measurements reported in the literature using the time- integrated spectral and polarization analysis approach based on their statistical results, aiming to establish a connection between the po- larization and jet properties of GRBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The sample and Methodol- ogy are presented in Section 2 and Section 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Our re- sults and their physical implication are summarized in Section 4 and Section 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The conclusion is presented in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Throughout the paper, the standard Λ-CDM cosmology with the pa- rameters H0 = 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='4 kms−1 Mpc−1, ΩM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='315, and ΩΛ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='685 are adopted (Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 3 Polarization measurement is a measurement where π ranges from 0% to 100% (0% ≤ π ≤ 100%) and includes the non-detection (0%), so measur- ing something consistent with zero is a measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Polarization detection, however, indicates a different meaning, where 0% < π ≤ 100% and excludes 0%, since we have excluded part of the parameter space equivalent to detec- tion measurements due to the fact that a non-detection of flux implies that we did not measure something.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2 THE SAMPLE A comprehensive database of GRB polarimetric observations has been created in a recent work (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2022a) by extensively search- ing for those GRBs in the literature whose polarization measure- ments have been reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' A total of 73 bursts with polarization detections were included in the database, covering a broad wave- length range from radio to optical, X-ray, and γ-ray emission (see Table 1 in Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The prompt emission data of these bursts were observed by different satellites (Fermi, Swift, BeppoSAX, and BATSE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' On the other hand, the diagnosis of jet composition usually requires a refined spectral analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Among these satellites, Fermi covers the broadest energy range in the observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Consequently, in order to fully evaluate all current spectral models we pay spe- cial attention to these Fermi-detected bursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The Gamma-ray Burst Monitor (GBM, 8 KeV-40 MeV, Meegan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2009) and the Large Area Telescope (LAT, 20 MeV- 300 GeV, Atwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2009), on- board the NASA Fermi Gamma-ray Space Telescope, together pro- vide unprecedented spectral coverage for seven orders of magnitude in energy (from ∼8 keV to ∼300 GeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Our statistical analysis in the current work includes the prompt γ-ray emission spectral anal- ysis and the connection between the spectrum and polarization are thus based on the Fermi-detected bursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' On the other hand, we fo- cus on the GRBs whose polarization measurements were recorded during the prompt emission in the γ-ray band in order to compare the polarization and spectral properties during the same time period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Following are the specific observational properties of the five fas- cinating bursts (see Table 1) that made up the smaller sample as a result of these selection criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' GRB 100826A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' On August 26, 2010 at 22:58:22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='898 (T0) UT, GRB 100826A, was detected by the Fermi/GBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' A t90 of (84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='993±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='724) s in the 10-1000 keV was measured using the GBM data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The GBM lightcurve exhibits a complicated shape with multiple-peak pulses, and the fluence (flux integrated over the burst duration) in the energy range of 10-1000 keV during the t90 dura- tion reported by the GBM team is (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='6388±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='001)×10−4 erg cm−2 (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' This burst has no redshift measurement, a value of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='3 is estimated using the Yonetoku correlation (Yonetoku et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Therefore, the isotropic-equivalent energy released from gamma- ray emission in the cosmological frame for this burst can be cal- culated as, Eγ,iso=(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='39+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='36 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='33)×1054 erg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The KED-to-PFD transition pattern belonging to the “HD” jet type is diagnosed for this burst by using a low-energy spectrum based on a detailed time-resolved spectral analysis during prompt emission (see Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Yonetoku et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (2011a) reported that a relatively higher polarization degree (πobs=27±11) with a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='9σ linear polarization signal significance was detected during the prompt emission (0-100 seconds after the trig- ger) of GRB 100826A using a GAmma-ray Polarimeter (GAP) on board a small Japanese solar-power-sail demonstrator, Interplanetary Kite-craft Accelerated by Radiation of the Sun (IKAROS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' GRB 110301A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' On March 01, 2011 at 05:08:43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='070 (T0) UT, GRB 110301A, was detected by the Fermi-GBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' A t90 of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='693± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='362) s in the 10-1000 keV was measured using the GBM data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The lightcurve exhibits a complicated shape with multiple-peak pulses, and the fluence in the energy range of 10-1000 keV from T0+0 s to T0+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='693 s reported by the GBM team is (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='5891±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='003)×10−5 erg cm−2 (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' This burst has no redshift measurement, a value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='36 is estimated by using the Yonetoku relation (Yone- toku et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' With this estimated value of redshift, the isotropic- equivalent energy released from gamma-ray emission in the cosmo- logical frame for this burst can be calculated as, Eγ,iso=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='4×1052 erg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' A KED-to-PFD jet is diagnosed for this burst by using a low-energy MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (2023) 4 Li & Shakeri spectrum based on a detailed time-resolved spectral analysis during prompt emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' A linear polarization signal (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='7σ) with a very high polarization degree (π=70±22) for this burst was claimed by Yone- toku et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (2012b) using the IKAROS/GAP data 0-7 seconds after the trigger (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' GRB 110721A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' On July 21, 2011 at 04:47:43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='761 (T0) UT, GRB 110721A, was detected by the Fermi-GBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' A t90 of (21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='822±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='572) s in the 10-1000 keV was measured using the GBM data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The lightcurve exhibits a single-peak pulse, and the fluence in the energy range of 10-1000 keV from T0+0 s to T0+21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='822 s as re- ported by the GBM team is (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='5891±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='003)×10−5 erg cm−2 (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' This burst has a value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='382 of redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' With this redshift, we calculate the isotropic-equivalent energy released from gamma-ray emission in the cosmological frame for this burst, Eγ,iso=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='0×1052 erg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' A peak-KED jet is diagnosed for this burst by using a low- energy spectrum based on a detailed time-resolved spectral analysis during prompt emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' A linear polarization signal (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='3σ) with a very high polarization degree (π=85+16 −22) for this burst was claimed by Yonetoku et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (2012b) using the IKAROS/GAP data 0-11 sec- onds after the trigger (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' GRB 140206A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' On 6 February 2014 at 06:36:12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='843 UT (T0), a long burst, GRB 140206B, was detected by Fermi-GBM and sev- eral other satellites (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=', INTEGRAL/IBIS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Its GBM lightcurve in the 10-900 keV exhibits three clearly separated emission episodes (G1, G2, and G3) as shown in the left panel of Figure 5, with T90 of 146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='690±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='419 s was measured by GBM data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' GRB 140206B is a very bright burst, and the fluence (flux integrated over the burst dura- tion) in the energy range of 10-1000 keV from T0+0 s to T0+146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='690 s reported by the GBM team is (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='23±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='003)×10−4 erg cm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' This burst has a value of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='73 for redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' With this redshift, we calculate the isotropic-equivalent energy released from gamma-ray emission in the cosmological frame for this burst, Eγ,iso=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='3×1054 erg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The spectral analysis has been performed in great detail in Li (2019a), and three different spectral components (KED-to-PFD) have been identified, a short-thermalized precursor early on, followed up with the main burst with a non-thermal emission later on, and in the last with a fainter burst with still a non-thermal emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Interestingly, immediately following up with the short-thermalized precursor, a clear polarization signal with 90% confidence was observed 4-26 seconds after the trigger as reported in Götz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (2014) using the INTEGRAL/IBIS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' This signal is a linear polarization with an up-limit polarization degree of π > 28 observed in the γ-ray energy band (Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' GRB 160802A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' GRB 160802A was detected by GBM on Au- gust 2, 2016 at UT 06:13:29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' A T90 of (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='384±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='362) s in the 10-1000 keV was measured using the GBM data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The prompt emis- sion light curve shows two clearly separated active periods, with a quiescent time interval between the two periods of about 10 sec- onds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The earlier period consists of overlapping pulses while the lat- ter period shows a clear single-peak pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The fluence in the energy range of 10-1000 keV from T0+0 s to T0+21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='822 s as reported by the GBM team is (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='8399±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='0057)×10−5 erg cm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' This burst has no redshift measurement, a value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='36 is estimated by using the Yonetoku relation (Yonetoku et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Using this estimated value of redshift, we further calculate the isotropic-equivalent energy re- leased from gamma-ray emission in the cosmological frame for this burst, Eγ,iso=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='2×1053 erg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' A peak-KED jet for each period is di- agnosed for this burst by using a low-energy spectrum based on a detailed time-resolved spectral analysis during prompt emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' A linear polarization signal (∼3σ) with a very high polarization degree (π=85±29) for this burst was claimed by Yonetoku et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (2012b) us- ing the AstroSat/GZTI data 0-20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='34 seconds after the trigger (Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 3 METHODOLOGY 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='1 Spectral Analysis Techniques In order to diagnose the jet properties for a given burst, a de- tailed time-integrated or time-resolved spectral analysis is re- quired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The spectral analysis is performed by a pure Python pack- age, namely, the Multi-Mission Maximum Likelihood Framework (3ML,Vianello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Moreover, a Bayesian approach and Markov Chain Monte Carlo (MCMC) iterations to explore the best parameter space was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Our spectral analy- sis includes the following main steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' First is to select detec- tors, sources, and background intervals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Second, we used the Bayesian block method (BBlocks, Scargle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2013) to bin the Time-Tagged Events (TTE) lightcurve of the brightest detector (with the minimum viewing angle), and the significance (S, Vianello 2018) for each BBlocks time bin was also calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Third, we use a typical GRB spectral model (Band function, Band et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 1993) to fit all the spectra selected by the BBlocks method and the best model parameters are obtained by adopting a fully Bayesian approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' For a detailed Bayesian spectral analysis and the reduction procedure applied to a GRB spectrum, we refer to Li (2019a,b, 2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Li & Zhang (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='2 Using the Yonetoku correlation to infer their redshift values In order to study the intrinsic properties in the cosmological rest frame, a redshift measurement for each burst is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Unfortu- nately, we currently have 3 bursts (GRB 100826A, GRB 110301A, and GRB 160802A) without known redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Several works to in- fer redshifts using empirical relations of GRB have been reported in the literature(Amati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Yonetoku et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' For example, (Yonetoku et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2004) discovered a new and much tighter relation- ship between the spectral peak energy (Ep) and the peak luminos- ity (Lp) using the combined data detected by the BeppoSAX and BATSE satellites, and claimed that one can use the Ep-Lp relation to estimating redshift without knowing distances in the BSTASE cata- log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' We, therefore, attempt applying the Yonetoku relation(Yonetoku et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2004) to estimate redshift values for these bursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Our procedure to estimate the pseudo-redshift using the Yonetoku relation includes the following steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' First, we perform a spectral fit to the peak spectrum (the brightest time bin was selected by using the Bayesian blocks method(Scargle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2013) at the highest statistical significance S) of each burst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The peak flux Fγ and peak energy Ep on the observer’s frame from the spectral fits are thus obtained, where Fγ is the observed peak flux integrated between (1-104) keV in units or erg cm−2 s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Second, in order to use the Ep,z-Lp,iso relation, a bolometric lu- minosity in a common cosmological rest-frame energy band (1-104 keV) is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' It can be obtained by using the spectral parameters to conduct a k-correction extrapolating the observed energy band to 1- 104 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' For a given burst, the k-correction factor (kc) can be derived using the following procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The observed flux Fobs (erg cm−2 s−1), in a fixed detector energy bandwidth [e1, e2] (for instance, for the Fermi-GBM observation, e1=8 keV, e2=40 MeV), can be written as: Fobs [e1,e2] = � e2 e1 EN(E)dE, (1) MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (2023) Time-averaging Polarimetric and Spectral Properties of GRBs 5 where E is in units of keV, and N(E) is a GRB photon number spec- trum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The total luminosity emitted in the bandwidth [e1,e2], defined in the cosmological rest-frame, is given by: L[e1(1+z),e2(1+z)] = 4ˆπD2 L(z)Fobs [e1,e2], (2) which DL(z) is the luminosity distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' To express the luminosity L in the cosmological rest-frame energy band, [E1=1 keV, E2 =104 keV], common to all sources, the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (2) can be rewritten as: L[E1,E2] = 4ˆπD2 LFobs � E1 1+z , E2 1+z � = 4πD2 Lk[e1,e2,E1,E2,z]Fobs [e1,e2], (3) where the k-correction factor, kc, is therefore defined as: kc = k[e1,e2,E1,E2,z] = Fobs � E1 1+z ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' E2 1+z � Fobs [e1,e2] = � E2/(1+z) E1/(1+z) EN(E)dE � e2 e1 EN(E)dE , (4) Last, with the k-correction factors known, the peak luminosity can be derived from the observed γ-ray flux Fγ according to L = 4ˆπd2 LFγkc, where dL is the luminosity distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Finally, as long as the prompt emission spectral properties can be obtained, the Yonetoku relation can be used to infer redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' With the above Steps, one can use Equation (2) in Yonetoku et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (2004) to estimate their redshift values, Lp 1052ergs−1 = (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='34+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='29 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='76)×105 �Ep(1+z) 1keV �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (5) Table 2 lists the spectral properties obtained from the peak spec- tral analysis and the estimated redshift values inferred from the Yo- netoku relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 4 RESULTS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='1 Spectral Properties and their inferred jet properties Five bursts were claimed to have a high-significance polarization detection (Table 1) and GBM data (Table 4) taken at their prompt emission, thereby providing an “ideal” sample to study the possi- ble connection between jet properties and polarization straightfor- wardly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' In practice, either time-integrated or time-resolved spectral analy- sis is frequently used to diagnose their jet properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Several meth- ods have been widely used to diagnose the jet properties based on their spectral analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The simplest method is to use the low-energy spectrum (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=', Preece et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 1998) to resolve the jet properties for a given pulse/burst (Method-I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' In this method, a single empirical model (such as the Band model) based on a time-resolved tech- nique is typically used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The KED jets, therefore, can be defined as all α indices, within uncertainties, in a given burst, obtained from time-resolved spectral analysis, are systematically above the synchrotron limit throughout the entire burst/pulse duration, being consistent with a matter-dominated fireball jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The PFD jets, on the other hand, are consistent with the scenario that all α indices, within uncertainties, in a given burst are below the synchrotron limit throughout the burst/pulse duration, this suggests a Poynting-flux- dominated jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The HD jets represent the moderate scenario, and can be presented that some α indices, obtained from time-resolved spectral analysis, are above the synchrotron limit (α >2/3) while some others are below the synchrotron limit (α <2/3) throughout the burst/pulse duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The HD jets can be further divided into two subcategories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (1) the peak-KED pattern since the thermal emission component (the spectra that violate the LOD line) is only detected around the peak of the pulse (thermal component dominates the peak of a pulse/burst);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' and (2) the KED-to-PFD (thermal to non-thermal component) transition pattern (Li 2019a), since the thermal emission component is detected at the beginning of the burst, and followed by non-thermal emission component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' However, it may be difficult to classify jets as either KED or PFD jets based on the spectral index alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' In contrast to an optically-thin synchrotron emission, a photo- spheric quasi-thermal component would, in fact, have a harder low- energy spectral index, but this does not guarantee that the jet is KED (Gill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Therefore, a more reliable approach is to find the best model (Method-II) by comparing various frequently-used spec- tral models using certain statistical information criteria, such as the Akaike Information Criteria (AIC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Akaike 1974), Bayesian Infor- mation Criteria (BIC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Schwarz 1978), and the Deviance Information Criterion (DIC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Spiegelhalter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Moreno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Af- ter the identification of the best spectral model, one can assess the jet properties using the spectral properties inferred from the best model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Moreover, we may also directly fit the spectral data with a physical model (such as the synchrotron emission model, Burgess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Method-III), so as to diagnose the jet properties and any potential connection with their polarization properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Our analysis in this task incorporates both Method-I and Method-II (primary method).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Method III will be used elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' We first perform time-integrated spectral analysis (treating the entire epoch of polarization observation as one time bin) by using various GRB spectral models, including power-law (PL), blackbody (BB), cutoff power law (CPL), Band function, PL+BB, CPL+BB, and Band+BB, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' We adopted both AIC and BIC to eval- uate different spectral models and select the preferred one, and the preferred model is the one that provides the lowest AIC and BIC scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' As such, we define PFD jets: a single non-thermal (Band-like) spectral compo- nent was found in the time-integrated spectral analysis, like GRB 080916C (Abdo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' KED jets: a dominate thermal (BB-like) spectral compo- nent was found in the time-integrated spectral analysis, like GRB 090902B (Ryde et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2010) and GRB 220426A (Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' HD jets: a hybrid spectrum of thermal (BB-like) and non- thermal (Band-like) components was observed in the time-integrated spectral analysis in a single burst, like GRB 110721A (Axelsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2012), and GRB 140206A (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=', Li 2019a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Our refined time-integrated spectral analysis suggests that the Band+BB model can best characterize the spectral shape of all the five bursts (see Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The global properties of our sample used in the spectral analysis are reported in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' These include the GRB name (column 1), observed duration T90 of burst4 (column 2), 10- 1000 KeV fluence (column 3), together with the used detectors (col- umn 4), the selected source (column 5) and background (column 6) intervals, and the best model (column 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The best-fit spectral param- eters of the sample with the Band+BB model are reported in Table 5, including GRB name (column 1), source duration (column 2), and corresponding significance (column 3), and Band component nor- malization K (column 4), low-energy power-law index α (column 5), peak energy Ep (column 6) of the νFν spectrum, and high-energy power-law index β (column 7);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' and BB component normalization K (column 8), and temperature kT (column 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' We next perform time-resolved spectral analysis (treating the entire epoch of polarization observation as divided into multiple- 4 The time interval during which 90% of the total observed counts have been detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (2023) 6 Li & Shakeri time events) for each individual BBlocks time bin using the Band model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Temporal evolution of α is presented in Figures 2-6 for GRB 110826A, GRB 110301A, GRB110721A, GRB 140206A, and GRB 160802A, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' With the BBlock time bins selected and the corresponding significance (S) values calculated, the data points are shown in different S ranges5: S < 10, 10 ≤ S ≤ 20, and S > 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' For a subset of GRBs, a mixture of thermal (blackbody contribution from the photosphere emission) and non-thermal (synchrotron emis- sion from relativistic electrons) components was observed in a sin- gle burst (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=', Li 2019a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Previous studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=', Burgess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2014) showed that the characteristic energy (Ep) of non-thermal emission is correlated to the characteristic energy (kT) of thermal emission with a power-law relation in form of Ep ∝ T q, where q ranges from ∼1-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Burgess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (2014) studied a set of bright Fermi single-pulse GRBs and claimed that one can use this correlation to identify whether the jet is dominated by kinetic (q ∼ 1) or magnetic energy (q ∼ 2) de- pending on the value of the exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' GRB 110721A was identified as the baryonic jet type in Burgess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (2014) with q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='24±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='11, which is also consistent with our finding in the current analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Our time-integrated spectral analysis indicates that all five bursts belong to the HD jet type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' This finding is quite interesting since only a subset of GRBs (a fairly low percentage) have an observed ther- mal component in their spectral analysis, as suggested by several statistical studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=', Li 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Recently, (Li 2022a) has made a great effort to collect a complete GRB sample in which all bursts were detected by Fermi/GBM with known redshift, and created a spectral parameter catalog based on their model-wise properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' He discovered that ∼ 5% (7/153) of the analyzed bursts were found to require a subdominant thermal component in their time-integrated spectral analysis, including GRB 110721A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Our results imply that high-degree polarization measurements may also be associated with a thermal component originating from photosphere emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Our time-resolved spectral analysis, on the other hand, further suggests that two bursts exhibit the peak-KED pattern (GRB 110721A and GRB 160802A) while the other three bursts (GRB 110301A, GRB 140206A, and GRB 160625B) display the KED-to-PFD transition pattern across the entire burst durations since all α indices are above the synchrotron limit early on (α >2/3), and then drop be- low the synchrotron limit later on (α <2/3), indicating a thermal- to-nonthermal transition signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Interestingly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' after carrying out a detailed time-resolved spectral analysis for a sample of the multi- pulsed bursts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Li (2019a) reported that the jet properties for a good fraction of the multi-pulsed bursts exhibit a transition from thermal to non-thermal component among pulses within a single burst,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' and claimed that such “transition" jet properties are clearly observed in four bursts (GRB 140206B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' GRB 140329B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' GRB 150330A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' and GRB 160625B),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' and the polarization properties in those transition bursts would also differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='2 The Observed Parameter Correlations: Polarization Degrees versus Other Relevant Quantities The most interesting result that draws our attention is that all five bursts in our target sample have a relatively high-degree polarization measurement and are associated with the “HD” jet properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' In the following discussion, we, therefore, pay special attention to this interesting observation and its theoretical interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 5 Note that the results obtained for those lower-S spectra may not be robust, since the spectral fits would not be well determined due to lack of enough photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Much evidence points towards the fact that correlation analysis plays a crucial role in the understanding of GRB physics as it pro- vides a crucial clue to revealing its nature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=', Amati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Yonetoku et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Liang & Zhang 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Dainotti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Xu & Huang 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Liang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Dain- otti & Amati 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Li 2022a, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Here, an attempt has been made to explore the correlations between the po- larization properties and several typical GRB observed quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' For instance, the polarization degrees π correlated with (i) the cos- mological rest-frame peak energy (Ep,z) of the νFν prompt emission spectrum, (ii) the isotropic-bolometric-equivalent emission energy Eγ,iso, (iii) the magnetization parameter σ0, (iv) the blackbody tem- perature kT, (v) the redshift z, and (vi) the corresponding energy fluence Sγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Using the same observed epoch during the prompt emis- sion phase, our target sample allows for a reasonable comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Our analysis includes the following steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (1) For a given burst in our target sample, we first select the same time interval for the prompt emission data as for the polarization measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (2) We then at- tempt to perform a spectral fit to the selected prompt emission data using PL, BB, CPL, Band, PL+BB, CPL+BB, and Band+BB func- tions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The Band+BB model has an AIC/BIC-statistic improvement of at least 10 with respect to the Band-alone and other models for these bursts, which suggests Band+BB as the preferred model that would fit the data and a thermal component existing in the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Since the thermal flux ratio (FBB/Ftot) for these bursts is less than 50% (see Table 5), the thermal components thus are sub- dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Interestingly, Chattopadhyay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (2019) analyzed the prompt emission and polarization data of 11 bright bursts detected during the first year of operation of CZTI, and reported that of these bursts, four bursts (GRB 160106A, GRB 160509A, GRB 160802A, and GRB 160910A) in their spectral analysis showed a deviation from the Band model and an additional thermal blackbody is needed in order to model their spectrum more precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (3) With the spec- tral analysis done in Step (2), we are able to obtain the spectral peak energy (Ep,z) and the blackbody temperature (kT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Eγ,iso can also be calculated in the cosmological frame with a redshift known (GRB 110721A and GRB 140206B), and with a k-correction applied by integrating the observed energy spectrum over 1 KeV/(1+z) to 10 MeV/(1+z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' We note here that for the remaining three bursts (GRB 100826A, GRB 110301A, and GRB 160802A) with an unknown redshift, we use the Yonetoku relation (Yonetoku et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2004) to es- timate their redshift values (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Using these hybrid- spectrum observed properties and following the method described in Gao & Zhang (2015) and Li (2020), we also calculate the mag- netization parameter σ0 for these bursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Finally, with these Steps completed, we therefore present π-Ep,z (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='7a), π-Eγ,iso (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='7b), π-kTz (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='7c), π-σ0 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='7d), π-z (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='7e), and π-Sγ (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='7f) plots in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Interestingly, these scatter plots all seem to exhibit a monotonic power-law decay in their log-log space (except for the π-σ0 correlation), with a similar decay slope ranging from -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='40 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Our results indicate that a higher Ep,z, Eγ,iso, and kTz tend to have a lower-degree polarization π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' However, we should note that the sample size is too small to be reliable enough to support the de- rived conclusion, so the results may not be statistically significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 5 PHYSICAL IMPLICATION There are several parameters to impact on the degree of polariza- tion in GRBs including the geometry of the jet, its angular struc- ture, the bulk Lorentz factor of outflow material, the magnetic field configuration and observer’s point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Here, we consider an MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (2023) Time-averaging Polarimetric and Spectral Properties of GRBs 7 ultra-relativistic axi-symmetric jet which lunched by a central en- gine weather a black hole or an rapidly rotating magnetar (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=', Usov 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Thompson 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Dai & Lu 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Wheeler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Zhang & Mészáros 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Metzger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Lei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Metzger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Bucciantini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Lü & Zhang 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' During the prompt emission, we have ultra- relativistic jet with bulk lorentz factor Γ ≫ 1 leading to strong beam- ing effect of GRB outflow materials where the Doppler factor can be approximated as δD ≈ 2Γ 1+(Γ ˜θ)2 , (6) Due to the relativistic beaming effect of GRB jets, the measured ra- diation energy of the bursts is smaller than its isotropic energy as Eγ = fbEγ,iso by a factor fb = ∆Ω/4π = (1 − cosθ j) ≈ θ2 j/2 where θ j is the half opening angle of the ejecta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' In principle, different GRBs can be viewed from different observing angles θobs with re- spect to the jet’s central axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Only those observers whose line-of- sight (LOS) intersects the surface of the jet can detect the GRBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' In the ultra-relativistic regime, the observed emission mainly receives from a region that is limited to a cone with angular size ˜θ ≲ 1/Γ around LOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' At the early time prompt emission when the LOS in- tersects the jet surface, if θobs/θ j ≲ 1−(Γθ j)−1 and Γθ j ≳ O(10), the jet’s edge remains invisible to the observer Gill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' In this case, the emission region can be approximated as an ex- panding thin spherical shell of width ∆ ≪ R/Γ2 (in the lab frame) in which particles cool relatively fast compared to the dynamical time scale of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' As the GRB jet has slowed down signifi- cantly when the opening angle θ j ≃ Γ−1 then the jet break happens and the edge effects become important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The flux density measured by a distant observer from each fluid element in an infinite thin-shell approximation for the prompt emission is given by (Granot 2005) Fν(t) = (1+z) 16π2d2 L(z) � δ3 DL′ ν′(r)d ˜Ω, (7) where dL(z) is the luminosity distance of the source, and d ˜Ω = d ˜φd(cos ˜θ) is the solid angle with ˜θ and ˜φ as the polar angle and the azimuthal angle measured from the LOS, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The co- moving spectral luminosity L′ ν′(r) for the synchrotron emission is L′ ν′(r) = L′ ν′(R) � 1−(ˆn′ · ˆB′)2� 1+α 2 , (8) where α = −dlog(Fν)/dlog(ν) is the spectral index, ˆn is the ob- server’s LOS in the comoving frame of the GRB jet and ˆB is the local direction of the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The spectral luminosity L′ ν′(R) in the comoving frame of the fluid in terms of frequency ν′ and the peak frequency ν′ p at which the most of the power is emitted is given by L′ ν′(R) = L′ ν′p � ν′ ν′p �−α , (9) here we consider a constant luminosity with a radius which the peak value L′ ν′p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' We assume to have synchrotron emission from accelerat- ing electrons in the magnetig field with isotropic velocity distribu- tion and the energy distribution as a power law ne ∝ γ−p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' In our scenario, we assume that each pulse originates from a sin- gle thin shell and Γ can in principle change for different pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The state of polarization of a radiation field can be expressed in terms of the Stokes parameters I (intensity), Q and U (linear polarizations), V (circular polarization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' In spite of linear polization, only a few mech- anisms can generate the high value of circular polarization in the usual scattering processes in GRBs, and the measured circular polar- ization has only been reported once in a GRB afterglow Wiersema et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Therefore we will not consider circular polarization in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Stokes parameters Q and U are differences in flux for two orthogonal directions on the sky which are coordinate dependent quantities (Rybicki & Lightman 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Westfold 1959), we define the local degree of linear polarization Π = � Q2 +U2/I where U I = Πsin2θp , Q I = Πcos2θp , θp = 1 2 arctan �U Q � , (10) and θp is the polarization position angle (PA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The direction of the polarization vector in the synchrotron emission is orthogonal to the LOS of the observer ˆn and the local direction of the magnetic field ˆB in the jet, ˆΠ = (ˆn× ˆB) |ˆn× ˆB)| , (11) The polarization measurements can help in order to probe the magnetic field structure inside the shock wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Moreover, the de- gree of the polarization depends on the GRB jet’s angular structure and the observer’s viewing angle from jet symmetry axis (Lazzati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The magnetic field structure in KED and PFD flows has a different origin and can be classified into three categories (Gill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Gill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2021): (i) a locally ordered magnetic field (Bord) with angular coherent length θ j > θB ≳ 1/Γ, (ii) a toroidal magnetic field (Btrod) which has an ordered axisymmetric configura- tion in the transverse direction with respect to the jet (iii) a tangent magnetic field which could be in principle parallel (B∥) or perpen- dicular (B⊥) to the local fluid velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' In the PFD the magnetic field is dynamically dominated and usually has a large coherence length such as Btrod which can be produced by a rotating central engine or in a high magnetized flow, other locally and globally ordered field configuration are also possible in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' On the other hand in KED we may have a tangled magnetic field structure with B⊥ or/and B∥ components, however generating such an anisotropic field con- figurations in shock waves seems to be challenging (Gill & Granot 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' A globally ordered magnetic field may naturally be advected from near the central source, while the random magnetic fields gen- erated in the shock dissipation region (Kumar & Zhang 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Geng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Gill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The magnetic field structures that are generated at relativistic collision-less shocks, due to the two-stream instabilities, are expected to be tangled within the shock plane (Medvedev & Loeb 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The degree of linear polarization generated in the synchrotron emission from an isotropic electron distribution with power-law en- ergy spectrum, and for a given direction of magnetic field is given by (Rybicki & Lightman 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Westfold 1959) Πlin max = α+1 α+5/3 = peff +1 peff +7/3, (12) where peff = 2α + 1 is the effective power-law index of the electron distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' peff = � � � 2, νc < ν < νm, slow cooling p, νm < ν < νc, fast cooling p+1, ν > max(νc,νm), either fast or slow cooling (13) and, therefore Πlin max = � � � � � 9/13, νc < ν < νm, fast cooling (p+1) (p+7/3), νm < ν < νc, slow cooling (p+2) (p+10/3), ν > max(νc,νm),either fast or slow cooling (14) MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (2023) 8 Li & Shakeri This polarization may originate from a very small region (point- like emitter) in which the magnetic field has a specific orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Only in the case of ordered magnetic field with the coherence length comparable or larger than the visible surface of the emitting region, the highest value of the polarization Πlin max in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (19) can be gen- erated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The photon index in the Synchrotron radiation is limited to −1/3 ⩽ α ≲ 3/2 which regarding Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (19) leads to the maximum degree of polarization 50% ≲ Π ≲ 75%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' In the left panel of Fig- ure (8), we see predicted polarization values using this theoretical model with observed data using our target sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Here α indices are obtained using the spectral analysis defined in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' We find that the observed data are well distributed along the line predicted by this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' In general, the measured polarization is obtained by integrating the local stokes parameters over the flux of the GRB jet as Π(tf ) = ¯ Q(tf ) I(t f ) = � dFν cos2θp � dFν , (15) assuming to have an axisymmetric flow and taking into account sym- metry consideration, we see that ¯U = 0 and consequently the instan- taneous total degree of the linear polarization is ¯Π = | ¯Q|/I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' We per- form an integration over the equal time surface (EATS) for a single pulse � ¯ Q(t)dt/ � I(t)dt in order to obtain pulse integrated polariza- tion of the prompt emission which leads to Πord Πmax = � ymax 0 dy(1+y)−2−α � dφΛ(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='φ)cos2θp � ymax 0 dy(1+y)−2−α � dφΛ(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='φ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (16) The above formula is valid for the prompt emission from an ultra- relativistic thin-shell for an on-axis observer (θobs = 0) where ymax = (Γθmax)2 and θmax defined as the maximum angle from LOS Granot (2003a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The factor Λ(y,φ) is an average over the magnetic field orientations in the plane of the ejecta as Λ(y,φ) ≡ ⟨(1−(ˆn′ · ˆB′)2) 1+α 2 ⟩ , (17) The polarization angle θp and Λ(y,φ) take different forms regarding the configuration of the magnetic field in the plane the GRB jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' In the case of an ordered magnetic field Bord we have : Λord(y,φ) ≈ � (1−y 1+y)2 cos2 φ+sin2 φ � 1+α 2 , (18) θp = φ+arctan � (1−y 1+y)cotφ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (19) The time-integrated linear polarization in the presence of an ordered magnetic field in the plane normal to the jet velocity is plotted as a function of the spectral index in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' As it is seen the polarization degree increases towards higher values of α and lower values of ymax which can cover the observed polarization of GRB 110721A, GRB 160802A, and GRB 110301A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Therefore for a configuration with the globally ordered magnetic field, high values of linear polarization even larger than 50% are obtainable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The degree of polarization for a magnetic field with locally tan- gled or random configuration is obtained by averaging over all direc- tions of the local magnetic field within the plane of the shock (Granot & Königl 2003a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Sari 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Gruzinov 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Nava et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The presence of a random magnetic field leads to negligible values of net linear polarization measured by an on-axis observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' In the case of a random field behind the shock wave only if the observer is off-axis and the circular symmetry is broken, non-zero net polarization is measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The total linear polarization arising from the whole jet which is subjected to a random field with a direction perpendicular to the jet velocity is given by Granot (2003a) Π⊥ Πmax = � y2 y1 dy(1+y)−2−α sin[2Ψ1(y)]G(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='α) Θ(1−ζ) � y1 0 dy H(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='α) (1+y)α+2 + � y2 y1 dy dy H(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='α) (1+y)α+2 � π−Ψ(y) π �,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (20) where Θ(1 − ζ) is the Heaviside step-function with ζ ≡ θobs/θ j as a parameter to define observer’s point of view,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' and G(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='α) = 1 2π � π 0 dφ �(1−y)2 (1+y)2 cos2 φ−sin2 φ �� 1− 4ycos2 φ (1+y)2 � α−1 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (21) H(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='α) = � π 0 dφ � 1− 4ycos2 φ (1+y)2 � 1+α 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (22) cosΨ(y) = (1−ζ2)y j −y 2ζ√yy j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (23) In above expressions y1,2 = (1∓ζ)2yj and y j = (Γθ j)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The variation of the linear polarization in the presence of a random field configura- tion measured by an off-axis observer is displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' In the left panel, the spectral indices are selected to be consistent with aver- age values reported in Table (5) for our target sample and for y j = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' In the right panel, yj is changed while α = 1, it is found that the ap- peared peak has a width in order of 1/√y j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (9), we see that the polarization degree is limited to small values for ζ < 1 while it is sharply increased for ζ ≈ 1 and finally reaches to an asymptotic limit at ζ > O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' It is seen that the Synchrotron radiation with B⊥ can potentially generate wide range of polarization values from low levels to moderate values which cover observed values associated to our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' In principle, various viewing angles θobs and different angular structures of the jet affect the measured fluence of GRBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Note that the fluence significantly decreases for a top-hat jet view- ing from outside the jet’s sharp edge, so high levels of polarization in off-axis jets may only be obtainable in very close bursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' In fact, the detectibility of GRB polarization needs high-fluence sources and usually, the fluence rapidly drops below the detector threshold for a large off-axis observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The time-resolved spectral analysis in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='1 showed thermal to non-thermal (KED-to-PFD) transitions in our sample where a sub- dominant component of the thermal emission during bursts is ob- served.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Observing hard values of the spectral indices during the bursts can be served as hints that LOS is not highly off-axis, since high latitude emission leads to a softer spectrum Lundman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' As it was reported in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='2, a correlation between the polariza- tion and the isotropic energy π-Eγ,iso (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='7b) has been observed within our sample, it is worth mentioning that higher polariza- tion values are recorded for closer bursts GRB 110301A (z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='36), GRB 110721A (z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='382), GRB 160802A (z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='90) and lower val- ues for farther sources GRB 140206B (z=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='73) and GRB 100826A (z=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='3) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='7e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The observed fluences of GRB 140206B and GRB 100826A is higher than other sources (see Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 4 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='7f) and due to their higher redshifts ζ can not obtain large values, however, low values of ζ would be enough to reproduce their measured polar- izations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The local degree of linear polarization for a tangled or random field configuration for a thin ultrarelativistic shell modeling of the prompt emission by assuming α = 1 is obtained by averaging over all local magnetic field directions as Πlin rnd = Πlin max (b−1)sin2 θB 2+(b−1)sin2 θB (24) MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (2023) Time-averaging Polarimetric and Spectral Properties of GRBs 9 where b ≡ 2⟨B2 ∥⟩/⟨B2 ⊥⟩ denotes the anisotropy of the magnetic field distribution as the ratio of the parallel B∥ to the perpendicular B⊥ components with respects to the shock direction, and θB is the an- gle between the LOS from the observer and the direction of the shock Sari (1999);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Gruzinov (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' In the case of a globally ordered magnetic field configuration aligned with the jet direction (B → B∥, b → ∞), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (24) returns back to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (19) and gives the maximum value of the linear polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The polarized emission may also originate from independent magnetic patches with various field orientation Li (2022a) where magnetic patches are locally coherent but distributed randomly in observed emission region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' In this case the measured polarization from different patches is estimated as Π = Πmax/ √ N, where N is the number of magnetic patches or equivalently multiple pulses where the coherence length of the magnetic field is as large as the emis- sion region in a single pulse and observed polarization is an average over multiple pulses (Gruzinov & Waxman 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Granot & Königl 2003b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The magnetic field which is generated within IS for KED jets has usually a coherence length much smaller than the angular size of the emission region which causes negligible net polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' It has been shown that even by taking into account the angular structure of the flow the polarization is limited to Π ≲ 20% for photospheric emis- sion of a relativistically expanding fireball Ito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Lund- man et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (2014a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Parsotan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The observed high val- ues of the polarization for GRB 110721A and GRB 160802A while they show the peak-KED pattern cannot be explained simply by the sub-photospheric dissipation model based on Comptonisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Be- cause the multiple scatterings at large optical depths region leads to wash out the directionality of polarization vectors (Lundman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2018b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' To explain the strong polarized signals, models invoking dis- sipation of ordered magnetic field are favored (Lyutikov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Zhang & Yan 2011b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' McKinney & Uzdensky 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' A structured jet photosphere model may also generate polarized photons via Comp- ton scattering but with a different energy-dependence compare to the synchrotron model in the ordered magnetic field (Chang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2014b,a, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The Jitter radiation emitted by ultra-relativistic electrons acceler- ating in a small-scale random magnetic field (Medvedev 2000), can also generate a hard energy spectrum with the photon index as high as α = +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Due to the random distribution of the magnetic field, jitter radiation is highly symmetric in the electron radiative plane, leading to the vanishing polarization degree for an on-axis observer (Mao & Wang 2013, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Mao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The maximum level of polarization is obtainable when the emitting plane is viewed from the edge on, it can even reach up to 90% (Prosekin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' However, for smaller off-axis viewing angles which can yield mea- surable fluences, jitter radiation causes almost negligible polariza- tion degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Meanwhile, regardless of the viewing angle the Jitter radiation cannot produce the observed high degree of polarisation close to the spectral peak energy of the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' To summarize, polarization features can be explained either by the synchrotron radiation in the ordered/random magnetic field (Granot 2003b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Granot & Königl 2003b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Nakar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2003), the jet structure (Lazzati & Begelman 2009), or the observer’s viewing angle with re- spect to the jet (Lazzati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2004), even in the case of thermal radi- ation from the jet photosphere (Lundman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2014b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' For a hybrid spectrum which include thermal and non-thermal components, we expect to see relatively high values of the polarization in the prompt emission which can be produced by synchrotron emission mecha- nism in the ordered magnetic field of the jet, and for random field configurations only for off-axis observers (Gill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' How- ever, the spectral properties of our target sample demonstrated that off-axis observations specially for the large viewing angle is not the case, and the observed values of the polarization most probably is a hint of the ordered magnetic field originating from the central en- gine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Since from PFD jets towards HD and KED jets, polarization washout effects are increased gradually due to thermal photons, we would expect that the inequality πKED ≲ πHD ≲ πPED is satisfied if other conditions are fixed for a given jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Due to the different de- grees of polarization predicted by different emission models in var- ious energy bands, it is essential to have a high-sensitivity gamma- ray polarimeter with a wide band-pass to detect energy-dependent polarization signals and constrain different models (Zhang 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' However, it should be noted that due to several free parameters in polarization models, upcoming more precise observations and theo- retical investigations are needed to discriminate between competing models in order to explain observed joint polarization and spectral properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 6 CONCLUSION Early polarization observations during the prompt emission phase play a crucial role in understanding the radiation mechanism and jet composition of GRBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Observations over the past few decades suggest that the jet composition of GRBs may have diverse prop- erties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' If the jet composition is matter-dominated (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=', a fireball), the GRB prompt emission spectra would include a bright thermal component originating from the fireball photosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Alternatively, if the jet composition is Poynting-flux-dominated, the GRB prompt emission spectra would include a dominant non-thermal compo- nent originating from the synchrotron radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Moreover, if the jet composition is hybrid-dominated, the GRB prompt emission spec- tra would include a thermal component originating from the fire- ball photosphere and a non-thermal component originating from the synchrotron radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' It is highly speculated that the prompt emis- sion is likely expected to be strongly polarized owing to its non- thermal origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Consequently, a different level of polarization de- grees (πKED ≲ πHD ≲ πPED) during the prompt emission phase is nat- urally expected due to the different types of jet composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' In this paper, we have collected a GRB sample in which all the bursts de- tected by Fermi/GBM and whose polarization detection in the emis- sion region was also reported in the literature, containing five inter- esting bursts (GRB 100826A, GRB 110301A, GRB 110721A, GRB 140206A, and GRB 160802A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Using the time-averaging polariza- tion observations and selecting the same epoch for the GBM data taken during the prompt emission phase, we then attempted to ex- plore the correlations between jet properties and polarization prop- erties of GRBs and aimed to confirm the validity of this correlation (πKED ≲ πHD ≲ πPED) from observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' We first performed a detailed time-averaged spectral analysis for each burst in our target sample by using several frequency-used GRB spectral models and selected the best one by using information crite- ria (AIC and BIC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The jet properties of GRBs can be classified into three categories based on their spectral analysis: the “KED”, “PFD”, and “HD” types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Using the spectral properties we then inferred their jet properties and discovered that all five bursts belong to the “HD”- jet type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The lack of the other two types of jets (KED and PED) prevents us from validating this correlation (πKED ≲ πHD ≲ πPED).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Hopefully, upcoming instruments will provide high-sensitivity po- larization observations in the future, leading to well-sampled, well- studied data sets, enabling such statistical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' We next conducted a time-resolved spectral analysis for each in- MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (2023) 10 Li & Shakeri dividual burst by dividing the emission period into multiple-time slices using the BBlocks method using the Band-alone model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Our refined time-resolved spectral analysis, on the other hand, further suggested that the “HD”-type has two subcategories: the peak-KED pattern and the KED-to-PFD transition pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' In our attempt to as- sess the jet properties of GRBs using Band-α evolution, we discov- ered that two bursts exhibit the peak-KED pattern (GRB 110721A and GRB 160802A) whereas the other three bursts show the KED- to-PFD transition pattern (GRB 110301A, GRB 140206A, and GRB 160625B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' All five bursts found in the “HD”-type imply that the pho- tosphere emission may also be a possible mechanism to power the high-degree polarization observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' We also made an attempt to explore the correlations between the polarization properties and several typical GRB observed quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Using the same observed epoch during the prompt emission phase, our target sample allows for a reasonable comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The corre- lations we attempted to study included the polarization degrees π correlated with (i) the cosmological rest-frame peak energy (Ep,z) of the νFν prompt emission spectrum, (ii) the isotropic-bolometric- equivalent emission energy Eγ,iso, (iii) the magnetization parameter σ0, (iv) the blackbody temperature kT, (v) the redshift z, and (vi) the corresponding energy fluence Sγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' As a result, we discovered that a higher Ep,z, Eγ,iso, and kTz tend to have a lower-degree polarization π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Lastly, we discovered that all five bursts in our target sample have a relatively high-degree polarization detection that seems to corre- late with the “HD”-jet type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' If it is an intrinsic characteristic of GRBs, this could provide a clue to studying the radiation mechanism and jet composition of GRBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' We have also discussed some physical interpretations of this interesting phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Since the configura- tion of the magnetic field inside the jet is one of the crucial param- eters to determine the polarization degree, we discussed two main configurations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' ordered and random fields), and their connec- tion to the jet composition is clarified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' We considered polarization patterns as a function of different dynamical parameters associated to the outflow materials, the spectral indices and the observer’s LOS with respect to the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Combining the spectral analysis and the polar- ization measurements allowed us to find out the detection of polar- ization values Π > 50% during prompt emission of GRB 160802A, GRB 110721A and GRB 110301A is a piece of strong evidence for the synchrotron emission mechanism in the presence of an ordered magnetic field which can be advected from the GRB central engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Regarding the different properties of our target sample, we conclude that geometrical effects and large off-axis observations are unlikely responsible for the measured polarizations assuming random mag- netic fields within the jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Finally, there are some caveats that are worth mentioning when applying our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (i) Spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' We have resolved the jet proper- ties based on the low-energy spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' However, it may be difficult to classify jets as either KED or PFD jets based on the spectral index alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Indeed, a photospheric quasi-thermal component would have a harder low-energy spectral index as compared to an optically-thin synchrotron, but that does not guarantee that the jet is KED (an ex- ample, see Gill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (ii) Polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The degree of polariza- tion ultimately probes the (local) structure of the B-field in the emis- sion region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' An ordered field would necessarily yield high polariza- tion whereas a tangled field would yield a very small polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' It is unclear, however, whether these field configurations are exclusive to a given jet configuration (or a particular level of magnetization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' In addition, the angular structure of the jet also plays an important role in governing the observed polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Thus, due to the large range of model parameters, it is difficult to attribute a given level of polarization to a given jet composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' More discussion is provided in a recent review article (Gill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (iii) Different instrument analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Currently, it is not clear why different instruments, namely, POLAR, IKAROS-GAP, and ASTROSAT/CZTI, are finding differ- ent levels of polarization for a small sample of GRBs (Chattopad- hyay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' There is no consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' POLAR is finding a rather low-level polarization, which is consistent with zero within 3σ of their quoted central values, whereas both IKAROS and AstroSAT are finding much higher levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Hard X-ray to soft gamma-ray po- larization measurements are very tricky and the analysis has to be carried out very carefully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' As such, some of these measurements are probably not representative of GRBs and need to be further verified by future more precise instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (iv) Time-resolved polarization analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' In the current analysis, none of the cases have shown time- resolved polarization measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Even though the GRBs in our target sample have time-resolved spectral indices, not having corre- sponding polarization measurements makes it difficult to ascertain the properties of the B-field and outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' We thank Ramandeep Gill, Jonathan Gra- not, Rahim Moradi, Mi-Xiang Lan, Asaf Pe’er, Jin-Jun Geng, Christoffer Lundman, Remo Ruffini, and ICRANet members for many discussions on GRBs physics and phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' This research made use of the High Energy Astrophysics Science Archive Re- search Center (HEASARC) Online Service at the NASA/Goddard Space Flight Center (GSFC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Data availability.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2003b, ApJ , 596, L17 —.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2005, ApJ , 631, 1022 Granot, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=', & Königl, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2003a, APJ, 594, L83 —.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2003b, ApJ , 594, L83 Gruzinov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 1999, ApJ , 525, L29 Gruzinov, A.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=', Nagataki, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=', Matsumoto, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2014, ApJ , 789, 159 Ito, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=', Nagataki, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=', Matsumoto, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=', Lei, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2018, ApJS , 236, 26 Li, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=', Xue, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='-S.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=', Ryde, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2022b, arXiv e-prints, arXiv:2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='02141 Liang, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=', & Zhang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2005, ApJ , 633, 611 Liang, E.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2005, A&A , 439, 245 Xu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=', & Huang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2012, A&A , 538, A134 Yonetoku, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2011a, ApJ , 743, L30 —.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2011b, ApJ , 743, L30 —.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2012a, ApJ , 758, L1 —.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2012b, ApJ , 758, L1 Zhang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2014, International Journal of Modern Physics D, 23, 1430002 —.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2018, The Physics of Gamma-Ray Bursts, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='1017/9781139226530 Zhang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=', & Mészáros, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2001, ApJ , 552, L35 Zhang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=', & Yan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2011a, ApJ , 726, 90 —.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2011b, ApJ , 726, 90 Zhang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=', Shao, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=', Yan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=', & Wei, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2012, ApJ , 750, 88 Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='-N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=', Kole, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=', Bao, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' 2019, Nature Astronomy, arXiv:1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='04207 MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (2023) 12 Li & Shakeri Table 1 A sample of GRB polarimetric observations GRB PD PA Energy band Time Significance Instrument Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' z (Fermi ID) (π%) (◦) (t-t0) (σ) (For polarization) 100826A(957) 27±11 159±18, 75±20 γ-ray 0-100s 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='9σ IKAROS-GAP Yonetoku et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (2011b) NA 110301A(214) 70±22 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (2014) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='73 160802A(259) 85±29 ∼-32 hard X-rays 0-20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='34s ∼3σ AstroSat-CZTI Chand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (2018) NA MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (2023) Time-averaging Polarimetric and Spectral Properties of GRBs 13 Table 2 Estimated values of redshift using the Yonetoku relation GRB tstart∼tstop S Eobs p Fobs p kc Lp z (s) (keV) (erg cm−2 s−1) (erg s−1) (Estimated) 100826957 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='208∼22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='288 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='97 459±20 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='42±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='10)×10−5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='85 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='27±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='10)×1053 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='3 110301214 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='876∼4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='126 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='60 126±6 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='36±0.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='79 385±39 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='04±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='80)×10−5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='95 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='05±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='80)×1053 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='90 MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (2023) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='Li & Shakeri ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='Table 3 Comparison of AIC/BIC between the best model and other models ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='GRB ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='04+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='03 110721200 0∼11 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='5 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='01+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='09 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='09)×10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='20+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='02 1620+234 −229 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='19+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='10 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='73+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='33 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='34)×10−5 33+2 −2 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='94+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='66 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='63)×10−6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='03+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='01 140206275 4∼26 170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='87+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='11 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='11)×10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='06+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='02 679+43 −43 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='32+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='08 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='78+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='53 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='52)×10−5 27+1 −1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='57+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='24)×10−6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='05+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='01 160802259 0∼20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='34 151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='7 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='89+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='21 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='21)×10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='03 515+44 −44 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='23+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='73 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='74 (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='11+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='22 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='21)×10−5 25+1 −1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='85+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='55 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='41)×10−6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='09+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='02 MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (2023) Time-averaging Polarimetric and Spectral Properties of GRBs 17 1 0 2 4 6 8 10 redshift 10 1 100 101 102 103 GRB 100826A Estimated value 0 2 4 6 8 10 redshift 10 1 100 101 102 103 GRB 110301A Estimated value 0 2 4 6 8 10 redshift 10 1 100 101 102 103 GRB 160802A Estimated value Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Estimated redshift using the Yonetoku relation for three bursts (GRB 100826A, GRB 110301A, and GRB 160802A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The yellow and cyan lines represent the left and right function of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (5), and their intersection point (purple color) is the estimated value of redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (2023) 18 Li & Shakeri 1 20 0 20 40 60 80 100 120 140 Time (s) 0 20 40 60 80 100 =27±11% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='5 (S > 20) (10 < S < 20) (S < 10) 100826A 101 102 103 104 105 Photon Energy - keV 100 101 102 103 104 keV × [keV 1S 1cm 2] Band fits Blackbody Band+Blackbody NAI7 NAI8 BGO1 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Left panel: prompt emission GBM light curve (overlaid in gray) and polarization observations in γ-ray/hard X-ray energy bands (cyan shaded area), as well as the temporal evolution of α based on the time-resolved spectral analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The horizontal dashed line represents the limiting value of α = −2/3 for electrons in the slow-cooling regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Right panel: the spectral data and its best-fit model (Band+BB) during the time epoch (see Table 1 and Table 4) of the matching polarization observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (2023) Time-averaging Polarimetric and Spectral Properties of GRBs 19 1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='0 Time (s) 0 20 40 60 80 100 =70±22% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='5 (S > 20) (10 < S < 20) (S < 10) 110301A 101 102 103 104 105 Photon Energy - keV 100 101 102 103 104 keV × [keV 1S 1cm 2] Band fits Blackbody Band+Blackbody NAI7 NAI8 NAIb BGO1 nai bgo Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Same as Figure 2 but for GRB 110301A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (2023) 20 Li & Shakeri 1 15 10 5 0 5 10 15 20 25 30 Time (s) 0 20 40 60 80 100 = (84+16 28)% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='5 (S > 20) (10 < S < 20) (S < 10) 110721A 101 102 103 104 105 Photon Energy - keV 101 102 103 104 keV × [keV 1S 1cm 2] Band fits Blackbody Band+Blackbody NAI6 NAI7 NAI9 BGO1 nai bgo Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Same as Figure 2 but for GRB 110721A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (2023) Time-averaging Polarimetric and Spectral Properties of GRBs 21 1 20 0 20 40 60 80 100 120 140 Time (s) 0 20 40 60 80 100 (Upper limit) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='5 (S > 20) (10 < S < 20) (S < 10) 140206A 101 102 103 104 105 Photon Energy - keV 101 102 103 104 keV × [keV 1S 1cm 2] Band fits Blackbody Band+Blackbody NAI0 NAI1 NAI3 BGO0 nai bgo Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Same as Figure 2 but for GRB 140206A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (2023) 22 Li & Shakeri 1 20 10 0 10 20 30 40 Time (s) 0 20 40 60 80 100 = 85 ± 29 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='5 (S > 20) (10 < S < 20) (S < 10) 160802A 101 102 103 104 105 Photon Energy - keV 101 102 103 104 keV × [keV 1S 1cm 2] Band fits Blackbody Band+Blackbody NAI2 BGO0 nai bgo Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Same as Figure 2 but for GRB 160802A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (2023) Time-averaging Polarimetric and Spectral Properties of GRBs 23 1 101 102 103 104 Ep, z[keV] 100 101 102 103 a 100826A 110301A 110721A 140206B 160802A 100 101 102 103 Power-law index= -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='41±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='24 1051 1052 1053 1054 1055 E , iso (erg) 100 101 102 103 b 100826A 110301A 110721A 140206B 160802A 100 101 102 103 Power-law index= -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='20±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='04 100 101 102 103 104 KT,z[keV] 100 101 102 103 c 100826A 110301A 110721A 140206B 160802A 100 101 102 103 Power-law index= -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='32±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='18 100 101 102 1+ 0 100 101 102 103 d 100826A 110301A 110721A 140206B 160802A 100 101 102 103 0 1 2 3 4 5 6 7 8 9 redshift 100 101 102 103 e 100826A 110301A 110721A 140206A 160802A 10 6 10 5 10 4 10 3 10 2 Fluence (erg cm 2) 100 101 102 103 f 100826A 110301A 110721A 140206A 160802A Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Scatter plots of polarization degree π versus several other observed quantities: (a) the cosmological rest-frame peak energy (Ep,z) of the νFν prompt emission spectrum, (b) the isotropic-bolometric-equivalent emission energy Eγ,iso, (c) the magnetization parameter σ0, (d) the blackbody temperature kT, (e) the redshift z, and (d) the corresponding energy fluence Sγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Data points with different colors indicate the different bursts in our target sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The solid lines (grey) are the best fit using the power-law model with 2σ (95% confidence interval) error region (shadow area).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (2023) 24 Li & Shakeri 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='0 100 101 102 100826A 110301A 110721A 160802A 140206A 100 101 102 lin max = + 1 + 5/3 ymax=1 ymax=2 ymax=5 ymax=10 ymax=102 110721A 160802A 110301A 100826A 140206A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='0 α Π/Πmax Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Left: The maximum degree of the linear polarization applying synchrotron emission model (Πlin max Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (19)) with observed data using α indices based on a time-integrated spectral analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' Right: Time integrated polarization degree in the presence of an ordered magnetic field Bord with in the plane of ejecta (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (19)) measured by an on-axis observer (θobs = 0), the evolution of the polarization is plotted in terms of α for different values of ymax = (Γθmax)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (2023) Time-averaging Polarimetric and Spectral Properties of GRBs 25 1 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='72 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='84 α=1 α=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='2 Random B⟂ yj = 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='5 ζ=θobs/θj Π/Πmax yj=103 yj=102 yj=10 yj=1 α = 1 Random B⟂ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='6 ζ=θobs/θj Π/Πmax Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' The time integrated polarization for a random magnetic field B⊥ which lies entirely in the plane of the shock (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (20)) as a function of the off-axis parameter ζ = θobs/θ j for different values of spectral index α (left) and yj = (Γθ j)2 (right) as labeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} +page_content=' (2023)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQfsfkS/content/2301.00576v1.pdf'} diff --git a/LtFOT4oBgHgl3EQf0TSa/content/tmp_files/2301.12935v1.pdf.txt b/LtFOT4oBgHgl3EQf0TSa/content/tmp_files/2301.12935v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9d34fc5bbbbe7542bd0ef06866e723df16a6f090 --- /dev/null +++ b/LtFOT4oBgHgl3EQf0TSa/content/tmp_files/2301.12935v1.pdf.txt @@ -0,0 +1,1569 @@ +ERA-Solver: Error-Robust Adams Solver +for Fast Sampling of Diffusion Probabilistic Models +Shengmeng Li 1 Luping Liu 2 Zenghao Chai 3 Runnan Li 1 Xu Tan 3 +Abstract +Though denoising diffusion probabilistic models +(DDPMs) have achieved remarkable generation +results, the low sampling efficiency of DDPMs +still limits further applications. Since DDPMs +can be formulated as diffusion ordinary differ- +ential equations (ODEs), various fast sampling +methods can be derived from solving diffusion +ODEs. However, we notice that previous sam- +pling methods with fixed analytical form are not +robust with the error in the noise estimated from +pretrained diffusion models. In this work, we +construct an error-robust Adams solver (ERA- +Solver), which utilizes the implicit Adams nu- +merical method that consists of a predictor and a +corrector. Different from the traditional predictor +based on explicit Adams methods, we leverage a +Lagrange interpolation function as the predictor, +which is further enhanced with an error-robust +strategy to adaptively select the Lagrange bases +with lower error in the estimated noise. Exper- +iments on Cifar10, LSUN-Church, and LSUN- +Bedroom datasets demonstrate that our proposed +ERA-Solver achieves 5.14, 9.42, and 9.69 Fenchel +Inception Distance (FID) for image generation, +with only 10 network evaluations. +1. Introduction +In recent years, denoising diffusion probabilistic models +(DDPMs) (Ho et al., 2020) have been proven to have poten- +tial in data generation tasks such as text-to-image genera- +tion(Poole et al., 2022; Gu et al., 2022; Kim & Ye, 2021; +Chen et al., 2022), speech synthesis(Huang et al., 2021; Lam +et al., 2022; Leng et al., 2022), and molecular conformation +formation(Hoogeboom et al., 2022; Jing et al., 2022; Wu +et al., 2022; Huang et al., 2022). They build a diffusion +process to add noise into the sample and a denoising process +to remove noise from the sample gradually. Compared with +1Microsoft Cloud+AI 2Zhejiang University 3Microsoft Re- +search Asia. Correspondence to: Xu Tan . +Implicit Adams +DPM-Solver +ERA-Solver (Ours) +Figure 1. We adopt the pretrained diffusion model from (Song +et al., 2020a) and visualize the error between the estimated noise +and ground-truth noise in training. We also provide the gener- +ated samples from the pretrained model based on implicit Adams +(traditional predictor-corrector method (Diethelm et al., 2002)), +DPM-Solver(Lu et al., 2022a), and our error-robust Adams solver +(ERA-Solver). Our solver is robust for the error from pretrained +model so as to generate samples with better quality. +generative adversarial networks (GANs)(Goodfellow et al., +2014) and variational auto-encoders (VAEs)(Child, 2021), +DDPMs have achieved remarkable generation quality. How- +ever, due to the properties of the Markov chain, the sampling +process requires hundreds or even thousands of denoising +steps. Such defects limit the wide applications of diffusion +models. Thus, it is an urgent request for a fast sampling of +DDPMs. +There have already existed many works for accelerating +sampling speed. Some works introduced an extra training +stage, such as knowledge distillation method (Salimans & +Ho, 2021), training sampler (Watson et al., 2021), or directly +combining with GANs (Wang et al., 2022), to obtain a fast +sampler. These methods require a cumbersome training +process for each task, and are black-box samplers due to +the lack of theoretical explanations. Denoising diffusion +implicit model (DDIM) (Song et al., 2020a) and Score- +SDE(Song et al., 2020b) revealed that the sampling can +be reformulated as a diffusion ordinary differential equa- +tion (ODE) solving process, which inspired many works to +design learning-free fast samplers based on numerical meth- +ods. PNDM (Liu et al., 2021) introduced the analytic form +arXiv:2301.12935v1 [cs.LG] 30 Jan 2023 + +70 +2 +60 +50 +40 +)03 +30 +20 +3 +10 +0 +0.9 +0.8 +0.7 +0.6 +0.5 +0.4 +0.3 +0.2 +0.1 +time tError-Robust Adams Solver +of a 4-order linear multistep method, which is also called ex- +plicit Adams method (Małgorzata & Marciniak, 2002), and +utilized the estimated noises observed at previous sampling +steps to sample efficiently, with a warming initialization +based on Runge-Kutta methods (Butcher, 1996). DPM- +Solver(Lu et al., 2022a) introduced exponential integrator +from ODE literature (Atkinson et al., 2011) and required ex- +tra network evaluations to observe intermediate noise terms +for approximating Taylor expansion (Mohazzabi & Becker, +2017) in the sampling process. +The existing learning-free methods (Liu et al., 2021; Lu +et al., 2022a; Song et al., 2020a) are based on the assump- +tion that the learned network has high accuracy in noise +estimations across all the sampling steps. However, we no- +tice that the noise estimated from the neural network is not +accurate enough and the error exists at almost every time +t, especially when time t approaches 0, as shown in Fig. +1. Existing sampling methods are not able to be robust for +noise estimation errors, since they consist of an analytic +sampling scheme with fixed coefficients to ensure sampling +convergence. For example, explicit Adams (Małgorzata & +Marciniak, 2002) consists of the analytic form with fixed +coefficients as formulated in Eq. 9. +In this paper, we aim at designing an error-robust diffusion +ODE solver to speed up the sampling process of DDPMs +while achieving good sampling quality. To this end, we +focus on implicit Adams solver (Małgorzata & Marciniak, +2006), a kind of numerical ODE solver, which involves +unobserved terms to achieve high-order precision and con- +vergence. In existing ODE literature(Atkinson et al., 2011), +predictor-corrector has been introduced to perform implicit +Adams solver, which avoids solving the implicit equation. +Explicit Adams usually acts as the predictor to predict the +unobserved term. However, traditional predictor-corrector +still suffers from the inaccurate estimation of diffusion noise +at each sampling step since it is composed of fixed coeffi- +cients, which is shown in Fig. 1. Instead of utilizing explicit +Adams as the predictor, we adopt the Lagrange interpola- +tion function (Sauer & Xu, 1995) that interpolates several +Lagrange function bases as the predictor. We maintain a +buffer of estimated noises observed at previous sampling +steps during sampling and adaptively select those estimated +noises with low estimation error as the Lagrange function +bases, to ensure accurate interpolation result and thus an ac- +curate predictor. In this way, we can obtain a diffusion ODE +sampler with not only high convergence (thanks to the im- +plicit Adams method(Małgorzata & Marciniak, 2006)) but +also good error robustness (thanks to the adaptive strategy +for the Lagrange interpolation function). +However, it is not easy to select noises with low estima- +tion error as the Lagrange function bases. That is because, +unlike the training stage, there exist no reference noises +at the sampling stage to judge how accurate the estimated +noise is. Thus, we further propose an approach to roughly +measure the accuracy of the estimated noise by calculating +the difference between the noise obtained by the predictor +(as the prediction) and the noise observed at the previous +sampling step (as the reference). Based on this measure- +ment, we propose a selection strategy for the buffer which +adaptively chooses estimated noises that are more accurate +to construct the Lagrange function bases, so as to result in a +more accurate predictor, and thus a better ODE solver. +Our contributions can be summarized as follows: +• We are the first to explore the potential of numerical im- +plicit Adams methods (Małgorzata & Marciniak, 2006) +to solve diffusion ODEs. We propose an error-robust im- +plicit Adams Solver (ERA-Solver), which is training-free +and can be extended to pretrained DDPMs conveniently. +• We propose to adopt the Lagrange interpolation function +(Sauer & Xu, 1995) that selectively interpolates several +Lagrange function bases with low errors of estimated +noises to ensure the ERA-solver is robust to the error in +estimated noise. +• Experiment results show that we achieve better results +on LSUN-Bedroom, LSUN-Church, and Cifar10 datasets +when compared with previous methods in 10 function +evaluations. On LSUN-Church and LSUN-Bedroom, we +achieve state-of-the-art FID results of 7.39 and 5.39 with +more function evaluations, compared with the previous +best results of 7.74 and 6.46. +2. Preliminary +2.1. Denoising Diffusion Probabilistic Models +Denoising diffusion probabilistic models (DDPMs)(Ho +et al., 2020) have demonstrated their great generation poten- +tial on various applications, such as text-to-image synthesis +(Poole et al., 2022; Gu et al., 2022; Kim & Ye, 2021), image +inpainting (Lugmayr et al., 2022; Liu et al., 2022; Kawar +et al., 2022), speech synthesis (Huang et al., 2021; Lam +et al., 2022; Leng et al., 2022), and molecular conformation +generation (Hoogeboom et al., 2022; Jing et al., 2022; Wu +et al., 2022; Huang et al., 2022). It involves a diffusion +process to gradually add noise to data, and a parameterized +denoising process to reverse the diffusion process, sampling +through gradually removing the noise from random noise. +The diffusion process is modeled as a transition distribution: +q(xt|xt−1) := N(xt; √αtxt−1, (1 − αt)I), +(1) +where α1, ..., αT are fixed parameters. With the transition +distribution above, noisy distribution conditioned on clean +data x0 can be formulated as follows: +q(xt|x0) = N(xt; √αtx0, (1 − αt)I), +(2) + +Error-Robust Adams Solver +where ¯αt = �t +s=1 αs. When t is large enough, the Markov +process will converge to a Gaussian steady-state distribution +N(0, I). +DDPMs also build the reverse process, i.e., the sampling +process, to the diffusion process above. Similar to VAEs, +DDPMs introduce the variational bound to derive optimiza- +tion objectives. After simplification (Luo, 2022), the main +objective of training optimization is formulated as a KL +divergence: +Lt−1 = DKL(q(xt−1|x0, xt)||pθ(xt−1|xt)), +(3) +where q(xt−1|x0, xt) has an analytical form derived from +the bayesian formula and can be expressed as a Gaussian +distribution parameterized with µt−1 = +1 +√αt (xt − +1−αt +√1−¯αt ϵ) +and σt−1 = 1−¯αt−1 +1−¯αt (1 − αt). +To simplify the optimization loss, the parameterized denois- +ing distribution is formulated as follows: +pθ(xt−1|xt) = N(xt−1; µθ(xt), σ2 +t−1I) +µθ(xt) = +1 +√αt +(xt − 1 − αt +√1 − ¯αt +ϵθ(xt, t)), +(4) +where ϵθ(xt, t) estimates the noise in noisy sample xt and +is called estimated noise in our paper. The variance σ2 +t−1 is +constant. In this way, the objective function Eq. 3 can be +rewritten as following: +Lt−1 = Eq[ +1 +2σ2 +t−1 +||µt−1 − µθ(xt)||2] += Ex0,ϵ[ +1 +2σ2 +t−1 +|| +1 +√αt (xt − 1 − αt +√1 − ¯αt +ϵ) − µθ(xt)||2] += Ex0,ϵ[ +1 +2σ2 +t−1 +(1 − αt)2 +αt(1 − ¯αt)||ϵ − ϵθ(xt, t)||2]. +(5) +Though DDPMs have good theoretical properties, they suf- +fer from sampling efficiency. It usually requires hundreds +or even thousands of network evaluations, which limits var- +ious downstream applications for DDPMs. There already +existed methods (Watson et al., 2021; Lyu et al., 2022; Lam +et al., 2022; Salimans & Ho, 2021) which depend on extra +training stage to derive fast sampling methods. The training- +based methods usually require tremendous training costs +for different data manifolds and tasks, which inspires many +works to explore a training-free sampler based on numerical +methods. +2.2. Numerical Methods for Fast Sampling +Denoising diffusion implicit model (DDIM) (Song et al., +2020a) is a classic training-free approach for fast sampling, +which introduces a non-Markovian process that allows the +sampling with any number of evaluation steps. We formu- +late the iteration time steps for solving diffusion ODEs as +{ti}N +i=0, where t0 is the beginning time and tN is the end +time. The sequence of sampling steps maintains the prop- +erty that ti>ti+1 and tN = 0. In particular, the last iterate +xtN represents the final generated sample. The denoising +iteration process of DDIM can be formulated as follows: +xti+1 = �¯αti+1(xt − √1 − ¯αtiϵθ(xti, ti) +√¯αti) ++ +� +1 − ¯αti+1 − σ2 +tiϵθ(xti, ti) + σtiz. +(6) +Furthermore, DDIM shares the same training objective with +DDPMs, which means the fast sampling scheme can be +applied to any pre-trained models. When the variance σ2 +ti is +set to 0, the sampling process becomes deterministic, which +can be regarded as the solving process of diffusion ODEs. +We use a term ϵti for the ease of description so that the +solving formula of diffusion ODE can be formulated (Song +et al., 2020a) as follows: +ϵti = ϵθ(xti, ti), +(7) +xti+1 = +√¯αti+1 +√¯αti +xti ++ ( +� +1 − ¯αti+1 − +� +¯αti+1(1 − ¯αti) +√¯αti +)ϵti. +(8) +Many numerical solvers, such as Runge-Kutta (Butcher, +1996) method and linear multistep method (Wells, 1982), +can be applied to solve diffusion ODEs. PNDM(Liu et al., +2021) combined Runge-Kutta and linear multistep method +so as to solve the manifold problem of diffusion ODEs. +Essentially, the linear multistep method can be regarded +as explicit Adams (Małgorzata & Marciniak, 2002), which +utilized previously estimated noises to calculate the ϵti. The +ϵti in Eq. 7 is reformulated as following: +ϵti = 1 +24(55ϵθ(xti, ti) − 59ϵθ(xti−1, ti−1) ++ 37ϵθ(xti−2, ti−2) − 9ϵθ(xti−3, ti−3)). +(9) +DPM-Solver(Lu et al., 2022a) introduced the method of +exponential integrator from ODE literature (Atkinson et al., +2011) to eliminate the discretization errors of the linear term +in the sampling process. Furthermore, DPM-Solver pro- +posed to utilize Taylor expansion (Mohazzabi & Becker, +2017) to approximate ϵti and contributed its analytical +forms for solving diffusion ODEs. +We notice that the estimated noises of pretrained diffusion +models are not precise across all sampling time, especially +when time ti is close to 0. Previous numerical methods +with analytical forms will suffer from the noise estimation +error of pretrained DDPMs. In this paper, we propose an +error-robust solver based on Adams numerical methods, +adaptively selecting noises with low estimation error. + +Error-Robust Adams Solver +Eq.(8) +Eq.(17) +Corrector +Eq.(11) +Predictor + Lagrange Buffer +Initial indexs +Selected Indexs +ERA-Solver + +Push +Selected +Not Selected +Initial Indexs +Error-Robust Selection +Figure 2. The pipeline of ERA-Solver. The sampling scheme is based on the predictor-corrector method for implicit Adams. Our predictor +is robust to the errors of the estimated noises from pretrained models. The sampling starts from normal Gaussian noise xt0 and performs a +denoising scheme (from xti to xti+1) iteratively to get the final generated image. +3. ERA-Solver +In this section, we first point out that the error in the esti- +mated noise ϵθ(xti, ti) by the network θ limits the previ- +ous numerical fast samplers and introduce implicit Adams +numerical solver (Sec. 3.1). Then, we apply predictor- +corrector sampling and leverage a Lagrange interpolation +function as the predictor (Sec. 3.2). We design an error +distance to measure the accuracy of the estimated noise and +enhance the proposed predictor with an error-robust strategy +to adaptively select the Lagrange bases with lower noise +estimation error (Sec. 3.3). The whole sampling process is +shown in Fig. 2. +3.1. Implicit Adams Methods +The sampling of DDPMs starts from a prior noise distribu- +tion xt0 ∼ N(0, I), and iteratively denoises xti to xti+1 +until time t reaches 0. In the sampling process, the most +time-consuming step is network evaluation. Assuming we +have a pretrained noise estimation model ϵθ, we need to +achieve good generation quality with as few evaluation times +as possible. +We notice that the noise estimation error exists across every +sampling time, especially when time ti is close to 0. It limits +previous numerical high-order solvers (Liu et al., 2021; Lu +et al., 2022a; Song et al., 2020a) since they are based on +the assumption that the network has no estimation errors. +Previous solvers usually involved observed noise terms to +achieve the analytic form of the sampling scheme that is +critical for sampling convergence. Thus, they are sensitive +to the errors of estimated noises from pretrained models. +In this paper, we explore the potential of implicit Adams +solver (Małgorzata & Marciniak, 2006). Different from +explicit Adams (Eq. 9), implicit Adams involves the unob- +served noise term and the ϵti in Eq. 7 is reformulated as +follows: +ϵti = 1 +24(9ϵθ(xti+1, ti+1) + 19ϵθ(xti, ti) +− 5ϵθ(xti−1, ti−1) + ϵθ(xti−2, ti−2)). +(10) +It can be noticed that xti+1 can be observed only when +ϵti is achieved, while the ϵti contains unobserved term +ϵθ(xti+1, ti+1), which makes it challenging to solve implicit +equations and may need more time-consuming iteration +steps. This greatly limits the implicit Adams method to be a +fast solver for diffusion ODEs. +Fortunately, in numerical ODE literature, the sampling effi- +ciency of implicit Adams can be improved with a predictor- +corrector sampling scheme (Diethelm et al., 2002). Specifi- +cally, the predictor makes a rough estimation of unobserved +term ¯ϵθ(xti+1, ti+1) and the corrector derives the precise +xti+1, which can reformulate Eq.10 as follows: +ϵti = 1 +24(9¯ϵθ(xti+1, ti+1) + 19ϵθ(xti, ti) +− 5ϵθ(xti−1, ti−1) + ϵθ(xti−2, ti−2)). +(11) +The traditional predictor-corrector utilizes explicit Adams +(Eq. 9) to perform predictor to make xti+1 observed so as +to derive ¯ϵθ(xti+1, ti+1). However, it still suffers from the +fixed form that is not robust to the noise estimation errors, +which can be observed in Fig. 1. +3.2. Predictor with Lagrange Interpolation Function +In this paper, we propose to utilize noises observed at pre- +vious sampling steps and construct the Lagrange interpo- +lation function as the predictor to predict unobserved term +ϵθ(xti+1, ti+1). In this way, we can design an adaptive +strategy to select Lagrange function bases to construct the +error-robust predictor. +Specifically, we maintain a buffer about all previously esti- +mated noises, which have been observed and need no extra + +Error-Robust Adams Solver +Selected +Not Selected +Sampling Process +Figure 3. Error-robust selection process. ∆ϵ is calculated based on +Eq. 15 instead of the training loss in Fig. 1. The sampling NFE is +set to 20. +computations, and its corresponding time: +{(tn, ϵθ(xtn, tn)), n = 0, 1, .., i}. +(12) +The maintained buffer is also called the Lagrange buffer +in this paper. Assume that the interpolation order is k, the +selected function bases to construct the Lagrange function +can be written as {(tτm, ϵθ(xtτm, tτm)), m = 0, ...k − 1}. +The corresponding Lagrange interpolation function can be +formulated as: +lm(t) = +k−1 +� +l=0,l̸=m +( t − tτl +tτm − tτl +), +Lϵ(t) = +k−1 +� +m=0 +lm(t) ∗ ϵθ(xtτm , tτm), +(13) +where τl belongs to the maintained Lagrange buffer and +has already been observed. At time ti+1, we can derive an +estimation about ϵθ(xti+1, ti+1): +¯ϵθ(xti+1, ti+1) = Lϵ(ti+1). +(14) +With this prediction, we apply the corrector process in Eq. +11 to get the ϵti and Eq. 8 to get the denoised sample xti+1. +It can be noticed that the proposed predictor makes use of +the observed noise estimations and involves no network eval- +uations. Furthermore, the Lagrange bases in the predictor +can be adaptively selected from those noise estimations with +low errors, which is more error-robust. We introduce this +error-robust selection strategy in the next subsection. +3.3. Error-Robust Selection Strategy +In this part, our goal is to design an error-robust selection +strategy for the maintained Lagrange buffer. When the +interpolation order is k, the intuitive selection approach is +to make a fixed selection of the last k estimated noises from +the maintained Lagrange buffer, which means τm = i − m +in Eq. 13. However, we notice that the noise estimation +Algorithm 1 ERA-Solver +1: Input: {ti}N +i=0, k, ϵθ +2: Instantiate: xt0 ∼ N(0, I), buffer Ω = ∅, ∆ϵ = λ +3: Ω = Ω ∪ {(t0, ϵθ(xt0, t0))} +4: for i in 0, 1, · · · , N − 1 do +5: +if i50. +We can notice +that our solver performs worse when NFE increases to 50, +especially on LSUN dataset (Tab. 2 and Tab. 1). It can be +understood that FID results fluctuate around a value, which +is verified on PNDM (Liu et al., 2021). +Why ERA-Solver performs worse on Cifar10. +From +Tab. 3, we can notice that ERA-Solver performs slightly +worse than DPM-Solver and PNDM when NFE increases, +which is different from the results on LSUN. The reason +behind it is the resolution of data. Cifar10 has a very low res- +olution (32 × 32) while LSUN has a higher resolution (256 +× 256), which means the LSUN dataset is more informative +than Cifar10. Thus, the model tends to have lower train- +ing error (noise estimation error) when trained on Cifar10, +leading to the slightly worse performance of ERA-Solver. +Rationality of selection function. +In this paper, we de- +sign the selection function for the error robustness of the +sampling solver. The changing trend of noise estimation +error with timestep (Fig. 1) inspires us to choose the power +function as an error-robust selection function and experi- +ments demonstrate its effectiveness. There may exist a better +selection function or a learning-based approach to selecting +Lagrange functions and we leave it for future work. +Why ERA-Solver has better error robustness. +We no- +tice that the previous numeric solvers depend on the assump- +tion that the pretrained diffusion model achieves accurate +10 +15 +20 +25 +30 +35 +40 +45 +50 +NFE +9.5 +10.0 +10.5 +11.0 +11.5 +12.0 +12.5 +13.0 +FID +ERS = 3.0 +ERS = 5.0 +ERS = 8.0 +Constant 0.2 +Constant 0.5 +Constant 0.8 +Constant 1.0 +Constant 2.0 +Figure 5. Generation quality measured by FID ↓. We compare our +error-aware scale (∆ϵ/λ in Eq. 17) and constant scale (replace +∆ϵ/λ with a constant) to demonstrate the effectiveness of the pro- +posed error measure based on the 3-order Lagrange interpolation +function. All FID results are evaluated on LSUN-Church with +5000 generated samples. +noise estimation. In this paper, we demonstrate that there +exists an obvious estimation error (Fig. 1) from pretrained +model, which limits the sampling efficiency of previous fast +solvers. +Combined with the Lagrange interpolation as the predictor +and an error-robust selection strategy, we show that the Im- +plicit Adams solver has the potential to be error-robust so as +to achieve better sampling efficiency. The predictor based on +Lagrange interpolation makes sure our solver has the conver- +gence property of the numerical solver (predictor-corrector +sampling scheme). The error-robust selection strategy adap- +tively introduces relatively accurate estimated noises and +mitigates the error accumulation in the sampling process. +More analysis can be found in Appendix. C. +6. Conclusion +In this paper, we propose an error-robust Adams solver +(ERA-Solver) that consists of a predictor and a corrector. +We leverage the Lagrange interpolation function to perform +the predictor, and further propose an error measure for the +sampling process and an error-robust strategy to enhance +the predictor. Experiments demonstrate that ERA-Solver +achieves better generation quality on LSUN-Church and +LSUN-Bedroom datasets at all NFEs. Our ERA-Solver +might be misused with powerful diffusion models (Rombach +et al., 2022; Kim & Ye, 2021) to generate adverse fake +content. We will restrict the usage of our solver and share it +with the fake detection community. +References +Atkinson, K., Han, W., and Stewart, D. E. Numerical so- +lution of ordinary differential equations. John Wiley & +Sons, 2011. + +Error-Robust Adams Solver +Bao, F., Li, C., Zhu, J., and Zhang, B. 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P., Kumar, +A., Ermon, S., and Poole, B. +Score-based genera- +tive modeling through stochastic differential equations. +arXiv:2011.13456, 2020b. +von Platen, P., Patil, S., Lozhkov, A., Cuenca, P., Lambert, +N., Rasul, K., Davaadorj, M., and Wolf, T. Diffusers: +State-of-the-art diffusion models. https://github. +com/huggingface/diffusers, 2022. +Wang, Z., Zheng, H., He, P., Chen, W., and Zhou, M. +Diffusion-gan: Training gans with diffusion. +arXiv +preprint arXiv:2206.02262, 2022. +Watson, D., Chan, W., Ho, J., and Norouzi, M. Learning fast +samplers for diffusion models by differentiating through +sample quality. In International Conference on Learning +Representations, 2021. +Wells, D. R. Multirate linear multistep methods for the so- +lution of systems of ordinary differential equations. Uni- +versity of Illinois at Urbana-Champaign, 1982. +Wu, L., Gong, C., Liu, X., Ye, M., and Liu, Q. Diffusion- +based molecule generation with informative prior bridges. +arXiv preprint arXiv:2209.00865, 2022. +Yu, F., Seff, A., Zhang, Y., Song, S., Funkhouser, T., and +Xiao, J. +Lsun: Construction of a large-scale image +dataset using deep learning with humans in the loop. +arXiv:1506.03365, 2015. +A. Ablation Study on Cifar10 +In this part, we provide ablation experiments on Cifar10 +dataset. We replace our error-robust selection strategy with +the fixed selection strategy that fixedly selects the last k +estimated noises previously saved in the Lagrange buffer at +every sampling step. The results are shown in Tab. 5. From +the table, we can see that our selection strategy achieves +better FID generation results. +We also conduct ablation studies for proposed error mea- +sures in the sampling process. We parameterize the power +function with various constants instead of our error mea- +sure. The comparison results are shown in Fig. 6. Our +error measure can enhance the selection strategy with better +generation results in general. +B. Additional Results on Celeba +The comparison results on Celeba are shown in Tab. 6. +When tested on a few NFEs like 10 and 12, our ERA-Solver +obtains 5.06 and 3.67 FID results, achieving 26.8% and +14.4% improvements compared with the previous best re- +sults of 6.92 and 4.20. It can also be observed that FID result +of ERA-Solver converges at about NFE 15. Compared with +DPM-Solver which converges at NFE 36, ERA-Solver has +better convergence speed, while achieving comparable gen- +eration quality (FID 2.75 vs 2.71). +C. Analysis of Error Robustness +Since data distributions like LSUN are high-dimensional +and complex, it is difficult to seek a ground truth for the pre- +dicted noise in the sampling process. Thus, we demonstrate +the error robustness of ERA-Solver from another perspec- +tive. Assume that we achieve a generated sample xgen +0 +from +a solver, we can utilize the diffusion process to add noise +to the sample, remapping the sample obtained from the de- +noising process back to the noise space to derive xgen +t +. We +measure the error robustness of the solver as the following: +||ϵ − ϵθ(xgen +t +, t)||2, +(18) +The estimated noise from pretrained model can be seen +as the stepping direction of the noisy sample (Song et al., +2020b). If a solver is not error-robust, its generated samples +will deviate from the original generation path in the sam- +pling process. Since the generation process can be seen as +the reverse process of the diffusion process, the deviation of +generated samples will increase the error in Eq. 18. +We select normal Implicit Adam solver (Małgorzata & +Marciniak, 2006), DPM-Solver (Lu et al., 2022a), and ERA- +Solver for comparisons and generate a batch of samples to +calculate Eq. 15 separately. Three solvers share the same +sampling NFE, random seed, and the pretrained model ϵθ. + +Error-Robust Adams Solver +Table 5. FID comparison between error-robust selection strategy +(ERS) and fixed strategy (fixed) based on various Lagrange func- +tion orders (k = 3, 4, 5, 6). +Method\ NFE +10 +15 +20 +50 +ERA-Solver-3 +fixed +5.95 +4.62 +4.24 +4.0 +ERS +5.79 +4.31 +4.07 +3.94 +ERA-Solver-4 +fixed +6.4 +4.46 +4.1 +3.98 +ERS +5.14 +3.86 +3.79 +3.91 +ERA-Solver-5 +fixed +17.21 +15.11 +17.47 +3.99 +ERS +6.26 +3.73 +3.69 +3.98 +ERA-Solver-6 +fixed +36.34 +51.58 +83.39 +118.38 +ERS +19.26 +4.16 +3.73 +4.04 +10 +15 +20 +25 +30 +35 +40 +45 +50 +NFE +4.0 +4.5 +5.0 +5.5 +6.0 +6.5 +FID +ERS = 10.0 +ERS = 5.0 +ERS = 8.0 +Constant 0.2 +Constant 0.5 +Constant 0.8 +Constant 1.0 +Constant 1.2 +Constant 1.5 +Figure 6. Generation quality measured by FID ↓. We compare our +error-aware scale (∆ϵ/λ) and constant scale based on the 4-order +Lagrange function. All FID results are evaluated on Cifar10 with +50k generated samples. +The results are shown in Fig. 7, from which we can see that +ERA-Solver achieves lower error in general. +D. Qualitative Results +We sample and visualize the generated samples from LSUN- +Church and LSUN-Bedroom datasets. We evaluate DDIM, +DPM-Solver, and our method based on 5, 8, 10, 12, 15, and +20 NFEs. Since PNDM (Liu et al., 2021) requires at least 13 +NFEs to perform sampling, we only compared our method +with DDIM (Song et al., 2020a) and DPM-Solver-fast (Lu +et al., 2022a) methods. We set λ = 5.0 and apply a 4-order +Lagrange interpolation function (k = 4) for comparison. +The comparison results are shown in Fig. 9 and Fig. 10. +Compared with DDIM, our ERA-Solver can obtain sharper +textures, benefiting from its high precision of the implicit +Adams scheme. Compared with DPM-Solver-fast, our ERA- +Solver can avoid excessive contrast and exposure, achieving +more natural texture details. +We also provide generated samples based on the 4-order +ERA-Solver with λ = 5.0 on Cifar10, as shown in Fig. 8. +E. Results on Stable Diffusion +We conduct generation comparison based on large-scale +latent diffusion mode, i.e., Stable Diffusion (Rombach et al., +2022). We apply the improved version (Lu et al., 2022b) of +DPM-Solver for better conditional generation. The PNDM +and improved DPM-Solver are all encapsulated in diffusers +(von Platen et al., 2022). We directly apply them by diffusers +to make sampling. For ERA-Solver, we set k = 4 and +λ = 10.0. The results are shown in Fig. 11 and Fig. 12. It +can be observed that ERA-Solver can generate promising +images when NFE is 15, which is faster than DPM-Solver +and PNDM. +We also provide computation time for each sampling pro- +cess, as shown in Tab. 7. It can be observed that at NFE +15, ERA-Solver consumes slightly more time (0.08s) than +DPM-Solver, which can be contributed to the cost of main- +taining the Lagrange buffer. The cost of the Lagrange buffer +will increase with NFE increasing. However, ERA-Solver +has been able to generate exquisite images on NFE 15. Thus, +the computation cost of the Lagrange buffer can be ignored. + +Error-Robust Adams Solver +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +time t +0 +50 +100 +150 +200 +250 +300 +350 +|| +(xgen +t +, t)||2 +DPM-Solver +ERA-Solver (Ours) +Normal Implicit Adams +Figure 7. The error comparison between three fast solvers. For ERA-Solver, we set k = 4 and λ = 5.0. The random seed and pretrained +model are shared with three solvers. +Table 6. Generation quality measured by FID ↓ on Celeba, varying the number of function evaluation (NFE). +Sampling method \ NFE +5 +10 +12 +15 +20 +40 +50 +100 +DDIM (Song et al., 2020a) +24.26 +13.4 +13.23 +11.63 +9.62 +6.87 +6.08 +4.5 +Analytic-DDPM (Bao et al., 2022) +\ +28.99 +25.27 +21.80 +18.14 +\ +11.23 +\ +Analytic-DDIM (Bao et al., 2022) +\ +15.62 +13.90 +12.29 +10.45 +\ +6.13 +\ +PNDM (Liu et al., 2021) +\ +\ +\ +10.91 +7.59 +4.06 +3.45 +2.97 +DPM-Solver-fast (Lu et al., 2022a) +\ +6.92 +4.20 +3.05 +2.82 +2.71 (NFE = 36) +ERA-Solver +23.32 +5.06 +3.67 +2.99 +2.75 +2.69 +2.73 +2.72 +Figure 8. Generated samples from ERA-Solver-4 (20 NFE). +Table 7. Computation time of sampling on Stable Diffusion, vary- +ing different solvers and NFE. +Sampling Method \ NFE +15 +25 +50 +PNDM (Liu et al., 2021) +2.69 +3.74 +6.13 +DPM-Solver (Lu et al., 2022a) +1.86 +2.76 +5.30 +ERA-Solver +1.94 +3.05 +6.01 + +Error-Robust Adams Solver +DDIM +DPM- +Solver- +fast​ +Ours +NFE=5 +NFE=8 +NFE=10 +NFE=12 +NFE=15 +NFE=20 +DDIM +DPM- +Solver- +fast​ +Ours +DDIM +DPM- +Solver- +fast​ +Ours +Figure 9. Generation quality comparison with 5, 8, 10, 12, 15, and 20 NFEs on LSUN-Church dataset. + +Error-Robust Adams Solver +DDIM +DPM- +Solver- +fast​ +Ours +NFE=5 +NFE=8 +NFE=10 +NFE=12 +NFE=15 +NFE=20 +DDIM +DPM- +Solver- +fast​ +Ours +DDIM +DPM- +Solver- +fast​ +Ours +Figure 10. Generation quality comparison with 5, 8, 10, 12, 15, and 20 NFEs on LSUN-Bedroom dataset. + +Error-Robust Adams Solver +PNDM +DPM-Solver +Ours +NFE=15 +NFE=25 +NFE=50 +Prompt: Cute and adorable ferret wizard, wearing coat and suit +Figure 11. Samples using the pretrained Stable-Diffusion (Rombach et al., 2022) with a classifier-free guidance scale 7.5 (the default +setting), varying different solvers and NFEs. The main part of input prompt is: “Cute and adorable ferret wizard, wearing coat and suit”. + +Error-Robust Adams Solver +NFE=15 +NFE=25 +NFE=50 +PNDM +DPM-Solver +Ours +Figure 12. Samples using the pretrained Stable-Diffusion (Rombach et al., 2022) with a classifier-free guidance scale 7.5 (the default +setting), varying different solvers and NFEs. The main part of input prompt is: “A beautiful mansion beside a waterfall in the woods”. + diff --git a/LtFOT4oBgHgl3EQf0TSa/content/tmp_files/load_file.txt b/LtFOT4oBgHgl3EQf0TSa/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..64d107fafdb72bf029fb9807186bc22202604546 --- /dev/null +++ b/LtFOT4oBgHgl3EQf0TSa/content/tmp_files/load_file.txt @@ -0,0 +1,1208 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf,len=1207 +page_content='ERA-Solver: Error-Robust Adams Solver for Fast Sampling of Diffusion Probabilistic Models Shengmeng Li 1 Luping Liu 2 Zenghao Chai 3 Runnan Li 1 Xu Tan 3 Abstract Though denoising diffusion probabilistic models (DDPMs) have achieved remarkable generation results, the low sampling efficiency of DDPMs still limits further applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'} +page_content=' Since DDPMs can be formulated as diffusion ordinary differ- ential equations (ODEs), various fast sampling methods can be derived from solving diffusion ODEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'} +page_content=' However, we notice that previous sam- pling methods with fixed analytical form are not robust with the error in the noise estimated from pretrained diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'} +page_content=' In this work, we construct an error-robust Adams solver (ERA- Solver), which utilizes the implicit Adams nu- merical method that consists of a predictor and a corrector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'} +page_content=' Different from the traditional predictor based on explicit Adams methods, we leverage a Lagrange interpolation function as the predictor, which is further enhanced with an error-robust strategy to adaptively select the Lagrange bases with lower error in the estimated noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'} +page_content=' Exper- iments on Cifar10, LSUN-Church, and LSUN- Bedroom datasets demonstrate that our proposed ERA-Solver achieves 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'} +page_content='14, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'} +page_content='42, and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'} +page_content='69 Fenchel Inception Distance (FID) for image generation, with only 10 network evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'} +page_content=' Introduction In recent years, denoising diffusion probabilistic models (DDPMs) (Ho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'} +page_content=', 2020) have been proven to have poten- tial in data generation tasks such as text-to-image genera- tion(Poole et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'} +page_content=' Gu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'} +page_content=' Kim & Ye, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'} +page_content=', 2022), speech synthesis(Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'} +page_content=' Lam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'} +page_content=' Leng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'} +page_content=', 2022), and molecular conformation formation(Hoogeboom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'} +page_content=' Jing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'} +page_content=' They build a diffusion process to add noise into the sample and a denoising process to remove noise from the sample gradually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'} +page_content=' Compared with 1Microsoft Cloud+AI 2Zhejiang University 3Microsoft Re- search Asia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'} +page_content=' Correspondence to: Xu Tan 8), high and low +number of subhaloes (𝑁sub > 120 and 𝑁sub < 85 respectively). +On average, our haloes share the same number of substructures and +concentrations as the Rhapsody-G sample (Wu et al. 2015; Hahn +et al. 2017; Martizzi et al. 2016). +We use the ΛCDM cosmology of the Rhapsody-New simulation +with density parameters Ωb = 0.049 for baryons, Ωm = 0.309 for +total matter and ΩΛ = 0.691 for the cosmological constant. The pri- +mordial spectral index, the amplitude normalisation and the Hubble +constant are 𝑛𝑠 = 0.9667, 𝜎8 = 0.8159 and 𝐻0 = 67.74 km/s/Mpc +respectively (Planck Collaboration et al. 2016). In this new cosmol- +ogy, we have a lower baryon fraction of 𝑓b = 0.1586 compared to the +Rhapsody-G’s value of 0.18. The updated value is also much closer +to more recent constraints from Planck Collaboration et al. (2021) of +𝑓b = 0.1564. +We generated the initial conditions using Music (Hahn & Abel +2011) for our nine clusters at 𝑧 = 49 from the minimum bound- +ing ellipsoid matrix retrieved from the cosmICweb database using a +traceback-radius of 2𝑅vir centered in a 1 ℎ−1Gpc box with an effective +resolution of 81923 particles3. All initial conditions were performed +1 where the ‘C’ denotes for the inclusion of anisotropic thermal conduction. +2 See https://cosmicweb.astro.univie.ac.at for more details +3 We share the same resolution as the Rhapsody-G 8K run. +MNRAS 000, 1–30 (2022) + +The Rhapsody-C simulations +3 +Table 1. Description of the Rhapsody-C runs presented in this work. In the +top part of the table we list the minimum cell size (Δ𝑥), the initial mass +per hydro cell (𝑚gas), the dark matter (𝑚dm) and minimum stellar particle +mass (𝑚∗,min). The middle part describes the physical models used in the +simulations studied in this work. In the bottom part, we list the properties +of each Rhapsody-C haloes: the internal halo ID in the Rhapsody-New +simulation, the number of substructures (𝑁sub), the virial mass (𝑀vir), radius +(𝑅vir) and concentration (𝑐vir) as well as the radius enclosing 500 times the +critical density of the Universe (𝑅500) and the total mass within (𝑀500). +Summary of the Rhapsody-C simulations +Δ𝑥 [ kpc] +𝑚dm [M⊙] +𝑚gas [M⊙] +𝑚∗,min [M⊙] +2.82 +1.54 × 108 +1.68 × 107 +6.58 × 106 +Sub-grid modelling and baryonic processes +Cooling, +AGN energy +AGN energy +Anisotropic +Label +SF, +deposition +accumulation +thermal +SN +scheme +threshold +conduction +NR +– +– +– +– +VW +✓ +volume-weighted +107 K +– +MW +✓ +mass-weighted +107 K +– +MC +✓ +mass-weighted +107 K +✓ +MW6 +✓ +mass-weighted +106 K +– +MW8 +✓ +mass-weighted +108 K +– +Halo Properties +ID +𝑁sub +𝑐vir +𝑀vir +𝑅vir +𝑀500 +𝑅500 +[1015M⊙] +[Mpc] +[1014M⊙] +[Mpc] +174742934 111 +6.02 +1.22 +2.82 +6.80 +1.37 +176970005 117 +6.69 +1.19 +2.78 +7.55 +1.41 +174824666 124 +6.35 +1.16 +2.77 +7.56 +1.42 +173505201 135 +5.68 +1.12 +2.80 +6.98 +1.38 +176144520 100 +6.80 +1.01 +2.64 +6.32 +1.33 +176061412 105 +9.52 +1.00 +2.63 +6.62 +1.35 +173917492 111 +8.64 +0.99 +2.62 +6.73 +1.36 +174743229 83 +6.50 +0.98 +2.62 +6.24 +1.33 +173587157 84 +5.45 +0.86 +2.51 +5.19 +1.25 +using second-order Lagrangian perturbation theory (LPT) with dark +matter and baryon perturbations at 𝑧 = 49. Compared to the original +Rhapsody-G simulations, we do not use the local Lagrangian ap- +proximation for the construction of the baryon density field. Baryons +and dark matter did not co-move prior to recombination and sub- +percent effects are expected at cluster scales (see e.g. Angulo et al. +2013; Hahn et al. 2021; Khoraminezhad et al. 2021). However, for +the simulations used here, we assume that baryons fully trace cold +dark matter perturbations. +2.2 Numerical approach +For our cluster zoom simulations, we use the Eulerian adaptive mesh +refinement Ramses code (Teyssier 2002) to follow the non-linear +evolution of the initial conditions. Gas dynamics are computed using +a second-order unsplit Godunov scheme for the ideal MHD equations +(Fromang et al. 2006; Teyssier et al. 2006) while collisionless dark +matter particles as well as stars and sink particles are evolved using a +particle-mesh solver. Our simulations use the method introduced by +Dubois & Commerçon (2016) for solving the anisotropic diffusion +of heat using an implicit finite-volume method which is independent +of the Courant time step constraint of the MHD scheme. +We employ a Lagrangian overdensity-based refinement strategy +that splits cells if they reach an overdensity of eight: the refinement +of the base grid by 𝑛 additional levels requires a density of 8𝑛 ¯𝜌. +Our simulation boxes of 1 ℎ−1Gpc on a side, reach a maximum +refinement level by maintaining a smallest cell size of physical +Δ𝑥 = 2.8 kpc. The dark matter 𝑁-body particle mass is 1.54×108M⊙ +and initial mass per hydro cell is 1.68 × 107M⊙. The high-resolution +Lagrangian ellipsoid patch, from which the 2𝑅vir sphere centred on +each cluster will form, is tagged using a passive scalar colour field +that is advected with the gas. Dynamic refinement is restricted to +the regions where this colour field is non-zero and no refinement +is allowed outside the zoom region. We thus focus most of the +computational resources on the forming cluster and its immediate +environment. +The rest of this section details the various physical ingredients +used in our high-resolution zoom-in simulations. See Section 2.2.1 +for gas cooling and heating as well as the star formation and stellar +feedback, Section 2.2.2 for the subgrid modelling of SMBH forma- +tion, evolution and AGN feedback ang finally in Section 2.2.3 we +describe the magnetic field evolution with the anisotropic thermal +conduction. The reader can skip to Section 3 and Section 4 for the +scientific results regarding the impact of the the various BH-related +sub-grid models and anisotropic thermal conduction respectively on +the stellar and gaseous content of a GC, or directly to Sections 5 +and 6 for properties of cluster galaxies and ICM as well as the the +evolution of our clusters along various scaling relations respectively. +2.2.1 Radiative gas cooling, metallicity and stellar evolution +Radiative gas cooling is calculated according to the tabulated rates +of Sutherland & Dopita (1993) for Hydrogen, Helium and metal +line cooling. The total gas metallicity is not evolved separately but +treated as a single species. It is advected with the MHD equations as +a passive scalar and is sourced by the supernovae feedback model. +We consider an UV background radiation according to the Haardt & +Madau (1996) model. An instantaneous reionisation takes place at +𝑧 = 10 to take into account an earlier reionisation in the particularly +overdense proto-cluster regions that we simulate. The unresolved cold +and dense gas that will constitute the inter-stellar medium (ISM) +of galaxies is approximated using a temperature floor given by a +polytropic equation of state, +𝑇floor = 𝑇∗ +� 𝑛H +𝑛∗ +�𝛾∗−1 +, +(1) +with 𝑛H the Hydrogen number density of the gas, 𝑛∗ = 0.1 cm−3 and +𝑇∗ = 104 K being respectively the star formation density threshold +and the ISM polytropic temperature with 𝛾∗ = 5/3 being the ISM +polytropic index. In practice, gas can be heated above the temperature +floor, but cannot cool below it. +Star formation occurs when the gas density exceeds 𝑛∗. A portion +of the gas in a cell is converted into a star particle that decouples from +the gas. We have a minimum stellar particle mass of 5.6 × 106M⊙. +The star particles are randomly drawn from Poisson process (Rasera +& Teyssier 2006) following a Schmidt law +�𝜌∗ = 𝜖∗ 𝜌gas / 𝑡ff, +(2) +with 𝜖∗ = 0.01 and 𝑡ff = �3𝜋/32G𝜌gas +�−1/2, the local free-fall time. +Stellar feedback is included using the model of Dubois & Teyssier +(2008) in which each newly formed star that traces a continuous +stellar mass distribution following the Salpeter (1955) initial mass +function and releases, after 20 Myr, a fraction 𝜂 = 0.1 of its mass +and metals with a yield of 𝑦 = 0.1. Therefore 𝑦𝜂 = 0.01 of the time- +integrated SFR is returned as metals in the ISM. In addition, each +MNRAS 000, 1–30 (2022) + +4 +A. Pellissier et al. +SN feedback event injects a thermal energy of 1051 erg into the sur- +rounding ISM. Compared to the original Rhapsody-G simulations, +we chose to enable the delayed cooling of the SN heated gas with +a dissipation time scale of 20 Myr. This additional sub-grid model +mimics the effect of non-thermal processes, such as turbulence or +CRs (Rodríguez Montero et al. 2022), which can dissipate energy on +longer time scales before being radiated away. The calibration of the +free parameters of the SN feedback listed above is able to reproduce +stellar masses consistent with abundance-matching results at masses +lower than 1012M⊙ for resolved haloes with at least 1000 particles +(see Section 5.1). +2.2.2 Black holes and active galactic nuclei +Ramses uses collisionless sink particles to model black hole growth +and evolution. The SMBH formation and evolution follow the model +of Biernacki et al. (2017), itself based on the precedent models +of Dubois et al. (2010) and Teyssier et al. (2011), and build on +a sink particle implementation developed within the context of +star-forming molecular clouds (Bleuler & Teyssier 2014). +Super-massive black hole seeding. The Phew clump finder +(Bleuler & Teyssier 2014), directly implemented in Ramses, de- +termines potential sites for SMBH sink particle formation by identi- +fying relevant peaks in the density field. We will briefly discuss the +main steps and free parameters of the sink seeding model that we +use. First, all density peaks above a threshold 𝜌peak are identified as +well as their connecting saddle points. To keep only relevant density +peaks, we merge all peaks that have a peak-to-saddle ratio lower than +3 to the neighbouring peak with which it shares the highest density +saddle point. This merging process, or noise removal, is halted when +a saddle density falls below the 𝜌saddle threshold. In short, a noise +removal is performed on the density field to select only the relevant +peaks above a density 𝜌peak which are later divided by the saddle den- +sity threshold 𝜌saddle into clumps to finally yield the sink formation +sites. The gas in the spherical region of radius equal to 4 (highest) +resolution elements Δ𝑥, defining the sink sphere, is investigated to +make sure that the gravitational field is compressive, strong enough +to overcome internal gas support and not only accelerated toward the +sink sphere centre but that this gas is contracting. As a proximity +check, we forbid the gas that is infalling to an already existing sink +to create another sink. While the choice for the initial seed mass is +arbitrary, we set it to be the same as our 𝑁-body dark matter particle +mass with 𝑚BH,seed = 108M⊙. +Gas accretion and black hole dynamics. Once SMBHs are formed, +they grow in mass at the (un-boosted) Bondi-Hoyle accretion rate +(Hoyle & Lyttleton 1939; Bondi & Hoyle 1944; Bondi 1952) capped +by the Eddington rate : +�𝑀acc = min � �𝑀Edd , �𝑀Bondi +� , +(3) +with : +�𝑀Bondi = 4𝜋𝜌∞𝑟2 +Bondi𝑣Bondi, +(4) +�𝑀Edd = 4𝜋G𝑀BH𝑚 𝑝 +𝜖𝑟 𝜎𝑇 𝑐 += 𝑀BH +𝑡𝑆 +, +(5) +where 𝜎𝑇 is the Thomson cross-section, 𝐺 the gravitational constant, +𝑀BH and 𝑚 𝑝, sink and proton mass respectively, 𝜖𝑟 = 0.1 is the +Shakura & Sunyaev (1973) radiative efficiency for a SMBH and +𝑡𝑆 ∼ 45 Myr is the Salpeter time. We also have 𝜌∞ = ¯𝜌/𝛼(𝑥sink) +with 𝛼 is the dimensionless density profile of the Bondi self-similar +solution (see Biernacki et al. 2017), ¯𝜌 the mean density inside the +sink sphere, 𝑥sink = 𝑟sink/𝑟Bondi and the sink radius and velocity +defined as follows : +𝑟Bondi = G𝑀BH +𝑣2 +Bondi +, +(6) +𝑣Bondi = +√︃ +𝑐2𝑠 + 𝑣2 +rel, +(7) +with 𝑣rel the relative velocity of the sink to the average gas velocity +inside the sink sphere and 𝑐𝑠, the local sound speed. While we use +MHD, we generically find high plasma beta values in our simulations. +Therefore, in the Bondi formula, the magneto-sonic speed effectively +reduces to the adiabatic sound speed . In addition to gas accretion, +SMBHs can also grow via mergers. In this work, we do not check +if two sinks form a bound system but directly merge if they are less +than one accretion radius apart, i.e. 4Δ𝑥. +The dynamics of a single SMBH cannot be resolved in cosmolog- +ical simulations. This can lead to spurious oscillations of the SMBH +in the potential well of its host halo, due to external perturbations and +the finite resolution effects, particularly during merger events. Bier- +nacki et al. (2017) implemented in Ramses a physically motivated +model based on Eddington-limited accretion. Their main assump- +tion is that the gas accretion rate onto the accretion disc is set by the +Bondi formula ( �𝑀Bondi) which corresponds to a large scale accretion +flow, while the accretion onto the SMBH is set by the Eddington +rate ( �𝑀Edd). The difference between the two rates therefore gives the +amount of gas not being accreted by the central SMBH. Instead, it +should be pushed away from the accretion disc by the Eddington radi- +ation pressure at a rate �𝑀dec = �𝑀Bondi − �𝑀acc, which we however do +not model explicitly in this work. We also stress that we are not using +radiation hydrodynamics in this work. This process of gas accretion +and ejection leads to an additional momentum exchange between the +gas and the sink particle, hence an additional drag force. This addi- +tional drag force is modelled by requiring a fixed center of mass of +the joint gas+sink system during the accretion and a conserved total +momentum. We implemented in Ramses a further modification to +the model of Biernacki et al. (2017) to move the SMBHs towards the +potential minimun (described in Section 3.2). +Active galactic nucleus feedback The accretion rate of gas onto the +SMBH sink particle is always computed from the cells in the sink +sphere (of radius 4Δ𝑥) using mass-weighting. Following Booth & +Schaye (2009), we do not inject the thermal AGN energy at each +time-step but store the rest-mass energy of the accreted gas until it +would be enough to raise the gas temperature inside the sink sphere by +Δ𝑇 = 107 K (unless specified otherwise, see e.g. studies of Teyssier +et al. 2011 and Le Brun et al. 2014 using this Δ𝑇 threshold strategy). +In other words, we inject this accumulated AGN energy when +𝐸AGN > 3 +2𝑚gaskB Δ𝑇 +(8) +in every gas cell of the sink sphere in a mass- or volume-weighted +way (see Section 3.3) with a maximum allowed temperature of the +AGN feedback set to 𝑇AGN = 1.5 × 1011 K. The rate at which this +thermal energy is released to the ambient gas is given by : +�𝐸AGN = 𝜖𝑐𝜖𝑟 �𝑀accc2, +(9) +where 𝜖𝑐 = 0.15 is the coupling efficiency (Dubois et al. 2012), i.e. +the fraction of radiated energy that couples to the surrounding gas, +and is calibrated on the local 𝑀BH − 𝑀∗ relation. +MNRAS 000, 1–30 (2022) + +The Rhapsody-C simulations +5 +2.2.3 Magnetic fields and anisotropic thermal conduction +To solve the MHD equations, the Ramses code uses the second-order +unsplit Godunov method based on the monotonic upstream-centred +scheme for conservation laws (MUSCL-Hancock method, van Leer +1977; Evans & Hawley 1988). The constrained transport approach is +used to evolve the induction equation +𝜕𝑩 +𝜕𝑡 = ∇ × 𝒖 × 𝑩, +(10) +where 𝒖 is the gas velocity and 𝑩 the magnetic field. The scheme +satisfies the solenoidal constraint ∇ · 𝑩 = 0 to machine precision +(Teyssier et al. 2006). The 2D Riemann problem at the cell edges is +solved using the approximate Harten-Lax-van Leer-Discontinuities +(HLLD) solver from Miyoshi & Kusano (2005) to compute time +averaged electromotive forces. +In the ideal MHD limit, the generation of magnetic fields from +a previously unmagnetised fluid is impossible. Therefore, the +magnetic fields must be seeded in our simulations. For simplicity, we +seed a uniform magnetic field along the box 𝑧 axis with a comoving +magnitude of 𝐵0 = 1.56 × 10−12 G, which ensures a divergence-free +initial field. +In the presence of a magnetic field, the conduction of heat in a +plasma becomes anisotropic since the motion of charged particles +perpendicular to the field lines is restricted. We use the implementa- +tion of Dubois & Commerçon (2016) using an implicit finite-volume +method for solving the anisotropic diffusion of heat through electrons +(Braginskii 1965) +𝜕𝜌𝜖e +𝜕𝑡 += −∇ · Qcond, +(11) +with 𝜖e the specific internal energy of electrons. The conductive heat +flux, Qcond, can saturate once the characteristic length scale of the +electron temperature gradient ℓ𝑇e is comparable to or less than the +mean free path of electron 𝜆e. Hence, following Sarazin (1986), we +introduce an effective conductivity which interpolates between the +unsaturated (Spitzer conductivity) and saturated regime by +Qcond,sat = − 𝑓sat 𝜅Sp∇𝑇e, += − 𝑓sat +� +−𝜅∥b (b · ∇) 𝑇e +� +− 𝑓sat (−𝜅iso∇𝑇e) , +(12) +with 𝑓sat = �1 + 4.2𝜆e/ℓ𝑇e +�−1, b = B/|B| the unit vector in the +direction of the local magnetic field, 𝑇e the electronic temperature, +, 𝜅iso and 𝜅∥ the isotropic and parallel conduction coefficient (with +respect to the magnetic field lines) respectively with 𝜅∥ = 𝜅Sp − +𝜅iso. In many astrophysical cases, 𝜅iso/𝜅∥ ≪ 1 since the Larmor +radius, 𝜆L, is much smaller than the mean-free-path of electrons, 𝜆e. +For instance, in a hot intra-cluster plasma with 𝑇e = 3 keV, 𝑛e = +10−2 cm−3 and 𝐵 = 1𝜇G, we have 𝜆L = 108 cm and 𝜆e = 1021 cm. +Here, we set a perpendicular conductivity coefficient of 1 per cent to +ensure numerical stability (Dubois & Commerçon 2016). +The electron energy is tracked separately from that of the ions as +described in Dubois & Commerçon (2016) and the rate of energy +transfer between the electron and ion temperatures is given by +𝑄e↔i = 𝑇i − 𝑇e +𝜏eq,ei +𝑛e𝑘B +𝛾 − 1, +(13) +with the equilibrium timescale +𝜏eq,ei = +3𝑚e𝑚p +8 +√ +2𝜋𝑛i𝑞4e ln Λ +� 𝑘B𝑇e +𝑚e +� 3 +2 +. +(14) +Both ion and electron adiabatic indexes are equal to 𝛾 = 5/3. +By modelling the anisotropic transport using Braginskii MHD +(equation 11), we follow the Spitzer ansatz that assumes a high +degree of electron-ion collisionality, which is a good assumption +in cluster cores (in which we are particularly interested here), but +would need to be corrected in cluster outskirts. Additionally, we +do not take into account the suppression of thermal conduction by +the ion mirror instability (Komarov et al. 2016) caused by magnetic +trapping of electrons by magnetic field strength fluctuations, or the +Whistler instability (Levinson & Eichler 1992; Pistinner & Eichler +1998; Roberg-Clark et al. 2016, 2018) where electron-whistler scat- +terings can significantly alter conduction at very sharp temperature +gradients such as in cold fronts (Komarov et al. 2018) or at tem- +perature scale lengths below the critical value 𝛽e𝜆e (Drake et al. +2021, with typical values of 𝛽e ∼ 100 and 𝜆e ranging from 0.1 kpc +in cool-cores to 1 kpc in cluster outskirts)4. In the recent idealised +simulations of Berlok et al. (2021) and Beckmann et al. (2022), it +was shown that whistler-based suppression of thermal conduction +has only a small impact on the ICM. In this work, we hence model +the upper limit of anisotropic thermal conduction within the ICM, +which is sufficient for our purposes to study the potential impact on +cosmological observables. +3 THE MODELLING OF SUPER-MASSIVE BLACK HOLES +The key ingredients of our SMBH formation and evolution models +are: (a) the conditions for the formation of the SMBH and the SMBH +seed mass, (b) the SMBH dynamics with a possible inclusion of a +dynamical friction model, (c) the SMBH growth by mass accretion +at the Bondi-Hoyle-Lyttleton rate limited to the Eddington rate and +finally (d) the induced AGN feedback which affects the surrounding +gas which couples back to all the previous model ingredients. Ramses +uses the so-called sink particle technique (Bate et al. 1995) to model +SMBH formation and evolution, which is a point mass which can +move through the fluid accretion and interact with it by the ejection +of mass, energy and momentum. +Motivated by the low efficiency of the AGN feedback model in +the Rhapsody-G simulations, our sub-grid models for the SMBH +formation, evolution and AGN feedback need to be revised. In this +section we test how the free parameters in the model influence the +cluster evolution. Respectively, for (a) we investigate in Section 3.1 +the effect of different SMBH seeding scenarii on the gaseous and +stellar content on one of our proto-clusters. Regarding (b), we will +present in Section 3.2 our new ‘tidal friction’ model which allow +to control SMBH orbits. Lastly, we study for (d) different AGN +feedback models in Section 3.1 which impact completely differently +cluster evolution. These analyses are all carried out on a fiducial halo +(173917492). In the simulations discussed in this section, we do not +implement yet the anisotropic thermal conduction. +3.1 Seeding of SMBHs +The specifics of SMBH seeding in simulations is an important aspect +of controlling the effect of AGN feedback in simulations. Different +models for black hole (BH) seeding are used in cosmological simu- +lations such as placing a BH particle in the centre of every massive +halo (Schaye et al. 2015; Weinberger et al. 2017; McCarthy et al. +2017) or models that use thresholds of local gas properties such as +4 where 𝛽e = 8𝜋𝑛e𝑇e/𝐵2 is the electron plasma beta, the ratio of electron +thermal to magnetic field pressure. +MNRAS 000, 1–30 (2022) + +6 +A. Pellissier et al. +metallicity, density, temperature and velocity (Dubois et al. 2014; +Tremmel et al. 2017; Habouzit et al. 2017; Dubois et al. 2021). +In this work, we generally adopt the same procedure as in +Rhapsody-G for black hole seeding, albeit with modified param- +eters. Following Biernacki et al. (2017), we use the ‘minimal’ Jeans +mass corresponding to the highest refinement level of our simula- +tion to define the initial SMBH sink particle mass 𝑀seed = 108M⊙ +which also correspond to our dark matter particle mass. Sink parti- +cle formation sites are identified on-the-fly using the Phew clump +finder algorithm which identifies density peaks with a given contrast +relative to the next saddle-point (see Bleuler & Teyssier 2014, for a +detailed description) which is directly implemented in the Ramses +code. The Phew parameters adopted for the original Rhapsody-G +simulations favoured the seeding of a sink particle in fewer but larger +patches of gas. Due to the stochastic nature of star formation and +supernova feedback that impact the local gas properties (hence the +SMBH seeding), we observed a large variability in the efficiency of +AGN feedback in this case. For the new suite of simulations dis- +cussed in this paper, we followed a more systematic investigation +into the impact of seeding on the proto-cluster region. In particular +we studied the following scenarios where we varied peak density and +saddle thresholds but kept all other parameters fixed: +• 𝜌peak = 0.5 ¯𝜌, 𝜌saddle = 2 ¯𝜌: Phew parameters as the original +Rhapsody-G setup, with ¯𝜌 = Ω𝑚𝜌𝑐 the mean matter density. +• 𝜌peak = 8 ¯𝜌, 𝜌saddle = 20 ¯𝜌. With a higher 𝜌peak value, only +the highest density regions in the simulation are probed. In that case, +smaller gas patches are selected but spatially more frequent. Thus, +it allows to seed more SMBHs in the simulation compared to the +original Rhapsody-G configuration. +• 𝜌peak = 8 ¯𝜌, 𝜌saddle = 200 ¯𝜌. We increase the saddle density +threshold by a factor of 10. As the result, a much lesser number of +peaks are merged which results in an increased number of SMBH +seeds. +• 𝜌peak = 8 ¯𝜌, 𝜌saddle = 15 ¯𝜌, with a lower saddle threshold which +induce more peak merging hence a lowered number of SMBH seeds +in the simulation. +We show the impact of these choices on the enclosed total stel- +lar mass in the proto-cluster region in Fig. 1 at 𝑧 = 2. At that time, +we find 21, 102, 168, 113 SMBHs of mean masses 3.5 × 108 M⊙, +1.8×108 M⊙, 1.6×108 M⊙ and 2.1×108 M⊙ inside the virial radius +respectively in the above-mentioned simulations. It demonstrates the +tight connection of the 𝜌peak parameter with the total number of +created sinks and the mean mass. The Fig. 1 clearly indicates the +resulting effect on the star formation suppression in the proto-cluster +environment: The simulations hosting a higher number of SMBHs +(being also spatially more frequent), shows a greater amount of AGN +feedback energy injected in haloes. As a result, this more profuse +AGN heating will reduce the gas cooling in haloes which decrease +the accretion of cold gas onto the central SMBH. The resulting mass +accretion rates are seen to be inversely proportional to the number of +SMBHs in our simulations. In consequence, the total stellar mass in +the proto-cluster is consistently reduced with an increasing number +of SMBHs in the simulations. We see that the total stellar mass for +the simulation using 𝜌peak = 8 ¯𝜌, 𝜌saddle = 200 ¯𝜌 is reduced by a +factor of 5 while the number of SMBHs is increased by the same +factor approximately. The total stellar mass in the proto-cluster can +be directly controlled by the number of SMBHs seeded in the simu- +lations. +Galaxy masses at 𝑧 = 2 were found to be in agreement with abun- +dance matching results with the use of 𝜌peak = 8 ¯𝜌, 𝜌saddle = 20 ¯𝜌 +101 +102 +103 +r[kpc] +1011 +1012 +M*( 0 where we see dramatically boosted mass growth. In the +case of a strong decay with 𝑓𝑑 = 1, SMBHs are essentially pinned +to the halo centers at most times. The most massive SMBH accretes +gas at high redshift and experiences, below 𝑧 = 5, frequent mergers +which leads to a very massive central SMBH by 𝑧 = 2. However, +the rapid mass growth of this central SMBH at high redshifts is also +responsible for a very efficient and early AGN feedback (which peaks +at 𝑧 ∼ 4), thus effectively strangulating the growth of the other black +holes. +To reach middle ground between the artificial pinning and the +large swinging of black holes, we found 𝑓𝑑 = 0.1 to yield reasonable +results, but more detailed investigations to tune this parameter might +be helpful in the future. In fact, observations of AGNs in dwarf +galaxies show that BHs are not located at the centers of their host +galaxies with an offset between tens of parsecs and a few kiloparsecs +(e.g. Shen et al. 2019; Reines et al. 2020; Mezcua & Domínguez +Sánchez 2020). A detection of an isolated stellar-mass black hole +MNRAS 000, 1–30 (2022) + +8 +A. Pellissier et al. +100 +fgas(< r) / (Ωb/Ωm) +z = 2 +101 +102 +103 +r [kpc] +1011 +1012 +M∗(< r) [M⊙] +fd = 0.0 +fd = 1.0 +fd = 0.1 +Figure 3. Gas depletion (top) and stellar mass (bottom) radial profile for the +simulations with no (black, 𝑓𝑑 = 0), strong (grey, 𝑓𝑑 = 1) or mild (blue, +𝑓𝑑 = 0.1) tidal descent measured at 𝑧 = 2. We show the universal baryon +fraction, Ωb/Ωm, with the horizontal grey line. We notice that simulations +including the SMBH orbital decay (grey and blue) show lower amount of gas +in the ICM (∼40 per cent). The greater is the SMBH decay, the stronger is the +stellar mass reduction inside the proto-cluster (40 per cent for 𝑓𝑑 = 0.1 and +60 per cent for 𝑓𝑑 = 1.0). +located ∼1.6 kpc away from the galactic center of the Milky Way +has been recently reported by Sahu et al. (2022). Recent simulations +(Pfister et al. 2019; Bellovary et al. 2021, 2019; Boldrini et al. 2020; +Ma et al. 2021) show that BHs in dwarf galaxies are expected to be +wandering around the central regions after the occurrence of mergers +or due to tidal stripping or dynamical friction heating. We observe +in the 𝑓𝑑 = 0.1 case a steadier mass growth of SMBHs which is +mainly driven by accretion of gas down to 𝑧 ∼ 3. As a result the +ICM is heated more gradually by AGN activity, cold gas clumps +can form and be later accreted onto SMBHs. As a consequence, we +observe at 𝑧 ≲ 3 a boosted gas accretion in the less massive black +holes, compared to the simulation with 𝑓𝑑 = 1, by almost an order +of magnitude. +Impact on gas and stars. Finally, in Fig. 3, we show the gas deple- +tion profile as well as the cumulative stellar mass in the proto cluster +region at 𝑧 = 2. At this time, the proto-cluster has a virial radius +of ∼500 kpc. Clearly, in the 𝑓𝑑 = 0 case, the AGN has not heated +the proto-ICM leading to a very high gas fraction at all radii. With +enforced orbital decay, the AGNs become active and we observe as +a consequence a stark reduction of the gas fraction. Thanks to the +tidal descent of SMBHs, AGN feedback is able to deplete the gas +from the central region and efficiently offset radiative losses in the +forming proto-cluster. In the lower panel of Fig. 3, we can see the +resulting reduction of the stellar mass formed. We observe, inside +the virial radius, a reduction by a factor of 40 and 60 per cent for +the simulations with 𝑓𝑑 = 0.1 and 𝑓𝑑 = 1 respectively. This result +is largely consistent with the recent work of Bahé et al. (2021) who +find that the magnitude of the AGN feedback suppression depends +on the ‘drift speed’ towards the center, where a slower SMBH drift +speed toward the halo center in their case also leads to systematically +higher stellar masses. +This new sub-grid model is a first step towards a more physical +solution to the ‘sinking problem’ of SMBH in numerical simulations. +We studied here, the dramatic effect it can have on the SMBH mass +growth, hence AGN activity, which induces amplified gas depletion +and a greater star formation quenching. The dimensionless 𝑓𝑑 pa- +rameter was tuned to reproduce the observed values of stellar masses. +We note here, that in our Rhapsody-C simulation benefiting from +this new sub-grid model, the Booth & Schaye (2009) boost (used +in the previous Rhapsody-G simulations) was dropped. Indeed, the +found SMBH accretion rates are already high enough once the sink +particles are more stably confined to the gas rich centre of haloes. +3.3 Delivering AGN feedback +AGN feedback is believed to proceed in two distinct modes (Best +& Heckman 2012). The quasar mode (or thermal) feedback occurs +when the gas accretion is comparable to the Eddington limit. A large +amount of radiation is emitted from the accretion region which is +able to photoionise and heat the gas in the BH vicinity. In con- +trast, the radio mode (or kinetic) feedback, preferentially triggered +during low-accretion-rate episodes, drives powerful well-collimated +radio-emmiting jets coinciding with cavities in the X-ray emission +(McNamara & Nulsen 2007; Fabian 2012). In some cases both mech- +anisms can be found in the same object (i.e., radio-loud quasar, see +e.g. Bañados et al. 2021). +3.3.1 Implementation +Thermal feedback is usually implemented in astrophysical codes +through the injection of energy or momentum in the surrounding +gas (e.g. Schaye et al. 2015; McCarthy et al. 2017; Tremmel et al. +2017). Radio-mode feedback is often implemented as a second sub- +resolution feedback channel once the accretion rate falls below a +threshold value (e.g. Dubois et al. 2014; Weinberger et al. 2017; +Henden et al. 2018). In this work, we focus purely on thermal feed- +back and will come back to the impact of kinetic AGN feedback in +future work. +Once an AGN event is triggered, in the thermal feedback model, +the released energy is assumed to thermalise within the ‘sink sphere’ +(defined as a sphere of radius 4 high-resolution elements i.e. 4Δ𝑥 +around the SMBH particle) thus effectively leading to an increase in +thermal energy in those cells. Even though these are operations at +the resolution level, multiple ways to distribute this energy among +those few cells are possible, with important consequences. We will +focus on two distinct weighting schemes here. +Mass-weighted (MW) injection. Here the total AGN energy 𝐸AGN +is injected at every fine time step proportionally to the gas mass in a +cell 𝑖 inside the sink sphere as +𝐸AGN,𝑖 = 𝐸AGN +𝜌𝑖Δ𝑥3 +𝑖 +� +𝑖 𝜌𝑖Δ𝑥3 +𝑖 +, +(19) +MNRAS 000, 1–30 (2022) + +The Rhapsody-C simulations +9 +In this case, energy is preferentially injected in denser regions (with +shorter cooling times). The MW scheme predominantly heats the +accretion region fuelling the central SMBH growth and thus reduces +future accretion. +Volume-weighted (VW) injection. Here the AGN energy is injected +at every fine time step proportionally to the volume of the cell 𝑖 inside +the sink sphere as +𝐸AGN,𝑖 = 𝐸AGN +Δ𝑥3 +𝑖 +� +𝑖 Δ𝑥3 +𝑖 +, +(20) +Compared to the MW scheme, here relatively more energy is given +to lower density cells, which for a given energy, leads to a higher +cell temperature. As a result stronger outflows through lower density +regions can be driven, and the immediate accretion gas supply is less +affected. +3.3.2 Validation in simulations +In Fig. 4 we show the effect of the mass-weighted (MW) compared +to the volume-weighted (VW) energy injection on both the ther- +modynamics of the intra-cluster gas and the stellar content of the +cluster over cosmic time. In both simulations, the energy accumu- +lation threshold Δ𝑇 is kept constant (cf. Table 1). We find that the +stellar content in the cluster has been reduced by a factor of ∼ 6−7 at +𝑧 = 0 (𝑅vir ∼ 2𝑅500) for the simulation using the VW AGN feedback +model compared to the MW model (top panel of Fig. 4). +Regarding the ICM, the volume-weighted entropy profiles of the +MW or VW simulations differ at all redshifts and become similar +only at 𝑧 = 0, except in the core (∼ 0.1𝑅500). Outside the core (i.e. +𝑟 > 0.1𝑅500), the entropy profiles of the MW simulation do not +significantly change out to the virial radius (i.e. 1.6𝑅500 and 1.9𝑅500 +at 𝑧 = 3 and 𝑧 = 0 respectively). At the same time, the VW simulation +shows a higher ICM entropy at high redshifts, which drops by 𝑧 = 0.5 +to values similar to the MW simulation. Similarly, the gas fraction +drops below the universal Ω𝑏/Ω𝑚 value for the VW simulation at +𝑧 > 0.5 whereas the MW simulation shows a higher amount of +gas. The higher entropy and lower gas fraction observed for the VW +simulation shows the efficiency of VW AGN deposition at heating +the ICM consistent with a strong star formation quenching. +From 𝑧 = 0.5, we observe an earlier flattening of the entropy profile +in the core of the VW simulation compared to the MW simulation, +but both settle with a similar core entropy of ∼ 100 keV cm2 at +𝑧 = 0. In other words, the VW AGN feedback model can prevent +gas cooling in the core from 𝑧 = 1 in contrast to the MW model. +The reason behind is that the MW model deposits the AGN feedback +energy preferentially in the dense accretion region and therefore has +difficulty to escape from the sink accretion sphere. Meanwhile, the +gas continues to cool outside the accretion region and star formation +proceeds at high rates. Thanks to the large reservoir of cold gas +surrounding the central SMBH, the AGN activity remains energetic +down to 𝑧 = 0 and can therefore gradually heat the cluster core until +it reaches a core entropy comparable to the VW simulation. +Despite the relative similarity of the entropy profiles at 𝑧 = 0, +the evolution of the AGN activity differs significantly. In the MW +simulation, the inefficient AGN feedback at early times fails to +regulate the star formation in the proto-cluster which leads to +over-massive cluster galaxies, whereas in the VW simulation, we see +a strong quenching of the star formation at earlier times as a result +of the VW deposition injecting more energy in the more diffuse gas. +These differences can be seen in the gaseous and stellar maps at +109 +1010 +1011 +1012 +1013 +M∗(< r) [M⊙] +MW +VW +10−1 +100 +K(r)/K500 +10−2 +10−1 +100 +r/R500 +10−1 +100 +fgas(< r) / (Ωb/Ωm) +z = 3.00 +z = 2.00 +z = 1.00 +z = 0.50 +z = 0.25 +z = 0.00 +Figure 4. Total enclosed stellar mass (top), ICM entropy (middle) and en- +closed gas fraction (bottom) radial profiles. Radii have been normalised to +𝑅500 corresponding to the radius enclosing 500 times the critical density at +the indicated redshift (we have 𝑅vir ∼ 1.7 − 2𝑅500). Solid and dotted lines +show the radial profiles of the MW and VW simulations respectively and line +color indicates the redshift at which the radial profile is computed. The en- +tropy profiles has been rescaled by the self-similar value 𝐾500 of Nagai et al. +(2007) to compare profiles at different redshifts more easily. The universal +baryon fraction is shown in the bottom pannel by the horizontal grey line. We +see the greater effect of the AGN with a volume weighted energy deposition +in quenching star formation at all radii. The entropy profiles indicate that the +VW AGN is more efficient in heating the ICM up to large radii at 𝑧 > 1. It +also allows an earlier transition to a NCC cluster by 𝑧 = 0.5, while the MW +simulation still shows in the core low entropy and high 𝑓gas values. +MNRAS 000, 1–30 (2022) + +10 +A. Pellissier et al. +𝑧 = 0 of the VW (left) and MW (center) simulations shown in Fig. 5. +Thanks to the heating at large radii enabled by the VW deposition +model, the pile up of cold gas observed in the MW simulations +has been prevented. Consequently, the stellar content in the cluster +has been greatly reduced in the VW simulation as well as a lower +number of cluster galaxies formed. +In this study, we do not vary the size of the AGN energy injection, +However using a VW deposition scheme, Dubois et al. (2012) showed +that the size of the AGN feedback deposition significantly impacts +the evolution of the SFR, galaxy and SMBH masses. Indeed, a larger +AGN injection region extends to less dense regions, hence far away +from the galaxies, which are more easily affected by AGN feedback +- which is less the case with a MW deposition. +3.3.3 On the energy accumulation threshold +Le Brun et al. (2014) showed that the entire gas profile can be varied +by tuning the energy accumulation threshold of the feedback model +of Booth & Schaye (2009). In contrast, Hahn et al. (2017) found that +none of the AGN models tested on a CC cluster of the Rhapsody-G +sample had a significant effect outside the core. +Similarly, we would like to test the robustness of the ICM +properties to changes in the AGN energy injection threshold, +Δ𝑇, over two orders of magnitude. We emphasise that this pa- +rameter does not control the total energy injected in an AGN +blast, but only its proportions: a higher value of Δ𝑇 results in a +less frequent but more energetic AGN blast and reciprocally, a +lower Δ𝑇 value induces more frequent and less energetic AGN blasts. +We simulate the same halo with the thermal AGN model using a +MW energy deposition and change the energy accumulation thresh- +old Δ𝑇. The simulations presented in this work all use Δ𝑇 = 107 K, +for this section we ran two additional simulations with Δ𝑇 = 106 K +(MW6) and 108 K (MW8) that we compare with the fiducial MW +simulation presented in the previous Section 3.3.2. We show in Fig. 6, +the evolution of the gas fraction as a function of the cluster mass mea- +sured in the 𝑅500 aperture. +We see that that the simulations with higher (MW8) or lower +(MW6) Δ𝑇 value shows a systematically higher gas fraction at all +cluster masses compared to the fiducial MW simulation. Hence, we +observe a non-linear dependence of the gas fraction on the energy +accumulation threshold. This is at odds with the findings of Le Brun +et al. (2014) who report a decreasing 𝑓gas with increasing Δ𝑇 values. +With a higher energy accumulation threshold, the AGN feedback in +the MW8 simulation show the highest amount of gas. Due to energetic +AGN blasts, gas is efficiently depleted from the core region. However, +as the AGN blasts are less frequent, the ICM cools efficiently and gas +condenses towards the cluster core between consecutive AGN blasts. +As a result, a cool core forms which can no longer be impacted by +the AGN activity. With a lower energy accumulation threshold, AGN +feedback events are not energetic enough (albeit more frequent) to +counterbalance the gas cooling outside the core in the MW6 simu- +lation. Consequently, it results in a greater amount of gas within the +𝑅500 region, compared to the MW simulation. +We see that a higher/lower Δ𝑇 does not systematically lead +to smaller/larger gas fractions in such a simulation. Moreover, +it does not significantly affect the overall amount of gas in the +cluster. Changes in the energy accumulation threshold only induce +a variation of 10 per cent at most of the cluster gas content. This is +considerably less compared to the overall 30 per cent gas fraction +reduction observed by Le Brun et al. (2014) when increasing by 0.5 +dex the Δ𝑇 value. This study is consistent with the findings of Hahn +et al. (2017) and shows that the Δ𝑇 parameter indeed has little impact +of the gas content of GCs in our simulations. However, compared +to Hahn et al. (2017), the differences that we observe originate +from the inclusion of our SMBH orbital decay model (presented in +Section 3.2) which keeps SMBHs close to their host halo centre, +resulting in a greater effect on the gas outside the cluster’s core. +3.3.4 Summary on the SMBH modelling +In this section, we discussed the response of the stellar and gaseous +cluster components to small changes of the SMBH and AGN feed- +back modelling. The seeding of SMBHs efficiently regulates the +star formation in the ICM: less massive SMBH seeds trigger more +spatially frequent SMBH formation hence more efficient AGN gas +heating and SF quenching. +Enabled by our new model using the tidal field information, the or- +bital decay of SMBH towards the potential minimum can be robustly +controlled. SMBH stable orbits enable a greater gas accretion. The +resulting enhancement of the AGN activity quenches the SF in the +ICM and deplete gas to large cluster radii. +Regarding AGN feedback, the energy injection scheme appreciably +impacts the ICM gas heating and controls its thermal evolution. When +the AGN feedback energy is delivered proportionally to the local gas +density, it remains confined to the core region but gradually pro- +gresses late to more intermediate radii. With a homogeneous AGN +energy injection, more energy is deposited in diffuse gas region and +thus can escape the cluster core and heat a large radii without pre- +venting the accretion of cold gas fuelling the central SMBH activity. +In that case, the ICM is almost entirely impacted by an early strong +heating which prevents the build up of cold gas and SF later on. +On the other hand, the amount of gas is relatively robust to changes +in the energy accumulation threshold (i.e. the AGN duty cycle and +energetics) of the purely thermal AGN model. +4 ANISOTROPIC THERMAL CONDUCTION +In the simulations of Hahn et al. (2017), the gaseous content of +the Rhapsody-G haloes was found to suffer from the over-cooling +problem with a too gas-rich ICM. None of their AGN feedback +models were able to impact the gas outside the cluster core to +bring it towards more realistic values. This suggested that AGN +feedback was likely not the sole solution to regulating galaxy cluster +thermodynamics. Thanks to improvements in our models, we have +seen in the previous section that AGN feedback is now able to +impact the gas on large scales but its impact remains extremely +dependent to the choice of parameters. +Anisotropic thermal conduction, in conjunction with AGN heating +and radiative cooling, likely plays an important role in setting +the gas properties of clusters (Kannan et al. 2017; Barnes et al. +2019). However, in the presence of thermal conduction, the heat +buoyancy instability (HBI) (Quataert 2008; Parrish et al. 2009) can +reorient the magnetic field lines to tangential configurations leading +to the suppression of conductive heat fluxes (Parrish et al. 2009; +Bogdanović et al. 2009). Simulations of Ruszkowski et al. (2011) +showed that turbulence can counteract the HBI and re-randomise the +magnetic field. The recent work of Beckmann et al. (2022) showed +in idealised massive galaxy cluster simulations that spin-driven +AGN feedback cannot counteract alone the HBI in the cluster center +and suggested that volume-filling turbulence is needed to restore +MNRAS 000, 1–30 (2022) + +The Rhapsody-C simulations +11 +VW +1 Mpc +MW +MC +29 +28 +27 +26 +25 +24 +23 +log10( +gas/gcm 3 ) +VW +1 Mpc +MW +MC +Figure 5. We show the maximal intensity maps of the gas (top panels) and stellar (bottom panels) density in a box of 5.9 Mpc (top) and 1.9 Mpc (bottom) on +the side at 𝑧 = 0 respectively. We have from left to right, the VW, MW and MC simulations. We can see the effect of the VW AGN feedback model at removing +the cold dense gas clumps compared to the MW model. We can also notice a similar effect, albeit lower, of the anisotropic thermal conduction in the MC +simulation at preventing the pile up of gas in dense clumps. As a result, the stellar content in the VW and MC simulations is greatly reduced compared to the +MW simulation. +significant thermal conduction. However, in isolated CC cluster +simulations, Yang & Reynolds (2016) found that magnetic tension +can suppress a significant portion of the HBI-unstable modes which +completely inhibit or significantly impair the HBI for realistic field +strengths on scales smaller than ∼ 50 − 70 kpc. Therefore, if thermal +conduction is not suppressed by the HBI, it can transport heat to +redistribute it in the ICM. It could help alleviating the SMBH/AGN +model parameter dependency found in the previous sections and +help to reach ICM regulation. +The MW simulation shows greater temperature gradients com- +pared to the VW, hence we expect that thermal conduction has a +larger effect in that case. We investigate to what extent anisotropic +thermal conduction (ATC) is able to offset radiative losses in the ICM. +In idealised adiabatic simulations, we have observed that anisotropic +thermal conduction can act as an efficient cooling or heating source +depending on the sign of the ICM temperature gradient (see Ap- +pendix A) where heat is transported from or to the cluster outskirts +in order to flatten out temperature inhomogeneities. In the previous +section, we have seen that a mass-weighted AGN feedback model +deposits its energy predominantly in the gas-rich regions. It causes +the AGN feedback energy to stay confined close to the central SMBH +with difficulty to escape the accretion region. We would like to deter- +mine whether ATC can transport the centrally injected AGN energy +on large distances to regulate the high cooling losses and star for- +mation rates. Ruszkowski et al. (2011) found that ATC was able to +noticeably reduce the effective radiative cooling driven gas accretion +in idealised cool core cluster simulations. In that context, we explore +the effect of ATC on the MW simulation with the shortest cooling +times in the core. We label the simulation with ATC and a mass- +weighted AGN deposition model as MC. The effective conductivity +is given in equation 12 for which we recall the canonical Spitzer +(1962) value, +𝜅Sp = 𝑛ekB𝐷𝑐, +(21) +where 𝐷𝑐 is the thermal diffusivity +𝐷𝑐 = 8 × 1031 +� 𝑘B𝑇e +10 keV +�5/2 � +𝑛e +5 × 10−3 cm−3 +�−1 +cm2 s−1. +(22) +We show in the top panel of Fig. 7 the effect of ATC on star +formation and the distribution of gas within the ICM. We also show +in the right panels of Fig. 5, the impact of ATC on the distribution +of stars and gas compared to the simulation without ATC (middle +panels) at 𝑧 = 0. With ATC, we observe the lower amount of gas +clumpiness in a more diffuse ICM which hosts less massive galaxies. +As seen in the profiles, the amount of stars in the MC simulation +has been reduced by ∼40 per cent already by 𝑧 = 3, before the peak +of the AGN activity. By smoothing out temperature gradients in the +ICM, ATC prevents the formation of cold gas clumps where stars +should form. Therefore, the lower amount of stars in the cluster is +MNRAS 000, 1–30 (2022) + +12 +A. Pellissier et al. +1014 +1015 +Mtot,500 [M⊙] +0.10 +0.11 +0.12 +0.13 +0.14 +0.15 +0.16 +0.17 +fgas,500 +MW6 +MW +MW8 +Eckert+19 - X-COP +Figure 6. Evolution of the cluster gas fraction as a function of its total +mass measured within 𝑅500 for the simulation using a AGN energy injection +threshold of Δ𝑇 = 106 (MW6, orange), 107 (MW, dark red) and 108 K (MW8, +black). The circles show the total gas fraction while the lighter-colored crosses +show only the X-ray emitting gas fraction (i.e. the gas with 𝑇 > 0.5 keV). We +compare our data to the hydrostatic gas fractions and total masses corrected +for the non-thermal pressure of the X-COP sample (Eckert et al. 2019) and +show the universal baryon fraction Ωb/Ωm = 0.1586 by the gray horizontal +line. Variations of the Δ𝑇 parameter by an order of magnitude, compared +to the MW simulation, increase by 10 per cent at most the gas fraction in +the 𝑀500 > 4 × 1914M⊙ range. The trend in gas depletion is observed to be +non-linear with respect to Δ𝑇 . +not due to an enhanced AGN activity but due to a suppression of +cold gas clump formation hence star formation. The entropy profiles +at 𝑧 = 3 shown in the middle panel of Fig. 7, reveal a higher core +entropy in the MC simulation as ATC transports heat to the central +region from the reservoir of hot gas at larger radii. In this case, +thermal conduction contributes in fact to the suppression of cold gas +accretion onto the central SMBH. In consequence, the AGN activity +declines and leads to a build-up of cold gas at later times, as we can +see from the gas fraction profiles at 𝑧 = 0, where the MC simulation +shows a higher amount of gas at all radii with a core contraction. +Kannan et al. (2017) found that ATC isotropises the injected AGN +energy and enhances its coupling with the ICM. They found that +the SFR is hence reduced by an order of magnitude while the overall +amount of AGN feedback energy deposited in the ICM is lower. They +also show that the earlier quenching comes with an earlier transition +to a NCC cluster. From the gas depletion profiles at 𝑧 = 0.25 shown +in the bottom panel of Fig. 7, we also witness an earlier transition to +a NCC cluster where the MC simulation shows a lower amount of +gas (and a higher entropy) in the core compared to the MW simula- +tion which does not include conduction. Additionally, we also found +lower SFRs in the galaxies within the halo by roughly a factor of +two which is however lower than the order of magnitude reduction +found by Kannan et al. (2017). In spite of these similarities, we do +not observe the reported strong quenching induced by a greater AGN +heating efficiency. We find that the simulation with conduction shows +a lower amount of injected AGN feedback energy by almost a factor +of 2. In our simulations, conduction reduces the AGN activity by +109 +1010 +1011 +1012 +1013 +M∗(< r) [M⊙] +MW +MC +10−1 +100 +K(r)/K500 +10−2 +10−1 +100 +r/R500 +10−1 +100 +fgas(< r) / (Ωb/Ωm) +z = 3.00 +z = 2.00 +z = 1.00 +z = 0.50 +z = 0.25 +z = 0.00 +Figure 7. Similarly to Fig. 4, we show the radial profiles for the simulations +with and without anisotropic thermal conduction, labelled MC and MW re- +spectively. Anisotropic thermal conduction allows to reduce the stellar content +in the intra-cluster medium by a factor of ∼2 already by 𝑧 = 3 due to the +transport of heat in the ICM that prevents the formation of cold gas clumps. +However, it also leads to the reduction of the AGN activity which causes +a build-up of cold gas at later times, as we can see from the gas depletion +profiles at 𝑧 = 0. +lowering the amount of cold gas available in the ICM which should +fuel the SMBH accretion. To summarize, heat transport smoothes +temperature inhomogeneities early-on which decreases star forma- +tion in the ICM. As a result, less cold gas clumps are available in the +ICM for SMBH accretion which results in weakening of the AGN +activity. Consequently, this decline in AGN feedback heating leads +to the contraction of the ICM at low redshifts. +MNRAS 000, 1–30 (2022) + +The Rhapsody-C simulations +13 +5 GALAXY PROPERTIES AND ICM PROFILES +5.1 The stellar mass-to-halo mass relation +We next study the properties of the galaxies in and around our +simulated clusters. Since our simulations have a high-resolution +region of twice the virial radius around each cluster at 𝑧 = 0, we +are probing a wide range of haloes mass which enable a statistical +comparison. A rightful test for the realism of our simulations is +to compare the stellar mass of our galaxies (both centrals and +satellites) as a function of their halo mass with results obtained +using the abundance matching technique (Shankar et al. 2017), the +Universe Machine with the semi-empirical model of Behroozi +et al. (2019), with cluster X-ray mass measurements (Kravtsov et al. +2018) and with the IllustrisTNG100 simulation (Nelson et al. +2018; Marinacci et al. 2018; Springel et al. 2018; Pillepich et al. +2018; Naiman et al. 2018). +In Fig. 8 we plot, for all haloes found in the nine Rhapsody-C +clusters at 𝑧 = 0, the total stellar mass 𝑀∗,200 within 𝑅200, the +radius enclosing 200 times the critical density of the Universe, as a +function of the total halo mass 𝑀200. The halo were identified using +the heavily modified version of Rockstar-Galaxies, a special +version of the original version of Rockstar (Behroozi et al. 2013), +designed to work with AMR tree and to compute a wide range of +observables during the sub-halo finding (for more detail see Hahn +et al. 2017) . +We show the different effect of the energy deposition scheme +of our thermal AGN feedback model in the left (MW) and middle +(VW) panels as well as the addition of thermal conduction (MC, right +panel). We observe a strong reduction of the star formation in the VW +simulations by almost an order of magnitude (see Section 3.3.2). As +the stellar masses are systematically lower for the VW simulations +compared to the IllustrisTNG100 simulation or the relations of +Shankar et al. (2017), Kravtsov et al. (2018) and Behroozi et al. +(2019), the stellar masses of the MW simulation are larger at 𝑀200 > +1013M⊙. On the other hand, the stellar masses of the MC simulations +(left panel), which use anisotropic thermal conduction in contrast to +the MW simulation, show stellar masses in good agreement with the +various studies. We see the effect of thermal conduction at regulating +the star formation in haloes (cf. Section 4). However, despite efficient +gas deletion (cf. right panel of Fig. 10), the VW simulations cannot +reproduce realistic galaxy properties since masses are systematically +too low. +5.2 Structure of the ICM +We perform a comparison of electron number density, pressure, +temperature and entropy in Fig. 9 with numerical simulations, SZ +and X-ray observations. We consider stacked profiles over cosmic +time of all our clusters for each type of simulations (NR, MW, +VW and MC, see details in Table 1) separately in order to quantify +the mean profiles. We selected only snapshots in the 0 ⩽ 𝑧 ⩽ 0.5 +redshift range to be consistent with the studies to which we compare +our data. +In the top left panel of Fig. 9, we compare the electron number den- +sity profiles with the low-𝑧 sub-sample of McDonald et al. (2017) +of (X-ray-selected) galaxy clusters at 𝑧 = 0 − 0.1. Additionally, we +compare our simulations to the publicly available data from the AC- +CEPT Chandra archival project described in Cavagnolo et al. (2009). +From the 242 ACCEPT galaxy cluster profiles and using the right +ascensions, declinations and redshifts, we match 141 objects to the +MCXC sample (Piffaretti et al. 2011) which gives access to X-ray ra- +dius and mass estimates. Cavagnolo et al. (2009) found a bi-modality +in the central entropy excess (𝐾0) distribution with two distinct pop- +ulation separated at 𝐾0 ∼ 30 − 50 keVcm2. We therefore classify +the ACCEPTxMCXC clusters as cool-core (CC) and non-cool-core +(NCC) if the core entropy excess 𝐾0 is respectively below or above +50 keVcm2. +The simulations using the MW AGN model (MW and MC) show a +denser core than the CC population of the ACCEPTxMCXC sample. +In contrast, the VW simulations agree almost perfectly with the +NCC ACCEPTxMCXC population within the 1𝜎 scatter shown by +the ribbon. The VW simulations approach the mean radial profile +of McDonald et al. (2017), but have a flatter core electronic density. +The non-radiative simulation shows the flattest core mean density +profile and is consistent with the NCC ACCEPTxMCXC clusters. +On the other hand, outside the core and especially at 𝑟 > 0.7𝑅500, +our simulations consistently show a steeper decrease of the electron +density with radius. +Our VW and NR simulations are consistent with both the CC +and the NCC ACCEPTxMCXC pressure profiles within scatter as +shown in the top right panel of Fig. 9. The VW and NR simulations +are in good agreement with the mean pressure profiles of Planck +Collaboration et al. (2013), the X-COP sample (Ghirardini et al. +2019) and the MUSIC simulated clusters (Gianfagna et al. 2021) out +to large cluster radii. On the other hand, the MW and MC simulations +show a factor 4 higher core pressure, but, meet at 𝑟 ⩾ 0.2𝑅500 the +VW and MC profiles. +In the ICM temperature profiles shown in the bottom left panel, +both ACCEPTxMCXC clusters and Ghirardini et al. (2019) show +large uncertainties. The CC and NCC populations of the ACCEP- +TxMCXC sample show flat temperature profiles with high ICM tem- +peratures out to 𝑅500. Ghirardini et al. (2019) also show, to a lesser +extend, higher temperatures outside the core compared to our simula- +tions. In the core region, we can see that the VW and NR simulations +approach the NCC mean profiles of the ACCEPTxMCXC sample and +the MW and MC simulations tend towards the NCC mean profile. +In the bottom right panel, the entropy slope of all our simulations +at large radii is consistent with the simulations of Voit (2005) and +the observations of the REXCESS sample (Pratt et al. 2009), but is +found to be steeper than the work of Ghirardini et al. (2019). The +ACCEPTxMCXC entropy profiles shows a shallower slope with a +higher normalisation (consistent with the high ACCEPTxMCXC +ICM temperatures). The VW simulations agree well with the +NCC ACCEPTxMCXC population within scatter while the MW +simulations better match the CC sub-sample and the relation of +Ghirardini et al. (2019). The NR simulations are somehow in +between but the MC shows low core entropy still compatible the CC +ACCEPTxMCXC clusters. +Overall, compared to the ACCEPTxMCMC CC and NCC mean +profiles, we can see that the simulations implementing a VW AGN +feedback model tend to produce GCs with NCCs. On the other hand, +the simulations implementing a MW AGN feedback model, tend to +produce more NCC clusters which is even accentuated by the addition +of thermal conduction. Our ICM radial profiles demonstrate that our +simulations are in relatively good agreement with both observations +and state-of-the-art simulations. We note, however, that the MC and +MW mean density profiles show a slight excess of gas in the core +compared to observations. +MNRAS 000, 1–30 (2022) + +14 +A. Pellissier et al. +Figure 8. Comparison of the 𝑀∗ − 𝑀 relations for the MW (right), VW (middle) and MC (right) simulations for all haloes in the nine Rhapsody-C clusters at +𝑧 = 0. Each color represents one of the nine simulations and we show the central galaxies as well as the satellite populations in transparency. We show the total +stellar mass as a function of the total mass within 𝑅200, the radius enclosing 200 times the critical density. Similarly, we also show the total stellar mass inside +𝑅200 for the haloes of the IllustrisTNG100 simulation with the 1𝜎 scatter ribbon. Note that at group and cluster scales (i.e. 𝑀200 ⩾ 1013M⊙), the relations +of Shankar et al. (2017), Kravtsov et al. (2018) and Universe Machine (Behroozi et al. 2019) indicate for lower stellar masses as they only take into account +the stellar mass of the central galaxies, while the IllustrisTNG100 relation and our data provide the total stellar mass in haloes i.e. the central galaxies and the +intra-cluster light. +r/R500 +10 5 +10 4 +10 3 +10 2 +10 1 +ne(r)E(z) 2 [cm 3] +r/R500 +10 2 +10 1 +100 +101 +102 +P(r)/P500 +10 2 +10 1 +100 +r/R500 +10 1 +100 +T(r)/T500 +10 2 +10 1 +100 +r/R500 +10 2 +10 1 +100 +K(r)/K500 +MW +VW +MC +NR +McDonald+17 - 0.0 +z + 0.1 +ACCEPTxMCXC [CC] +ACCEPTxMCXC [NCC] +Ghirardini+19 +Planck 2014 +Gianfagna+21 +Pratt+10 +Voit+05 +Figure 9. Mean ICM radial profiles of the electronic number density (top left), dimensionless pressure (top right), temperature (bottom left) and entropy (bottom +right) of our VW, MW, MC and NR simulations for 𝑧 ⩽ 0.5 in comparison with the matched sample of ACCEPT (Cavagnolo et al. 2009) and MCXC (Piffaretti +et al. 2011) and the sample of McDonald et al. (2017). We show the best fit thermodynamic profiles of Ghirardini et al. (2019), the pressure profiles of Planck +Collaboration et al. (2013) and Gianfagna et al. (2021) as well as the outer entropy slopes of Pratt et al. (2009) and Voit (2005) +MNRAS 000, 1–30 (2022) + +14 +MW +13 +12 +200 +11 +*W) +log10 +10 +9 +8 +11 +12 +13 +14 +15 +log10VW +11 +12 +13 +14 +15 +log1n (M200 / Mo )MC +UNIVERSE MACHINE +ILLUSTRISTNG100 +Kravtsov+18 +Shankar+17 +11 +12 +13 +14 +15 +log1n (M200 / M)The Rhapsody-C simulations +15 +6 EVOLUTION ALONG CLUSTER SCALING RELATIONS +Well-calibrated scaling relations between observed (X-ray, SZ or +optical) quantities and the total mass of GCs are not only important +to understand the physical processes that give rise to these relations, +but are a crucial ingredient for cosmology (Giodini et al. 2013). +Hydrodynamical simulations can model the complex processes +of structure formation with the inclusion of baryonic physics in a +cosmological context. Having access to the true cluster mass, such +simulations can be used to explore possible biases in the mass +estimation methods and can help to obtain a definitive measure of the +true cluster mass scale to enhance cosmological parameter analysis +using cluster counts (see Pratt et al. 2019, for a review). However, +it is first important to estimate the degree to which numerical +models impact and potentially bias cluster scaling relations before +directly confronting them with observational results. We studied +independently in Section 3.3 and Section 4 the impact of the AGN +deposition schemes and the addition of ATC, respectively, on the +stellar and gaseous content of a single Rhapsody-C halo. However, +the effect of such numerical schemes on the global properties of the +whole Rhapsody-C sample still needs to be assessed. In this section, +we extend the analysis to the full Rhapsody-C sample and quantify +the changes in the cluster scaling relations with the variation of +sub-grid baryonic models. +We summarise the simulations details in Table 1, where we label +the various simulations NR, VW, MW and MC for convenience. In +short, all simulations share the same resolution and numerical strat- +egy, with the same modelling for gas cooling, star formation, stellar +feedback, black hole seeding and growth (except for the adiabatic run, +NR). The only differences are in the AGN energy injection scheme +and whether ATC is included. +6.1 Synthetic X-ray observables +We chose a different methodology from Hahn et al. (2017) to compute +the X-ray observables from the simulation. Instead of using simple +weighting schemes, we produce a synthetic X-ray spectrum from +which the temperature (and gas density) of the ICM can be estimated +as closely as possible to the observer’s methodology. For each cell +in the range 0.15 ⩽ 𝑟/𝑅500 ⩽ 1,8 we read the gas density, tempera- +ture and metallicity from the simulation and compute the emissivity +𝜖(𝑇, 𝑍) (atomic lines and continuum) using tabulated emission mod- +els from the Astrophysical Plasma Emission Code (APEC, version +3.0.9, Smith et al. 2001). In each spectral bin, we compute the photon +emission rates9 𝜙𝑖 of all the cells 𝑖, +𝜙𝑖 = 𝜖(𝑇𝑖, 𝑍𝑖) 𝑛e,𝑖 𝑛H,𝑖 Δ𝑥3 +𝑖 . +(23) +We sum the individual spectra of all gas cells in the core-excluded +region inside 𝑅500 to produce amockX-ray spectrum(moredetails on +the computation of X-ray observables are given in Appendix B). We +then fit the obtained spectrum, using MCMC via the emcee python +library (Foreman-Mackey et al. 2013), with a single temperature +APEC model generated with PyAtomDB (Foster & Heuer 2020). The +X-ray luminosity is obtained by integrating over the spectra measured +from the simulation (not from the best-fit model) in the soft X-ray +8 We exclude the core (𝑟 < 0.15𝑅500) from the analysis to avoid being biased +by the presence of any cool-core or central AGN activity. See in Appendix C +for more details. +9 We omit any redshift or column density dependence +𝑌 , 𝑋 +𝛽ss +𝛾ss +𝑌0 +𝑋0 +𝑇X, 𝑀 +2/3 +2/3 +5.0 keV +5.0 × 1014M⊙ +𝐿X, 𝑀 +1 +2 +4.0 × 1044 erg/s +5.0 × 1014M⊙ +𝐿X,bol, 𝑀 +4/3 +7/3 +1.0 × 1045 erg/s +5.0 × 1014M⊙ +𝐿X, 𝑇X +3/2 +1 +4.0 × 1044 erg/s +5.0 keV +𝐿X,bol, 𝑇X +2 +1 +1.0 × 1045 erg/s +5.0 keV +𝑀gas,X, 𝑀 +1 +0 +5.0 keV +5.0 × 1014M⊙ +𝑌SZ, 𝑀 +5/3 +2/3 +40 kpc2 +5.0 × 1014M⊙ +𝑌X, 𝑀 +5/3 +2/3 +3.0 × 1014M⊙keV +5.0 × 1014M⊙ +Table 2. Scaling relations in the form 𝑌 ∝ 𝑋 𝛽 +ss 𝐸 (𝑧)𝛾ss expected from the +self-similar theory. We note𝑌 the integrated Comptonization Y parameter and +𝑀 the total cluster mass, 𝑇X, 𝐿X and 𝐿X,bol, the X-ray temperature and the +soft band and bolometric luminosity. The last two columns list the pivot values +used for the fitting the (non-self-similar) scaling relations (equation 24). +(𝐿X,500) and bolometric band (𝐿X,bol,500) i.e. 0.5–2 keV and 0.0– +100.0 keV respectively, with no instrument spectral response. +6.2 X-ray scaling relations +We use the publicly availabe Bayesian regression scheme LIRA of +Sereno (2016b,a) for our cluster scaling relation analysis. We con- +sider a power-law function of the form 𝑌 ∝ 10𝛼𝑋𝛽𝐸(𝑧)𝛾 that de- +scribes the average scaling relation of a given cluster observable 𝑌 +with another cluster observable 𝑋. For the fitting procedure, we fo- +cus on logarithms of the cluster observables of 𝑌 and 𝑋 which are +normalised by their respective pivot values 𝑌0 and 𝑋0 : +log 10 +� 𝑌 +𝑌0 +� += 𝛼 + 𝛽 log 10 +� 𝑋 +𝑋0 +� ++ 𝛾𝐸(𝑧) ± 𝜎𝑌 |𝑋, +(24) +where 𝛼 is the normalisation, 𝛽 is the slope, 𝜎𝑌 |𝑋 is the intrinsic +scatter of 𝑌 at fixed 𝑋 and 𝐸(𝑧) = 𝐻(𝑧)/𝐻0 is the expansion func- +tion, which describes the evolution of the Hubble parameter with +redshift for a given cosmology. We fix its evolution with redshift to +the self-similar expectation, i.e. 𝛾 = 𝛾ss, and we chose pivot values +to be the average sample values which we list in Table 2. We ad- +ditionally fit the intrinsic scatter of 𝑌 at fixed 𝑋. As we are using +‘true’ observables computed directly from the simulation, we do not +assume any selection effects or prior distributions on the regression +parameters. +6.2.1 The 𝑓gas − 𝑀tot relation +We start our study of the cluster scaling relations with the ratio of the +gas mass to the total cluster mass. This quantity is a crucial ingredient +for cosmology because in combination with external information on +the baryon density parameter Ωb, it has provided some of the earliest +and most robust constraints on the cosmic matter density Ωm and +dark energy (Ettori et al. 2009; Allen et al. 2011; Mantz et al. 2022, +e.g. for recent measurements). Moreover, a constant gas fraction is +a key assumption in the self-similar model of Kaiser (1986) from +which the cluster scaling relations ensue but observations indicates +for a mass-dependence (Pratt et al. 2009; Lovisari et al. 2015; Eckert +et al. 2016). +In this section, we discuss how this quantity evolves with mass +when different baryonic physical models are considered. Interest- +ingly, we will see in the following sections that when discussing +the properties of other cluster scaling relations, their outcomes can +already be predicted by the mean of the gas fraction evolution. A +detailed characterisation of this dependence will be investigated in +MNRAS 000, 1–30 (2022) + +16 +A. Pellissier et al. +1014 +1015 +Mtot,500 [M⊙] +0.08 +0.10 +0.12 +0.14 +0.16 +0.18 +0.20 +fgas,X,500 +1014 +1015 +Mtot,500 [M⊙] +fgas,500 +1014 +1015 +Mtot,500 [M⊙] +fb,500 +MW +VW +MC +NR +Lovisari+15 +Eckert+19 - X-COP +Figure 10. Similarly to Fig. 6, we show here the fraction of the X-ray emitting gas (left), gas fraction (middle) and baryonic fraction (right) as a function of the +mass, all measured within 𝑅500. The horizontal grey line show our cosmic baryon fraction of 0.1586. We distinguish the simulation types (MW, VW, MC or +NR) by using different colors (blue, red, purple and black respectively) while individual Rhapsody-C haloes have different symbols. The colored solid lines and +shadding show the fitted relations reported in Table 3 with the 1𝜎 statistical error. The X-ray emitting gas fraction is an increasing function of the total mass +for full-physics simulations except in the non-radiative case (NR), which does not implement physical processes that deplete gas from cluster central regions. In +the middle panel, when we consider all gas and not only the hot gas, the 𝑓gas,500 values show a constant evolution with mass, i.e. time, except in the case of the +VW simulations. The early and strong AGN heating happening at low cluster masses efficiently depletes gas from the 𝑅500 region and quenches star formation. +However, in the VW simulation, due to strong cooling losses at high masses (later times), gas condenses to the cluster centre and the baryonic fraction rises to +values comparable to the other simulations (NR, MW, MC). On the other hand, the origin of the offset between the gas fraction of the simulation without (MW) +and with ATC (MC) comes from the ability of the thermal conduction to prevent the formation of stars in the ICM, leading to a gas rich ICM. +future work. +We show in the left panel of Fig. 10 the X-ray emitting gas fractions +measured inside 𝑅500, 𝑓gas,X,500, of all Rhapsody-C haloes for all +MW (blue), VW (red), MC (purple) and NR (black) simulations as a +function of their total mass enclosing 500 times the critical density. +This X-ray emitting gas fraction is the ratio of the mass of the hot gas, +i.e. with 𝑇gas > 0.5 keV, to the total mass inside 𝑀500. We compare +our data to the hydrostatic gas fractions and total masses corrected +for the non-thermal pressure of the X-COP sample (values are taken +from Table 2 of Eckert et al. 2019) and with the relation of Lovisari +et al. (2015). At the high-mass end, our simulations are in good +agreement with the results of Eckert et al. (2019) but systematically +show higher gas fraction than the result of Lovisari et al. (2015), +which use hydrostatic mass estimates. +Each of the simulation types occupies a different place in the +𝑓gas,X,500 − 𝑀500 plane. The NR simulations systematically show +the highest 𝑓gas,X,500 values, as non-gravitational processes that +could be responsible for any gas depletion are not included. On the +other hand, the radiative runs reveal systematically lower 𝑓gas,X,500 +for lower mass haloes. The VW runs show the steepest increase with +mass to reach comparable values to the NR haloes, but their values +are still lower compared to the NR 𝑓gas,X,500 values. The MW and +MC simulations also show positive slopes (although shallower than +for VW) with increasing mass to reach different (X-ray emitting) +gas fractions values at the highest halo masses, with the MW ones +being the lowest. We list the slopes and normalisations we found +in Table 3 along scaling relations derived in observational studies. +The slopes of our MW and VW simulations are in agreement with +the slopes of Sun et al. (2009); Lovisari et al. (2015); Ettori (2015); +Table 3. Fitted normalisation 𝛼 and slope 𝛽 parameters using LIRA for haloes +with 𝑧 ≲ 1.5 along their standard deviations. In the bottom part of the table, +we give the slopes found by observational studies. +𝑓gas,X,500–𝑀500 +𝛼 +𝛽 +MW +0.873 ± 0.004 +0.145 ± 0.025 +VW +0.973 ± 0.004 +0.221 ± 0.021 +MC +0.951 ± 0.009 +0.103 ± 0.039 +NR +1.030 ± 0.006 +0.015 ± 0.027 +Sun et al. (2009) +0.135 ± 0.030 +Lovisari et al. (2015) +0.16 ± 0.04 +Ettori (2015) +0.198 ± 0.025 +Eckert et al. (2016) +0.21 ± 0.11 +𝑓gas,500–𝑀500 +𝛼 +𝛽 +MW +0.930 ± 0.004 +0.015 ± 0.021 +VW +1.010 ± 0.003 +0.189 ± 0.018 +MC +0.991 ± 0.007 +−0.016 ± 0.031 +NR +1.070 ± 0.006 +0.007 ± 0.026 +𝑓b,500–𝑀500 +𝛼 +𝛽 +MW +1.100 ± 0.002 +0.007 ± 0.014 +VW +1.060 ± 0.003 +0.167 ± 0.018 +MC +1.090 ± 0.006 +−0.047 ± 0.026 +NR +1.070 ± 0.006 +0.007 ± 0.026 +Eckert et al. (2016). On the other hand, the MC simulations using +thermal conduction show shallower slope but is still consistent with +the analysis of Lovisari et al. (2015). As expected, a zero slope +is found for the NR simulation that do not include radiative processes. +In the middle panel of Fig. 10, we now plot the gas fraction 𝑓gas,500 +MNRAS 000, 1–30 (2022) + +The Rhapsody-C simulations +17 +without any cut in the gas temperature. This quantity does not reflect +what X-ray observations measure but can help to understand the +origin of the different slopes found in the 𝑓gas,X,500 − 𝑀500 relations. +We can see that the NR, MW and MC simulations now show a fairly +constant fraction of gas within 𝑅500. MW haloes have ∼10 per cent +lower values compared to their non-radiative counterparts, which is +not the case if we look at the baryonic fraction 𝑓b,500 in the right +panel of Fig. 10. This suggests that the amount of gas ‘missing’ in +the MW simulations, compared to the NR simulations, has been +converted into stars as 𝑓b = 𝑓gas + 𝑓stars. To a lesser extent, we see the +same behaviour for the MC simulations, which indicates that ATC +suppresses star formation at the expense of a denser ICM. +However, haloes in the VW simulations, which benefit from early +and strong AGN activity, show the steepest increase in both 𝑓gas,X,500 +and 𝑓gas,500 with mass (i.e. cosmic time). This demonstrates the +ability of the VW AGN feedback model to efficiently deplete the +gas from the 𝑅500 region in lower mass haloes, where the hot gas +can escape more easily in a shallower potential well. Hence, at earlier +times, the ICM temperatures rose to high values that significantly held +back both the infall of gas (due to a high central gas pressure) and the +formation of cold gas clumps, that can fuel both star formation and +SMBH gas accretion. This explains why at low masses (i.e. earlier +times), VW haloes also show low baryonic fractions. However, at +later times, while this relatively hot gas is slowly radiatively cooling, +it condenses toward the cluster centre to meet 𝑓b,500 values similar to +the other simulations (see the right panel of Fig. 10). This indicates +that VW haloes host cooling flows at late times consequent to early +and efficient AGN activity. +Our radiative simulations suggest a non constant evolution for the +X-ray emitting gas fraction 𝑓gas,X,500 which increase with mass. In +agreement with the self-similar model, we find a roughly constant +evolution of the total gas fractions with mass (i.e. cosmic time), +with the exception of the VW simulations which show efficient gas +depletion due to energetic AGN feedback events that deplete gas from +the central regions of lower mass haloes. +At our resolution, no baryons are expelled beyond 𝑅500 in the MW +and MC simulations in contrast to the VW simulations. Although the +VW model has an effect on the baryon content of our haloes, the +galaxies properties are off (as we can see in Fig. 8) while galaxies of +the MW and MC reproduce more realistic properties. +6.2.2 The 𝑇X − 𝑀tot relation +In Fig. 11, we show a comparison of the X-ray temperatures of the +Rhapsody-C clusters as a function of their mass with various scal- +ing relations from the literature, both simulations and observational +studies, plotted in their respective studied mass ranges10. As ob- +servational mass estimates may be biased with respect to the true +three-dimensional spherical masses in simulations such as ours, we +differentiate them in Fig. 11 by using dash-dotted lines for studies that +use hydrostatic mass estimates. Indeed, we can see that, at a given +temperature, hydrostatic mass based studies systematically indicate +a lower total mass. +In the top panel of Fig. 11, without making any distinction +between the simulation types, we plot the evolution of each +Rhapsody-C halo along published scaling relations as they grow +10 We only show the scaling relation at 𝑧 = 0 for the redshift-dependent +scaling relations of Giles et al. (2016); Lieu et al. (2016); Mantz et al. (2016); +Le Brun et al. (2017); Bulbul et al. (2019); Henden et al. (2019); Lovisari +et al. (2020) +Mtot,500 [M⊙] +100 +101 +E (z)−2/3 T ce +X,500 [keV] +Lieu+16 - XXL [ci] +∗ Le Brun+17 - cosmo-OWLS +∗ Cui+18 - The 300 [mw] +∗ Henden+19 - FABLE +∗ Biffi+14 - MUSIC [ci] +Reichert+11 +∗ Barnes+17a - MACSIS +∗ Barnes+17b - C-EAGLE +Bulbul+19 +Lovisari+20 +1014 +1015 +Mtot,500 [M⊙] +100 +101 +E (z)−2/3 T ce +X,500 [keV] +Figure 11. Core excluded X-ray temperature as a function of the total mass +within 𝑅500. In the top panel we show the evolution of all our Rhapsody- +C haloes along published scaling relations. We compare our data to the +studies using hydrostatic mass estimates or true/weak-lensing masses using +dash-dotted and solid lines, respectively. The scaling relations are plotted +for the same mass considered in each of these works. In the top legend, we +specify in brackets if the measurements include cluster cores (ci) and we +indicate simulation works with asterisks. In the top panel, we distinguish +each Rhapsody-C halo with a unique symbol and color, while in the bottom +panel, the color stands for the type of simulations with black, blue, red and +purple used for the NR, MW, VW and MC simulations, respectively. We plot +in the bottom panel the best fit scaling relations obtained for our haloes and +give the values of the slopes and intercepts found inside the brackets located +in the legend. +in mass. In the high mass end, i.e. for 𝑀500 ⩾ 5 × 1014M⊙, our +results are in good agreement with studies using ‘unbiased’ mass +measurements i.e. weak lensing mass estimates for observational +studies (Lieu et al. 2016) or true total masses measured from +simulations (Biffi et al. 2014; Lieu et al. 2016; Le Brun et al. 2017; +Cui et al. 2018; Henden et al. 2019). Accounting for a hydrostatic +mass bias of ∼20 per cent brings our data into agreement with the +MNRAS 000, 1–30 (2022) + +18 +A. Pellissier et al. +Table 4. Similarly to Table 3, we show the fitted normalisation (𝛼) and slope +(𝛽) for the 𝑇 ce +𝑋,500–𝑀500 scaling relation for haloes with 𝑧 ≲ 1.5. In the +second and last part of the table, we show the slopes found by the analyses +based on numerical simulations and observations respectively, for which we +compare our data to in the upper panels of Figs. 11. We highlight both in +Figs. 11 and this table, the simulation works with asterisks. +𝑇 ce +𝑋,500–𝑀500 +𝛼 +𝛽 +MW +−0.059 ± 0.003 +0.683 ± 0.016 +VW +−0.037 ± 0.003 +0.621 ± 0.014 +MC +−0.075 ± 0.005 +0.699 ± 0.021 +NR +−0.060 ± 0.004 +0.724 ± 0.018 +∗ Barnes et al. (2017a) +0.58 ± 0.01 +∗ Barnes et al. (2017b) +0.47 ± 0.07 +∗ Biffi et al. (2014) +0.56 ± 0.03 +∗ Cui et al. (2018) +0.627 ± 0.007 +∗ Henden et al. (2019) +0.64 ± 0.02 +∗ Le Brun et al. (2017) +0.577 ± 0.006 +Bulbul et al. (2019) +0.83 ± 0.10 +Lieu et al. (2016) +0.56 ± 0.12 +Lovisari et al. (2020) +0.66 ± 0.06 +Reichert et al. (2011) +0.57 ± 0.03 +observational studies using hydrostatic mass estimates (Reichert +et al. 2011; Bulbul et al. 2019; Lovisari et al. 2020) or simulations +estimating mass with a mock X-ray analysis (Barnes et al. 2017a,b). +However, a more precise characterisation of the hydrostatic mass +bias measured in our simulations will be the subject of a future paper. +Comparison with observations Compared to the literature, our +simulations indicate slightly steeper slopes than most of the stud- +ies with the exception of the observations by Bulbul et al. (2019). +While being consistent in the high mass range (𝑀500 ⩾ 5×1014M⊙), +Lieu et al. (2016) indicates for higher X-ray temperatures in the lower +mass range compared to our results, which might be induced by the +inclusion of the cluster core in the X-ray temperature measurements +(see Appendix C). +Comparison with simulations Similarly, while being in agreement +with Cui et al. (2018) and Henden et al. (2019), we report system- +atically slightly steeper slopes compared to the other simulation +works but our data agree well within scatter with these studies in the +high mass range. The shallower slope of Biffi et al. (2014) could be +induced by the inclusion of the core in the temperature estimation +(see Appendix C). However, we are aware that our spectral fits for the +temperature estimation in the low mass range can be slightly biased +low (as discussed in Appendix B), which could explain the steeper +slopes we find. As discussed more extensively in Appendix B, we +show that the mass-weighted temperature estimates are a factor of +∼2 lower than the ones resulting from our spectral fit. However, +assuming such a factor of 2 lower temperatures shifts the scaling +relation of Cui et al. (2018) to even lower temperatures. +We quantitatively compare our scaling relation slopes with the +above-mentioned studies in Table 4. We observe that the inferred +slopes and normalisations are rather insensitive to the physical mod- +els used for our simulations as the temperature reflect the depth of +the cluster’s potential well. The core-excluded X-ray temperatures are +similar for simulations with or without ATC (MC and MW respec- +tively). Therefore, thermal conduction does not play an important +role in offsetting the X-ray temperatures outside the core in our sim- +ulations. While the VW simulations indicate a shallower slope with a +higher normalisation, they converge to the same core-excluded X-ray +temperature values in the high mass range. We see again the effi- +ciency of the VW AGN heating in raising the ICM temperature to +higher values, especially in lower mass haloes where the potential +well is shallowest. Most importantly, we see that all simulations con- +verge to the same temperatures with similar scatter for masses above +5 × 1014M⊙. On average, the slopes agree with a ∼2 per cent steeper +value than the self-similar expectation and no significant effect of the +AGN models or ATC on our 𝑇X − 𝑀tot scaling relation is seen. +Surprisingly, the non-radiative simulations are able to reproduce the +same core-excised temperatures as the full-physics simulations. Be- +sides the fact that the temperature is less affected by feedback pro- +cesses as it reflects more the cluster potential well, this results also +indicates that non-gravitational processes mostly affect the core. In +the radial range 0.15 ⩽ 𝑅/𝑅500 ⩽ 1, radiative cooling, thermal con- +duction, AGN and SF feedback do not play a major role in offsetting +the ICM core-excluded X-ray temperatures. We note that only the +VW AGN feedback, being the most effective, is able to heat the gas +at these radii in the lower mass regime. +From Table 4, we see that different slopes and normalisations for +the 𝑇X − 𝑀tot scaling relation are found in the literature. These nor- +malisation differences can be attributed to the method used to infer +cluster masses, which might be biased compared to the true mass (e.g. +due to the hydrostatic bias or biased weak lensing estimates). The +observational studies of Sun et al. (2009) and Lovisari et al. (2020) +showed that the slopes remain consistent for low mass groups to mas- +sive galaxy clusters. This consistency implies that non-gravitational +processes are not affecting the 𝑇X − 𝑀 scaling relation in a different +manner in distinct mass (or temperature) ranges. Bulbul et al. (2019) +actually found the steepest slope in their observations. They explain +this apparent tension by the fact that they simultaneously fit the mass +and redshift trend of the scaling relation, in contrast to the assumed +self-similar redshift evolution in other studies. Lovisari et al. (2020) +claimed that it could also be explained if their SPT-SZ masses suffer +from a mass-dependent bias (similar to the Planck mass estimates). +We observe slightly steeper slopes compared to the simulation +works of Biffi et al. (2014); Barnes et al. (2017a,b) and Le Brun +et al. (2017) but our results agree with the studies of Cui et al. (2018) +and Henden et al. (2019). The discrepancy could originate from the +higher temperature found in lower mass haloes in those works. It can +be attributed to the method used to estimates X-ray temperatures (see +discussion in Appendix B and C) but also from the efficiency of the +feedback model to deplete and heat the gas in lower mass halo. +6.2.3 The 𝐿X − 𝑀tot relation +The X-ray luminosity mass scaling relation is important as it can +relate one of the ‘cheapest’ X-ray observables to the total cluster +mass. To fully exploit the data from large galaxy cluster samples +provided by X-ray surveys such as e-ROSITA (Liu et al. 2021), which +collects too few photons to infer any spectra or construct any mass +profiles, it is of great use to have a well calibrated 𝐿X − 𝑀tot scaling +relation and an accurate determination of its scatter. However, the +X-ray luminosity measurement depends on the energy band and +the aperture from which it is derived as well as the flux extraction +method. As a consequence, among all the X-ray scaling relations, +it is the one that shows the largest scatter. Moreover, due to its +density squared dependence, the X-ray luminosity can be easily +biased by the presence of gas-rich substructures, a cool core and +non-gravitational processes (Reichert et al. 2011) which motivates +MNRAS 000, 1–30 (2022) + +The Rhapsody-C simulations +19 +Mtot,500 [M⊙] +1043 +1044 +1045 +E (z)−2 Lce +X,500 +� +erg s−1� +Mantz+16 - WtG +∗ Barnes+17b - C-EAGLE +Bulbul+19 +1014 +1015 +Mtot,500 [M⊙] +1043 +1044 +1045 +E (z)−2 Lce +X,500 +� +erg s−1� +Mtot,500 [M⊙] +1044 +1045 +E (z)−7/3 Lce +X,bol,500 +� +erg s−1� +∗ Biffi+14 - MUSIC [ci] +∗ Henden+19 - FABLE +Reichert+11 +∗ Barnes+17a - MACSIS +Lovisari+20 +1014 +1015 +Mtot,500 [M⊙] +1044 +1045 +E (z)−7/3 Lce +X,bol,500 +� +erg s−1� +Figure 12. As Fig. 11, but for the core-excluded X-ray soft band (left) and bolometric (right) luminosities measured in the radial range 0.15 ⩽ 𝑅/𝑅500 ⩽ 1 as a +function of halo mass inside 𝑅500. We notice the large differences in slope and intercept between the published scaling relations. +the exclusion of the core from analyses. +In figure 12, we show the core excluded soft-band (0.5 − 2 keV) +and bolometric (0.01 − 100 keV) X-ray luminosities as a function of +the cluster mass for all our haloes. The X-ray luminosity is rather sen- +sitive to non-gravitational processes and to the ICM clumpiness, and +this can explain why such a diversity of slopes and normalisations +is observed. As we can see in Table 5, published studies show on +average a pronounced deviation, which is on average 50 and 30 per +cent greater than the expected self-similar scaling for soft-band and +bolometric luminosities respectively. In the bottom panels of Fig. 12, +we make a distinction between simulations that incorporate galaxy +formation physics (MW, VW, MC) and those without (NR). The X- +ray luminosity is more sensitive to the physical models used in the +simulations, hence on radiative processes, compared to the temper- +ature that do not show such large discrepancy between the different +simulations. Therefore, the calibration of scaling relations using the +Table 5. Similarly to Table 4 for the scaling of 𝐿ce +X,500 and 𝐿ce +X,bol,500 with +𝑀500. The scaling relations listed here are shown in Fig. 12. In constrast to +observations, we denote numerical works with an asterisk. +𝐿ce +X,500–𝑀500 +𝐿ce +X,bol,500–𝑀500 +𝛼 +𝛽 +𝛼 +𝛽 +MW +−0.186 ± 0.007 +1.230 ± 0.038 +−0.165 ± 0.009 +1.255 ± 0.049 +VW +−0.182 ± 0.004 +1.440 ± 0.023 +−0.203 ± 0.005 +1.497 ± 0.028 +MC +−0.090 ± 0.009 +1.150 ± 0.043 +−0.095 ± 0.014 +1.112 ± 0.062 +NR +0.048 ± 0.007 +0.920 ± 0.033 +−0.010 ± 0.007 +1.146 ± 0.033 +∗ Barnes et al. (2017b) +1.33 ± 0.13 +∗ Biffi et al. (2014) +1.45 ± 0.05 +∗ Barnes et al. (2017a) +1.88 ± 0.05 +Mantz et al. (2016) +1.65 ± 0.14 +∗ Henden et al. (2019) +1.97 ± 0.10 +Bulbul et al. (2019) +1.60 ± 0.17 +Reichert et al. (2011) +1.52 ± 0.04 +Lovisari et al. (2020) +1.82 ± 0.25 +X-ray luminosity is more complex than relations using the X-ray tem- +perature or the X-ray analogue of the Sunyaev-Zeldovich𝑌 parameter +(see later in Section 6.3). +MNRAS 000, 1–30 (2022) + +20 +A. Pellissier et al. +Comparison with observations For the soft-band luminosity, we +systematically find shallower slopes compared to the relations of +Mantz et al. (2016) and Bulbul et al. (2019). With the exception of +the NR simulations which show a 25 per cent higher value, our slopes +for the bolometric luminosity do not significantly change from the +ones derived using the soft-band luminosities. With the exception +of the VW simulations which agree within scatter with the slope +of Reichert et al. (2011), we find even greater discrepancy with +observation for the bolometric luminosity. When setting the redshift +evolution of the scaling relation to the self-similar value (which is +the choice we have made for this work), Lovisari et al. (2020) find a +slope of 1.45 ± 0.10 which agrees only the VW simulations. +If we account for a mass bias of 20 percent (which is also shown to +be a good fit for the 𝑇X −𝑀 scaling relation), we can bring our data in +agreement with the study Reichert et al. (2011) which use hydrostatic +mass estimates but widen the gap with the scaling relation of Bulbul +et al. (2019) and Lovisari et al. (2020) which show lower luminosities +(higher mass) at fixed mass (X-ray luminosity). We observe a higher +normalisation than Mantz et al. (2015). +Comparison with simulations Our radiative simulations have +slopes that are consistent with the simulations of Barnes et al. +(2017b) for the soft-band luminosity but only the VW simulations +agree with the slope of Biffi et al. (2014) for the bolometric +luminosity. Barnes et al. (2017a) and Henden et al. (2019) find +50 per cent steeper slopes compared to our radiative simulations. +The Macsis simulations (Barnes et al. 2017a) indicate a 50 per +cent steeper slope on average compared to our radiative simulations. +Some of this discrepancy can be attributed the ability of their +AGN feedback model to efficiently heat gas in lower mass haloes, +lowering their X-ray luminosity. Regarding normalisation, assuming +a hydrostatic mass bias of 20 percent increases the offset with +the C-Eagle and the Macsis simulations. This discrepancy could +originate from the difference in the energy injection of the thermal +AGN feedback model (which use different Δ𝑇 values and the number +of heated neighbour particles). The Music simulations (Biffi et al. +2014) show, at fixed total mass, lower X-ray luminosities compared +to our data. +By looking at the differences in slope and normalisation between +our simulation types, we observe that the X-ray luminosity follows +the same trend with mass as the X-ray emitting gas fraction (see +the left panel in Fig. 10) with the steepest slope being for the VW +simulations, then in descending order, MW, MC and finally NR. +This illustrates once more the relation between the X-ray luminosity +and the halo gas content, which is driven by the different physical +models (AGN and ATC) that the simulations use. The shallower +slope of the NR compared to both the self-similar scaling and the +other simulations originates from the absence of galaxy formation +physics or radiative gas cooling, which could boost the X-ray +luminosity. The higher NR normalisation can be explained by the +higher gas content in haloes as no star formation (which should turn +cold and dense gas into stars) or feedback processes (which could +deplete the gas in haloes) occur. +6.2.4 The 𝐿X − 𝑇X relation +We now look at the X-ray scaling relations in Fig. 13, which relates +the core-excluded X-ray temperatures and luminosities, and collect +their parameters (slope and intersect) in Table 6. As for the 𝐿X − 𝑀 +Table 6. Similarly to Table 5 for the scaling of 𝐿ce +X,500 and 𝐿ce +X,bol,500 with +𝑇 ce +X,500. +𝐿ce +X,500–𝑇 ce +X,500 +𝐿ce +X,bol,500–𝑇 ce +X,500 +𝛼 +𝛽 +𝛼 +𝛽 +MW +−0.096 ± 0.009 +1.569 ± 0.096 +−0.066 ± 0.011 +1.215 ± 0.125 +VW +−0.126 ± 0.006 +2.339 ± 0.076 +−0.132 ± 0.007 +2.202 ± 0.084 +MC +0.014 ± 0.014 +1.545 ± 0.112 +0.012 ± 0.020 +1.123 ± 0.155 +NR +0.132 ± 0.008 +1.173 ± 0.067 +0.099 ± 0.008 +1.383 ± 0.065 +Giles et al. (2016) +2.63 ± 0.15 +∗ Biffi et al. (2014) +2.29 ± 0.07 +Pratt et al. (2009) +2.34 ± 0.13 +∗ Barnes et al. (2017a) +3.01 ± 0.04 +∗ Henden et al. (2019) +3.02 ± 0.15 +scaling relations, we observe a large offset between recent numerical +and observational works both in normalisation and slope. +Comparison with observations The relations of Giles et al. (2016) +and Pratt et al. (2009), from observations of the XXL and REXCESS +clusters respectively, are rather shifted to higher temperatures at fixed +soft-band X-ray luminosity. We note however that Giles et al. (2016) +use core-included measurements, unlike Pratt et al. (2009), which +can significantly bias the X-ray luminosity (see Appendix C). While +the VW simulations agrees with the slopes of the Giles et al. (2016) +and Pratt et al. (2009), our haloes follows on average 40 per cent +shallower scaling relations as we can see in Table 6. +Comparison with simulations All haloes agree, within scatter, +with the values found by the Music and Fable simulations while +the Macsis simulations indicate a slightly lower normalisation +i.e. higher temperatures at fixed bolometric luminosity. When +comparing quantitatively the slopes for the 𝐿X,bol − 𝑀 scaling +relations with these simulations, we found shallower slopes by more +than a factor 2 on average. Only our VW simulations agree with the +slope of Biffi et al. (2014) which however consider core-included +bolometric luminosities. +In more detail, we systematically find significantly shallower +slopes compared to the literature as we can see in Fig. 13 and Table 6. +The VW simulations, however, show the steepest slopes compatible +with the studies of Pratt et al. (2009) and Biffi et al. (2014) for +the soft-band and bolometric luminosities respectively. The steeper +evolution of the X-ray luminosity with temperature of the VW +simulations is the result of both the more effective AGN feedback +at lower halo masses and the gas enrichment at high halo masses +(see Fig. 10), which significantly boosts the X-ray luminosities. The +effect the different radiative models explored in this work is the most +visibly seen for the 𝐿X −𝑇X relations. Although this scaling relation +might be the most straightforward to derive from observations, +its calibration remains challenging as it demonstrates the most +significant sensitivity on the physical models used in simulations. +6.2.5 The 𝑀gas,X − 𝑀tot relation +We show in Fig. 14 the evolution of the X-ray emitting gas mass (i.e. +the gas with 𝑇 > 0.5 keV) with the cluster total mass enclosed within +𝑅500. We see that the ICM mass correlates well with the total cluster +mass with a relatively small scatter. The study of observed relaxed and +disturbed clusters by Lovisari et al. (2020) showed that this relation +is quite insensitive to the dynamical state of the clusters, however +with a higher scatter for their disturbed sample. In the upper panel of +Fig. 14, we can see that our haloes show a relatively higher gas mass +for haloes with 𝑀500 < 5 × 1014M⊙. As we can see in Table 7, our +MNRAS 000, 1–30 (2022) + +The Rhapsody-C simulations +21 +T ce +X,500 [keVM] +1043 +1044 +1045 +E (z)−1 Lce +X,500 +� +erg s−1� +Giles+16 [ci] - XXL +Pratt+09 - REXCESS +100 +101 +T ce +X,500 [keVM] +1043 +1044 +1045 +E (z)−1 Lce +X,500 +� +erg s−1� +T ce +X,500 [keV] +1044 +1045 +E (z)−1 Lce +X,bol,500 +� +erg s−1� +∗ Biffi+14 [ci] - MUSIC +∗ Henden+19 - FABLE +∗ Barnes+17a - MACSIS +100 +101 +T ce +X,500 [keV] +1044 +1045 +E (z)−1 Lce +X,bol,500 +� +erg s−1� +Figure 13. A pure X-ray scaling relation – the core excluded X-ray soft band (left) and bolometric (right) luminosities as a function of the core excluded X-ray +temperature. We keep the same figure properties as in Fig. 11. +simulations indicate slopes consistent with the observations of Mantz +et al. (2016) (1.04) and the C-Eagle simulations (1.07, Barnes et al. +2017b), albeit shallower compared to the observations of Bulbul et al. +(2019) and Lovisari et al. (2020) but also the cosmo-OWLS (Le Brun +et al. 2017) Macsis (Barnes et al. 2017a) and Fable (Henden et al. +2019) simulations. +The dependence of the gas fractions on the physical models of our +simulations obviously translates in this scaling relation. Therefore, +similarly to the findings of Section 6.2.1, we see a ∼10 per cent steeper +evolution of the gas mass with the total cluster mass for the VW +simulation while the NR, MW and MC show relatively similar slopes. +6.3 Sunyaev–Zeldovich scaling relations +The Sunyaev-Zel’dovich (SZ) effect (Zeldovich & Sunyaev 1969; +Sunyaev & Zeldovich 1970) - which is the distortion of the cosmic +microwave background (CMB) spectrum by the inverse Compton +scattering of the low-energy CMB photons with free electrons in the +ICM - provides an unique view of the ICM baryons. By probing the +line-of-sight integral of the ICM thermal pressure support, it yields +an ideal proxy for the gas mass in a galaxy cluster and therefore the +total mass. +We compute 𝑌SZ,500, the integrated Comptonization parameter 𝑌 +within 𝑅500, directly from the simulation using the cell gas temper- +ature, 𝑇𝑖, and electronic density, 𝑛e,𝑖 = 𝜌gas,𝑖/(𝜇emp), as +𝑌SZ,500 = +𝜎𝑇 +mec2 +𝑟𝑖⩽𝑅500 +∑︁ +𝑖 +kB𝑇𝑖𝑛e,𝑖d𝑉𝑖, +(25) +where 𝜎𝑇 , me, c and d𝑉𝑖 are respectively the Compton cross +section, the electron mass, the speed of light and the volume of the +considered gas cell. +The 𝑌SZ,500 quantity does not show any particular scatter as the +ICM pressure profiles in clusters tend to be universal within 𝑅500 +(Arnaud et al. 2010). It is less sensitive to the gas density than X-ray +MNRAS 000, 1–30 (2022) + +22 +A. Pellissier et al. +Mtot,500 [M⊙] +1013 +1014 +Mgas,X,500 [M⊙] +∗ Henden+19 - FABLE +Mantz+16 - WtG +∗ Le Brun+17 - cosmo-OWLS +∗ Barnes+17a - MACSIS +∗ Barnes+17b - C-EAGLE +Bulbul+19 +Lovisari+20 +1014 +1015 +Mtot,500 [M⊙] +1013 +1014 +Mgas,X,500 [M⊙] +Figure 14. As Fig. 11, but for the evolution of the (X-ray emitting) gas +mass to the total cluster mass inside 𝑅500 compared to both numerical and +observational studies. +observables due to its linear dependence, and hence core exclusion +is not necessary. Therefore measure 𝑌SZ,500 in the 𝑟 ⩽ 𝑅500 range. +In Fig. 15 we show the 𝑌SZ,500 − 𝑀tot,500 scaling relations. We +see that the 𝑌SZ,500 parameter is tightly connected to the cluster +mass, where we observe the lowest scatter compared to the X-ray +scaling relations. Indeed, this parameter probes the mass-weighted +temperature, which is much less sensitive to the gas clumpiness (as +opposed to the emission measure weighted temperature of X-ray +quantities). +Our Rhapsody-C haloes agree well with all previously published +scaling relations from both numerical simulations (Cui et al. 2018; +Henden et al. 2019; Barnes et al. 2017a; Le Brun et al. 2017) and +observational works (Planck Collaboration et al. 2014; Nagarajan +et al. 2019). Due to the very low scatter in the 𝑌SZ − 𝑀 scaling +relation, its use seems well suited for studies that aim to constrain +cosmological parameters. In a more quantitative comparison, we can +see in Table 7 that Planck Collaboration et al. (2014); Nagarajan et al. +Mtot,500 [M⊙] +100 +101 +102 +103 +E (z)−2/3 YSZ,500 +� +kpc2� +Nagarajan+18 +∗ Cui+18 - The 300 +∗ Henden+19 - FABLE +∗ Le Brun+17 - cosmo-OWLS +∗ Barnes+17a - MACSIS +Planck14 Baseline +1014 +1015 +Mtot,500 [M⊙] +100 +101 +102 +103 +E (z)−2/3 YSZ,500 +� +kpc2� +Figure 15. Evolution of the integrated Compton 𝑌SZ,500 parameter of the +Sunyaev-Zel’dovich effect as a function of the total halo mass computed +within 𝑅500. We keep the same properties as of Fig. 11. We see in the upper +panel the tight evolution of the Rhapsody-C haloes along published scaling +relations irrespective of the physical models used in our simulations. In the +bottom panel, we see that the models used in our simulations induce a very +slight change in the slope of our scaling relations. +(2019); Cui et al. (2018); Nagarajan et al. (2019) have slope values in +agreement with our simulations (within errors), but the latter shows +a 17 per cent lower slope value on average. On the other hand, the +simulations of Le Brun et al. (2017) and Henden et al. (2019) indicate +slightly steeper slopes. For observational studies, this difference can +be understood by 𝑌SZ,500 being a mass-dependent observable and +therefore less constrained at lower masses, which can explain the +difference in the slopes of Nagarajan et al. (2019) and the Planck +Collaboration et al. (2014) baseline that probe different cluster mass +ranges. +Between our simulations, the slopes of the radiative simulations are +in very good agreement and the non-radiative simulations. We find +similar slopes for all simulations with a difference being at most 4 per +MNRAS 000, 1–30 (2022) + +The Rhapsody-C simulations +23 +cent between the MW and VW simulations. Although the difference +in the AGN feedback model between the MW and VW simulations +yields disparate X-ray scaling relations, the difference here weakens +for the 𝑌SZ − 𝑀 scaling relation. We can understand this slight differ- +ence as the VW simulations produce higher ICM temperatures (and +hence higher pressures) at lower halo masses due to a more efficient +AGN gas heating at early times. The slightly shallower slope observed +for simulations with conduction (MC) compared to the simulations +without (MW) is a consequence of the higher fraction of cooling gas +at high halo masses which results in a lower pressure support, hence +lower 𝑌SZ,500 values at higher halo mass (cf. Section 6.2.1). There- +fore, we see here that the physical models used in our simulations +do not play a particular role in offsetting the 𝑌SZ,500 − 𝑀tot scaling +relation. +Again, the slight normalisation and slope changes can be under- +stood in the same way as the conclusions of Section 6.2.1: the simu- +lations with ATC show a shallower evolution with mass than simu- +lations without, as the ICM is more quiescent with a suppressed star +formation, and the higher normalisation of VW simulations can be +ascribed to their higher ICM pressure support caused by their AGN +feedback model. +We also measure the X-ray analogue of 𝑌SZ,500 by taking the +product of the mass of the X-ray emitting gas (𝑀gas,X,500) and +the X-ray temperature from our spectral fit (𝑇ce +X,500). In Fig. 16 +we show our data along other 𝑌X − 𝑀 scaling relations and we +observe a increased scatter than from the SZ scaling relation, as +this 𝑌X parameter is more sensitive to the internal structure of the +ICM11. Our slope values remain rather constant and very similar +to the slopes found for the 𝑌SZ,500 − 𝑀tot,500 relation (see Table 7) +and overall, our data is consistent with the results of the studies +shown in Fig. 16, but our slopes indicate a ∼6 per cent lower value +on average. In the lower mass range, where differences between +published scaling relations become more obvious, our haloes follow +the relations of the numerical studies of Le Brun et al. (2017); +Barnes et al. (2017a) and Henden et al. (2019) while the former +indicates somewhat lower 𝑌X,500 values. On the other hand, the +C-EAGLE simulations (Barnes et al. 2017b) indicate a 10 per cent +shallower slope with higher 𝑌X,500 values, especially at low halo +masses. +To summarize, we have seen in this section that the evolution +of our haloes along cluster scaling relations are not significantly +affected by changes in the AGN feedback model or the inclusion +of ATC. Yet, in this study we focused on massive systems whereas +the effect of feedback processes might be more pronounced in the +group regime. Despite their profound impact on the cluster gaseous +and stellar components, the global properties of our haloes and their +evolution with mass are not significantly affected by the baryonic +processes in the ICM (with the exception of the X-ray luminosity, +which is relatively sensitive to the ICM thermodynamical state). +This finding is good news for cluster cosmology, which relies on +scaling relations to derive cluster total masses. Numerical simulations +are proven to be a reliable and suitable tool for such calibrations. +11 For instance, a completely smooth ICM will give equality between 𝑌X,500 +and 𝑌SZ,500, as in this case we have ⟨𝑛2⟩ = ⟨𝑛⟩2 (which shows the respective +dependence of the 𝑌X,500 and 𝑌SZ,500 on the gas number density). +Mtot,500 [M⊙] +1013 +1014 +1015 +E (z)−2/3 Y ce +X,500 [M⊙ keV] +∗ Henden+19 - FABLE +∗ Le Brun+17 - cosmo-OWLS +∗ Barnes+17a - MACSIS +∗ Barnes+17b - C-EAGLE +Bulbul+19 +Lovisari+20 +1014 +1015 +Mtot,500 [M⊙] +1013 +1014 +1015 +E (z)−2/3 Y ce +X,500 [M⊙ keV] +Figure 16. Similarly to Fig. 15, we show the X-ray analogue of the core- +excluded integrated SZ signal, i.e. 𝑌X = 𝑀gas,X × 𝑇 ce +X , as a function of the +total halo mass. Compared to 𝑌SZ − 𝑀, we observe a greater scatter, expected +from the sensitivity of X-ray observables. However, it shows the lowest scatter +compared to the other X-ray scaling relations (𝑇X − 𝑀, 𝐿X − 𝑀 or 𝐿X −𝑇X). +Our data is in fair agreement with published studies even at lower halo masses +where the scatter is the greatest. Similarly to Fig. 15, the difference in the used +physical models only induce a slight difference in the inferred slopes. +7 SUMMARY AND CONCLUSIONS +We presented the Rhapsody-C suite, a series of zoom-in magneto- +hydrodynamical simulations of massive galaxy clusters (𝑀vir ∼ +1015M⊙) with a physical resolution of 2.8 kpc. The simulations in- +clude radiative gas cooling, star formation, feedback from supernovae +(SN) and active galactic nuclei (AGN) as well as anisotropic thermal +conduction (ATC). This work was motivated by the Rhapsody-G +suite (Wu et al. 2015; Hahn et al. 2017; Martizzi et al. 2016), which +suggested shortcomings in the thermal AGN model and the need for +additional sources of energy injection. Hence, in this paper we re- +visit thoroughly the seeding of super massive black holes (SMBHs), +introduce a new model for their dynamical evolution, and consider +different AGN energy deposition schemes as well as the anisotropic +transport of heat within the intra-cluster medium (ICM). We study +MNRAS 000, 1–30 (2022) + +24 +A. Pellissier et al. +Table 7. Same as Table 4 for the X-ray emitting gas mass (𝑀gas,X,500) integrated 𝑌SZ parameter and its X-ray analogue (𝑌X). +𝑀gas,X,500–𝑀500 +𝑌SZ,500–𝑀500 +𝑌X,500–𝑀500 +𝛼 +𝛽 +𝛼 +𝛽 +𝛼 +𝛽 +MW +−0.030 ± 0.002 +1.075 ± 0.009 +−0.034 ± 0.003 +1.812 ± 0.014 +−0.022 ± 0.004 +1.757 ± 0.022 +VW +0.018 ± 0.001 +1.105 ± 0.007 +0.029 ± 0.003 +1.737 ± 0.014 +0.047 ± 0.003 +1.725 ± 0.017 +MC +0.008 ± 0.003 +1.050 ± 0.013 +−0.031 ± 0.004 +1.752 ± 0.019 +−0.000 ± 0.006 +1.748 ± 0.027 +NR +0.043 ± 0.002 +1.006 ± 0.008 +0.027 ± 0.005 +1.784 ± 0.023 +0.050 ± 0.004 +1.730 ± 0.021 +∗ Barnes et al. (2017a) +1.25 ± 0.03 +∗ Barnes et al. (2017b) +1.69 ± 0.07 +∗ Barnes et al. (2017a) +1.84 ± 0.05 +∗ Barnes et al. (2017b) +1.07 ± 0.05 +∗ Cui et al. (2018) +1.62 ± 0.31 +∗ Barnes et al. (2017b) +1.57 ± 0.07 +∗Le Brun et al. (2017) +1.32 ± 0.01 +∗ Henden et al. (2019) +1.88 ± 0.05 +∗ Henden et al. (2019) +1.88 ± 0.05 +∗ Henden et al. (2019) +1.25 ± 0.04 +∗Le Brun et al. (2017) +1.948 ± 0.018 +∗Le Brun et al. (2017) +1.948 ± 0.018 +Mantz et al. (2016) +1.04 ± 0.05 +Nagarajan et al. (2019) +1.51 ± 0.31 +Bulbul et al. (2019) +2.01 ± 0.20 +Bulbul et al. (2019) +1.26 ± 0.10 +Planck Collaboration et al. (2014) +1.79 ± 0.065 +Lovisari et al. (2020) +1.85 ± 0.10 +Lovisari et al. (2020) +1.25 ± 0.05 +the impact of each of the models on the cluster stellar component and +examine how they shape the intra-cluster gas. We next investigate the +evolution of our simulated clusters over cosmic time with a range of +cosmological observables that serve as mass proxies when the above- +mentioned sub-grid models are changed. We focus in this analysis on +the total cluster mass versus X-ray temperature, luminosity, gas mass +and the integrated Comptonization parameter. The main findings of +our analysis are as follows : +• The star formation in the proto-cluster can be efficiently +controlled by the seeding of the SMBHs in the ICM. With a low +number of SMBH seeds, the AGN heating cannot prevent the +gas from over-cooling in the proto-cluster. Seeding less massive +but more numerous SMBHs enables an efficient, more fre- +quent AGN heating in time and space. Our simulations indicate that +this might be one of the most crucial parameters to regulate feedback. +• We develop a new model for the SMBH dynamics, which +is made publicly available for the Ramses code. It consists of +decaying the SMBHs towards the local potential minimum along +the steepest gradient with a magnitude that depends on the tidal +forces experienced by the SMBH during its evolution. This new +model makes the accretion of gas onto SMBHs easily tunable by +keeping the SMBHs relatively close to the potential minimum. +Consequently, the amount of AGN feedback energy injected into the +ICM can be controlled and it is shown to have a significant effect on +reducing the gas content in the proto-cluster as well as quenching +star formation. The abundance (see point above) and the locations +of the SMBHs therefore appear critical to regulate AGN feedback. +• By changing only the AGN energy injection scheme (i.e. +volume- or mass-weighted energy deposition), we observe a +dramatic change in both the distribution of gas and in star formation. +Mass-weighted deposition preferentially injects the AGN feedback +energy into the dense accretion regions, which then has difficulty +escaping and is thermalised by the cold gas reservoir surrounding +the central SMBH. As a result, a build-up of cold gas occurs in +the ICM and a high star formation rate follows. At late times, this +larger cold gas reservoir fuels AGN activity that can increase the gas +entropy in the core to produce a non cool-core (NCC) state at 𝑧 = 0. +On the other hand, volume-weighted deposition injects more +energy in more diffuse regions, which allows the AGN feedback +energy to escape the accretion region early to heat the gas over +large distances. Star formation is dramatically quenched, the gas +in the core is efficiently depleted and the transition to an NCC +proceeds from 𝑧 = 1. When volume-weighted deposition is used, +the more diffuse ICM leads to the cessation of AGN activity at +lower redshifts because of a low cold gas supply. Thus, with the +decline of the AGN activity, the ICM gradually cools to settle into a +similar ICM thermodynamical state as the simulation implementing +mass-weighted AGN energy deposition at 𝑧 = 0. +In spite of this relative similarity between the two simulations at +𝑧 = 0, the star formation and AGN activity histories greatly differ, +as do the galaxy masses and the ICM clumpiness. +• Anisotropic thermal conduction appears to reduce star forma- +tion in the ICM by almost a factor of 2 by flattening out temperature +gradients in the ICM. ATC leads to an earlier transition to an NCC +cluster thanks to the transport of heat within the ICM. However, in +our simulations, we do not observe enhanced AGN activity but the +opposite: ATC delays the AGN activity by preventing the formation +of cold gas in the ICM that would otherwise fuel the SMBH cold +gas accretion. +• The cluster gas fractions are not appreciably altered by changes +in the energy accumulation threshold of the thermal AGN feedback. +The observed slight gas depletion does not linearly scale with this +threshold. It suggests that purely thermal feedback cannot shape the +ICM on large scales. +• The evolution of our simulated cluster observables over cosmic +time is in relatively good agreement with both observational and +numerical studies. Among the X-ray scaling relations, the 𝑇X − 𝑀 +relation is rather insensitive, especially in the high halo mass regime +(𝑀500 ⩾ 5 × 1014M⊙), to the use of the different galaxy formation +models (radiative gas cooling, ATC, mass- or volume-weighted AGN +energy deposition) as they show no noticeable difference with the +scalings derived for adiabatic simulations. The same conclusions +hold for the mean Sunyaev-Zeldovich flux scaling relations (and its +X-ray analogue) with much lower scatter. This suggests that galaxy +formation physics does not play a particular role in significantly +offsetting the global cluster observables, especially at the high mass +end. +To aid the astrophysical and cosmological interpretation of current +and future galaxy cluster surveys, we have increased the complexity +of the Rhapsody-C high-resolution cosmological simulations by +including anisotropic thermal conduction and various SMBH/AGN +models, as well as generating more sophisticated X-ray observables +compared to the Rhapsody-G simulations. Despite the apparent +sensitivity of the ICM and cluster galaxies to the numerical models +used, the evolution of the simulated cluster observables over cosmic +MNRAS 000, 1–30 (2022) + +The Rhapsody-C simulations +25 +time is remarkably insensitive to changes in our astrophysical +models. A notable exception of the X-ray luminosity, which is +sensitive to clumping. The power of the Rhapsody-C simulations is +to have a sample of clusters sharing a similar mass at 𝑧 = 0 but with +diverse assembly histories, shapes and richness. We are therefore in +a position to study the scatter around the mean scaling relations and +relate it to the astrophysical processes that shape each cluster. This +will be investigated in future research. +In this work, by looking at the 𝑇X − 𝑀tot scaling relation, we +observed that accounting for a ∼20 per cent mass bias can make our +data consistent with studies based on hydrostatic mass estimates. A +detailed study of the energy budget in our simulated galaxy cluster +is needed to quantify the level of non-gravitational energy and non- +thermal pressure support. Moreover, thanks to the implementation +of the thermal conduction of Dubois & Commerçon (2016) which +includes a separate treatment of electrons and ions, we are able +to resolve differences in their distribution and energetics. We will +investigate these aspects in future work. +ACKNOWLEDGEMENTS +We are grateful to Pawel Biernacki for helpful discussions about +the modelling of super-massive black holes in Ramses, to Yohan +Dubois regarding the thermal conduction scheme, as well as Chris- +tian Garrel for his invaluable help with the LIRA code. We are in- +debted to Lorenzo Lovisari and Yohan Dubois for thorough feed- +back on an early version of the manuscript, and we thank Ricarda +Beckmann, Sandrine Codis, Frédéric Bournaud, Aoife Boyle, and +Sunayana Bhargava for useful discussion and comments. +AP and OH acknowledges funding from the European Research +Council (ERC) under the European Union’s Horizon 2020 research +and innovation programme (grant agreement No. 679145, project +‘COSMO-SIMS’). This work was granted access to the HPC re- +sources of TGCC under the allocation A0040410487 made by +GENCI. +This work made use of the following open source software: Matplotlib (Hunter +2007), NumPy (Harris et al. 2020), SciPy (Virtanen et al. 2020), emcee +(Foreman-Mackey et al. 2013), PyAtomDB (Foster & Heuer 2020), APEC +(Smith et al. 2001), Astropy (Price-Whelan et al. 2018). +DATA AVAILABILITY +The simulation data and post-processed data can be made available +per reasonable request to the authors on an individual basis. +REFERENCES +Abbott T. M. C., et al., 2022, Phys. Rev. D, 105, 023520 +Allen S. W., Evrard A. E., Mantz A. B., 2011, ARA&A, 49, 409 +Anders E., Grevesse N., 1989, Geochimica Cosmochimica Acta, 53, 197 +Angulo R. E., Hahn O., Abel T., 2013, MNRAS, 434, 1756 +Arnaud M., Pratt G. W., Piffaretti R., Böhringer H., Croston J. H., Pointe- +couteau E., 2010, A&A, 517, A92 +Bañados E., et al., 2021, ApJ, 909, 80 +Bahé Y. 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Depending on the temperature gradient in the core (⩽ 200 kpc), +anisotropic thermal conduction can act as a cooling or heating source and +endeavours to smooth out substructures in the ICM. +Wu H.-Y., Evrard A. E., Hahn O., Martizzi D., Teyssier R., Wechsler R. H., +2015, MNRAS, 452, 1982 +Yang H. Y. K., Reynolds C. S., 2016, ApJ, 818, 181 +Zeldovich Y. B., Sunyaev R. A., 1969, Ap&SS, 4, 301 +APPENDIX A: ANISOTROPIC THERMAL CONDUCTION +AS A HEATING OR COOLING SOURCE +To understand the effect of anisotropic thermal conduction (ATC) on +the ICM before adding physics related to galaxy formation. We run +our simulation at a lower resolution corresponding to an effective +resolution of 40963 cells for the full simulation box, yielding a DM +particle mass of 8.22 × 108 ℎ−1Mpc. +We show in Fig. A1, the gas depletion profiles for an adiabatic sim- +ulation (grey line) and where only ATC was added (blue). Similarly, +we compare a simulation where the gas is allowed to radiatively cool +(black) to the same simulation when ATC is switched on (red). +In the adiabatic case, as the cluster forms, the gas in the center +is adiabatically heated and the higher pressure support prevents the +gravitation collapse of gas from the cluster outskirts. +However when thermal conduction is added, we observe that while +the cluster forms and the more gas collapse towards the center, heat +generated during this gas compression is now transported outwards +were the gas temperature is lower. As the result more low-entropy +gas fall inwards and the cluster centre gets denser. With thermal +conduction, the heat generated by the gas compression during the +cluster evolution is transported to the colder outskirts which lowers +the pressure support in the cluster center. Consequently, gas flows +inwards, the cluster core contracts and a higher gas fraction can +be observed at all radii in Fig. A1. In that case, thermal diffusion +is actually behaving like a gas cooling source driving a cooler and +denser cluster core. +We now allow the gas to radiatively cool. Because this process is +quadratically sensitive to the gas density, as the gas in the center is +getting denser, it also cools at faster rates, which leads to a runaway +instability: the cooling catastrophe. To prevent the overcooling of +the gas, we set an artificial limit by imposing a pressure floor below +which no gas can cool but can only increase its density along the +adiabat 𝑃gas ∝ 𝜌𝛾 +gas. In this configuration, we observe the formation +of a much lower entropy and higher density core compared to the +adiabatic case which translates into a high fraction in the central +100 kpc. By comparing the gas depletion profiles of the radiative +simulations in Fig. A1, we can see by comparing the black and red +line that thermal conduction can deplete gas from the core to the +outskirts. In the cooling-only simulation (black), we see the fraction +of gas peaking at 60 kpc, defining the extent of the core, which drops +to reach the universal baryon faction at 700 kpc. With the addition +of thermal conduction (red), the core extends now to 100 kpc with a +lower amount of gas. The gas fraction decreases less steeply towards +the Ω𝑏/Ω𝑚 value at a greater distance of 2 Mpc. In that config- +uration, thermal conduction drives the transport of heat from the +outskirts toward the overcooling core: it now acts as a heating source. +Interestingly, we see that thermal conduction can act as a cooling +source by transporting heat from the core to larger radii, or, as a +heating source by transporting heat inwards. This behaviour is dic- +tated by the sign of the temperature gradient in the inner cluster +region. More generally, thermal conduction is effective at flattening +out temperature substructures in the ICM. +APPENDIX B: ESTIMATING THE X-RAY TEMPERATURE +Having access to the true density and pressure of the gas inside a +simulation cell, we can estimate the true gas temperature from our +simulations. However, the comparison between real and simulated +data is complicated by different problems like sky background, pro- +jection effects and instrumental noise. Additionally there is a possible +mismatch between the spectroscopic temperature estimated from X- +ray observations and the temperatures usually defined in numerical +studies. The former is a mean projected temperature obtained by fit- +ting a single (or multitemperature) thermal model to the observed +photon spectrum while the later fully exploits the three-dimensional +thermal information of gas cells/particles. The average gas temper- +ature in simulated cluster can be obtained by a weighted sum of all +cell/particle temperatures 𝑇𝑖 as +𝑇𝑤 = +� +𝑖 𝑤𝑖𝑇𝑖 +𝑤𝑖 +, +(B1) +where 𝑤𝑖,vw = d𝑉𝑖 in case of a volume-weighted temperature +(𝑇vw), with d𝑉𝑖 is the AMR cell volume determined by the +local resolution of the simulation, or with 𝑤𝑖,mw = 𝑛𝑖d𝑉𝑖 for a +mass-weighted temperature (𝑇mw) with 𝑛𝑖 the gas cell density. This +mass-weighted temperature gives a more ‘physical’ average as it +emphasizes dense regions that participate more to the X-ray emission. +However, X-ray astronomy suffers from well-known biases +due as its intensity depends quadratically on the density since +both Bremsstrahlung and the collisional excitation responsible +for the metal line emissions results from two-body processes. +For this reason, X-ray observations are especially biased by the +dense regions such as the cluster core or the presence of gas-rich +substructures which motivate the definition of a ‘emission-weighted’ +temperature (𝑇emw) where the weighting function is proportional +to the emissivity of each gas element 𝑤𝑖,emw = 𝑛2 +𝑖 Λ(𝑇𝑖) with the +cooling function Λ(𝑇) mainly being the so-called bolometric cooling +MNRAS 000, 1–30 (2022) + +28 +A. Pellissier et al. +function Λ(𝑇) ∝ √𝑇𝑖 which implicitly assume that bremsstrahlung +(free-free) emission is the dominant mechanism at high X-ray +temperatures (> 3 keV)12. Mazzotta et al. (2004) found that the +above-defined emission-weighted temperature over-estimates the +projected spectroscopic X-ray temperatures of thermally complex +clusters and propose another spectroscopic-like temperature 𝑇sl +with 𝑤𝑖,sl = 𝑛2 +𝑖 𝑇3/4 +𝑖 +which better approximates the spectroscopic +temperature in Chandra and XMM-Newton observations. This +weighting function, beside being biased toward the densest regions +of the clusters such as in the emission-weighted case, will also be +biased toward the coolest regions. +To circumvent the shortcomings of simple weighting schemes, we +instead produce an X-ray spectrum from which we derive the gas +temperature and density using a single thermal as close as possible +to the observers’ methodology. The simulated ICM spectrum is ob- +tained by computing the continuum and line emission of a gas cell +with 𝑇 ⩾ 0.5 keV and a metallicity 𝑍, with the publicly available +AtomDB atomic database13. As such, from the continuum and line +emissivity 𝜖𝑖(𝑇𝑖, 𝑍𝑖), we compute the rate of emitted photon Φ𝑖 of +the cell 𝑖 +𝜙𝑖 = 𝜖𝑖(𝑇e,𝑖, 𝑍𝑖) +∑︁ +𝑖 +𝑛e,𝑖 𝑛H,𝑖d𝑉𝑖, +(B2) +which allow us to produce a mock X-ray spectrum by summing of +the individual rest frame spectra of each gas cell. +In X-ray observation, the most common method to obtain the ICM +temperature and density is to fit the observed spectra by a single +temperature APEC model. Therefore, we follow a similar method- +ology and chose to perform fits using a Monte Carlo Markov Chain +(MCMC) sampling method thanks to the emcee python library +(Foreman-Mackey et al. 2013). We fit our data to a single temperature +spectra generated using the PyAtomDB library. We show the result of +a such fits in the upper panel Fig. B1 . +We note that the presence of cold gas at high redshifts can produce +a low-energy bump in the spectrum which complicate the fitting +procedure. As the result, it could induce the fit to converge faster +to high values of the density (i.e. higher spectrum normalisation). +In order to maximize the likelihood, the MCMC chain will later try +converging to lower temperatures (i.e. steeper cutoff) to compensate +for the overestimated gas density. Consequently, the spectral fit can +slightly underestimate the temperature of haloes in the case of a +high fraction of cold the gas, which is predominantly the case at +high redshifts (𝑧 ⩾ 1). +To overcome the overestimation of the gas density (and a un- +derestimation of the temperature), we first fit the gas density +0.20–2.00 keV band first as the X-ray flux is not very sensitive on the +temperature and metallicity in this band (as long as the metallicity +is low, i.e. 𝑍 ≲ 0.5). We use the posterior distribution of this first +MCMC sampling as the prior distribution of the gas density for a +second MCMC while using flat priors for the gas temperature and +metallicity. The fits in Fig. B1 result from this two-step MCMC. +However, the low energy bump cannot be constrained with a single +temperature model and a double (or multi) temperature model would +be more suited. +12 Nevertheless, at lower temperatures, metal lines participate significantly +to the X-ray emission which becomes temperature and metallicity dependent. +13 We use the abundances of Anders & Grevesse (1989) as well as APEC +equilibrium line and continuum fits files from the 3.0.9 version of AtomDB - +https://atomdb.org/ +100 +101 +E [keV] +1048 +1049 +1050 +1051 +1052 +1053 +Φ [ph s−1] +Data +Best fit ; z = 0 +Best fit ; z = 1 +1014 +1015 +Mtot,500 [M⊙] +100 +101 +E (z)−2/3 T ce +X,500 [keV] +SL +SF +MW +VW +Figure B1. Top panel: ICM photon emission rate directly computed from +the simulation in the core-excised 𝑅500 sphere (black) along the best fit +APEC model resulting from our MCMC sampling for the same halo at 𝑧 = 0 +(orange) and 𝑧 = 1 (dark orange). At low redshifts, our fitting procedure +performs well. We note that the gas metallicity responsible for the emission +lines is not well constrained but is not relevant for our analysis as the lines +does not significantly participate to the X-ray flux. For the 𝑧 = 1 spectrum, +the high fraction of cold gas (i.e. 𝐸 ⩽ 1 keV) lead to a slight overestimation +of the density (i.e. higher normalisation) and an underestimation of the gas +temperature (i.e. steeper spectrum). +Bottom panel: Scaling of the core-excluded temperature with the total mass +inside the 𝑅500 using different temperature estimates. 𝑇vw (blue) and 𝑇mw +(red) are a weighted average using respectively the cell volume and the cell +density. 𝑇sl (green) uses the cell emissivity and the Mazzotta et al. (2004) +weights of hot (𝐸 ⩾ 0.5 keV) gas cells only. We show the temperatures +resulting from the spectral fits 𝑇sf in orange. We can see that our 𝑇sf estimates +approach well the spectroscopic-like temperature. +MNRAS 000, 1–30 (2022) + +The Rhapsody-C simulations +29 +1014 +1015 +Mtot,500 [M⊙] +1043 +1044 +1045 +1046 +E (z)−2 Lce +X,500 +� +erg s−2� +ci +ce +Figure C1. Mass versus X-ray luminosity for all haloes with 𝑧 ⩽ 1.5 for the +all simulations type combined (NR, MW, VW and MC, see details in Table 1. +The blue symbols show the core-included (ci) X-ray luminosity within 𝑅500 +and the black symbols shows the luminosity in the core-excluded region +corresponding to the [0.15 − 1] 𝑅500 aperture. The see that the exclusion +of the core reduces dramatically the scatter and indicates for 40 per cent the +X-ray luminosities on average. +We show in the bottom panel of Fig. B1, the differences between +the different gas temperature estimates (𝑇vw, 𝑇mw, 𝑇sl and 𝑇sf). To +compute the average within 𝑅500 of 𝑇sl and 𝑇sf, only the hot X-ray +emitting gas (𝐸 ⩾ 0.5 keV) is considered while 𝑇vw and 𝑇mw use the +information of all cells with no cut in minimum gas temperature. +We see that the 𝑇sl and 𝑇sf are similar and shows that the formula +of Mazzotta et al. (2004) gives a good estimate of the spectroscopic +temperature, especially in the lower mass range. However, these two +X-ray’ estimates are, on average, 10 and 20 per cent higher than 𝑇mw +and 𝑇vw respectively. This shows that accounting for the bias induced +by the X-ray emitting gas offsets to higher temperatures the simple +mass- or volume-weighted averages, which also do not have any cut +in minimum gas temperature. +𝑇mw is 10 per cent higher compared to 𝑇vw at higher halo masses but +shows the steepest slope as we can see in Fig. B1 (we have for haloes +with 𝑧 ⩽ 1.5 slopes of 0.700 ± 0.019, 0.723 ± 0.018, 0.726 ± 0.013 +and 0.689 ± 0.014 when using 𝑇sl, 𝑇sf, 𝑇mw and 𝑇vw respectively). +APPENDIX C: ON THE CORE INCLUSION +The presence of AGN activity in the core of GCs can introduce +a strong variability of the X-ray luminosity as the central density +fluctuates and unrealistically high luminosities can be obtained. We +compare in Fig. C1 the core-included (ci) and core-excluded (ce) +X-ray luminosities computed for two apertures: the entire cluster +emission interior to 𝑅500 and in the [0.15 − 1] 𝑅500 aperture respec- +tively. +The inclusion of the core typically boost the X-ray luminosity with a +much greater scatter as it is greatly sensitive to the thermodynamic +state of the cluster core (high gas density and possible AGN heat- +ing). On average, the exclusion of the core decreases by 40 per cent +the X-ray luminosity and shows a steeper slope of (1.169 ± 0.033, +1014 +1015 +Mtot,500 [M⊙] +0.85 +0.90 +0.95 +1.00 +1.05 +1.10 +1.15 +1.20 +1.25 +T ce +w,500 / T ci +w,500 +Tsl +Tmw +Tvw +Figure C2. Ratio of the core-excluded to the core-included temperature versus +mass for all haloes (NR, MW, VW and MC combined) with 𝑧 ⩽ 1.5. We +show the difference induced by the core exclusion on different temperature +estimates: 𝑇vw (blue) and 𝑇mw (red) are a weighted average using respectively +the cell volume and the cell density while 𝑇sl (green) uses the cell emissivity +and the Mazzotta et al. (2004) weights of hot (𝐸 ⩾ 0.5 keV) gas cells only. +compared to 0.681 ± 0.070 for the ci). +It also significantly reduces the scatter and hence is more suited for +galaxy cluster sample studies where clusters can have very different +central states, consistent with the finding of Pratt et al. (2009) and +Mantz et al. (2018). +The inclusion of the core does not significantly impacts the 𝑇 − 𝑀 +scaling relation.14 For instance, we find slopes of 0.717 ± 0.020 for +the core-included and 0.700 ± 0.019 for the core-excluded 𝑇sl − 𝑀 +scaling relation which are consistent.15 In Fig. C2, we see that the VW +temperatures are widely insensitive to the core inclusion/exclusion +because the volume inside 0.15× 𝑅500 represents only a tiny fraction +of the total 𝑅500 sphere. +We can see that, in halo masses lower than ∼ 2 × 1014M⊙, the +exclusion of the core tends to increase the MW temperatures as +the measurement is biased by the presence of dense and cold gas +in lower mass haloes (i.e. higher redshifts). On the other hand, for +𝑀500 > 3 × 1014M⊙ the MW temperature is on average 2 per cent +lower when the core is excluded from the measurement . While +showing a larger scatter, the ratio of the ce to ci 𝑇sl can be as low as +0.85 with a median value of 0.94. +On average, we see that the ce temperature estimates are more +biased low at higher halo masses which can help to explain why very +slightly steeper slopes are found in the ci 𝑇 − 𝑀 scaling relation. +14 As the measurement of the temperature from a MCMC spectral fit is +expensive, we only have measurement of 𝑇sf in the core-excluded region. +Therefore, we only discuss here the 3 other estimates (𝑇vw, 𝑇mw and 𝑇sl) for +which we have both ci and ce measurements. +15 Similarly, we find for the 𝑇mw − 𝑀 relation slopes of 0.745 ± 0.014 and +0.726 ± 0.013for the core-included and core-excluded estimates respectively +and for the 𝑇vw − 𝑀 relation, 0.706 ± 0.014 and 0.689 ± 0.014 respectively. +MNRAS 000, 1–30 (2022) + +30 +A. Pellissier et al. +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–30 (2022) + diff --git a/NNE0T4oBgHgl3EQf0QK5/content/tmp_files/load_file.txt b/NNE0T4oBgHgl3EQf0QK5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ba0cd356f887fc91522e4bcbaafeaf68898d9814 --- /dev/null +++ b/NNE0T4oBgHgl3EQf0QK5/content/tmp_files/load_file.txt @@ -0,0 +1,2770 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf,len=2769 +page_content='MNRAS 000, 1–30 (2022) Preprint 10 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='0 Rhapsody-C simulations – Anisotropic thermal conduction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' black hole physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' and the robustness of massive galaxy cluster scaling relations Alisson Pellissier,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='2★ Oliver Hahn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='4 Chiara Ferrari2 1AIM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' CEA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Université Paris-Saclay,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Université Paris Diderot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Sorbonne Paris Cité,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' F-91191 Gif-sur-Yvette,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' France 2Université Côte d’Azur,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Observatoire de la Côte d’Azur,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Laboratoire Lagrange,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Bd de l’Observatoire,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' CS 34229,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 06304 Nice cedex 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' France 3Department of Astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' University of Vienna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Türkenschanzstraße 17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 1180 Vienna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Austria 4Department of Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' University of Vienna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Oskar-Morgenstern-Platz 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 1090 Vienna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Austria Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' in original form ZZZ ABSTRACT We present the Rhapsody-C simulations that extend the Rhapsody-G suite of massive galaxy clusters at the 𝑀vir ∼ 1015M⊙ scale with cosmological magneto-hydrodynamic zoom-in simulations that include anisotropic thermal conduction, modified supermassive black hole (SMBH) feedback, new SMBH seeding and SMBH orbital decay model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' These modelling improvements have a dramatic effect on the SMBH growth, star formation and gas depletion in the proto-clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' We explore the parameter space of the models and report their effect on both star formation and the thermodynamics of the intra-cluster medium (ICM) as observed in X-ray and SZ observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' We report that the star formation in proto-clusters is strongly impacted by the choice of the SMBH seeding as well as the orbital decay of SMBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Feedback from AGNs is substantially boosted by the SMBH decay, its time evolution and impact range differ noticeably depending on the AGN energy injection scheme used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Compared to a mass-weighted injection whose energy remains confined close to the central SMBHs, a volume-weighted thermal energy deposition allows to heat the ICM out to large radii which severely quenches the star formation in proto-clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' By flattening out temperature gradients in the ICM, anisotropic thermal conduction can reduce star formation early on but weakens and delays the AGN activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Despite the dissimilarities found in the stellar and gaseous content of our haloes, the cluster scaling relations we report are surprisingly insensitive to the subresolution models used and are in good agreement with recent observational and numerical studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Key words: methods: numerical – cosmology: large-scale structure of Universe – galaxies: clusters: intra-cluster medium – X-rays: galaxies: clusters – conduction 1 INTRODUCTION Forming the nodes of the cosmic web, clusters of galaxies are the largest virialised structures in our Universe and their matter content reflects that of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Originating from the highest peaks in the initial cosmic density field (Kaiser 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Bardeen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 1986), their spatial distribution and abundance carry the imprints of the process of structure formation and are heavily sensitive to the un- derlying cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Therefore, they represent veritable crossroads of astrophysics and cosmology as they provide valuable information from the physics driving structure formation to the nature of dark matter and dark energy (Voit 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Allen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Kravtsov & Borgani 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Weinberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Counting galaxy clusters (GCs) as a function of their mass and cosmic time provides an excellent (late Universe) probe of cosmo- logical parameters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='g Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2022) including dark energy, the summed neutrino masses (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='g Mad- ★ E-mail: alisson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='pellissier@oca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='eu havacheril et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2017) and modifications of gravity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='g Wilcox et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' However, the predictive power of GCs as cosmological probes is limited principally by our ability to accurately measure their masses using X-ray, Sunyaev-Zeldovich (SZ) or gravitational lensing analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Mass estimations require high-quality data and rely on various assumptions that are challenged by the presence of possible biases caused by several factors, such as deviations from hydrostatic equilibrium, triaxiality, or instrumental features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Hence, cluster cosmological surveys depend heavily on well-calibrated scaling relations that relate directly observed quantities - so-called mass proxies - such as the X-ray luminosity, to the underlying cluster mass (see Pratt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2019, for a recent review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' GCs grow hierarchically over cosmic time as gravity pulls baryonic and dark matter to form collapsed structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' They also grow in mass via major mergers that represent the most energetic phenomena since the Big Bang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Moreover, the feedback from supernovae (SN) and active galactic nuclei (AGN) in cluster galaxies injects a substantial amount of energy into the intra-cluster © 2022 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='02684v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='CO] 6 Jan 2023 2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Pellissier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' medium (ICM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Such processes continually shape the baryonic components of clusters and can inject up to 1064 ergs of gravitational potential energy during one cluster crossing time (∼ Gyr), primarily dissipated by shocks into heating of the intra-cluster gas to high (X-ray emitting) temperatures (Markevitch & Vikhlinin 2007), but also through large-scale ICM motions generating cluster-wide turbulence (Hitomi Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2016, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' A fraction of this energy can also be channeled into non-thermal plasma components such as cosmic rays (Brunetti & Jones 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Bykov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2019) and magnetic field amplification (Donnert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2018) as revealed by the presence of extended radio emission (Ferrari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Feretti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' van Weeren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' These energetic processes are expected to contribute to the deviation of cluster properties from self-similar predictions, which only account for gravitational evolution in scale-free cluster evolution (Kaiser 1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Galaxy cluster observables are therefore a complex interplay of both cosmology and astrophysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' They require detailed understanding for precisely calibrating cluster scaling relations to fully exploit the power of galaxy clusters as cosmological probes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' In this context, numerical simulations provide valuable informa- tion, as they can follow the evolution of galaxy clusters with ex- actly known properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' They can capture the effects of physical processes during cluster formation and predict the resulting observ- ables self-consistently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' However, astrophysical processes related to galaxy formation cannot be resolved in hydrodynamical cosmologi- cal simulations due to limited computational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Astrophys- ical processes occurring below the typical resolution are accounted for by so-called sub-grid models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Significant recent advances in the development and calibration of efficient sub-grid models has led to cosmological simulations that can reproduce a large number of the observed galaxy properties (see Vogelsberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2020, for a re- view).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' However, reproducing the galaxy cluster entropy profiles and the cool-core/non-cool-core dichotomy remains challenging (Rasia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Hahn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Barnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2017a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' In cosmological simulations of galaxy clusters, special attention has been dedicated to the effects of feedback from stars and AGN feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' These ad- vances have yielded a range of simulations from several independent groups that reproduce various cluster observables, such as the X-ray and SZ scaling relations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Barnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2017a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Le Brun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Truong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Henden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' The outcomes of such simulations can rely heavily on the param- eter choice of the sub-grid AGN feedback models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Le Brun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' (2014) showed that the entire baryon gas profile can be varied by tuning the energy accumulation threshold Δ𝑇 of the feedback model of Booth & Schaye (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' In the Rhapsody-G simulations of Hahn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' (2017), changes in the AGN feedback model had no significant impact on the gas outside the cluster core region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' This suggests that the sub-grid modelling of astrophysical processes needs to be improved or that the addition of new physics is necessary to increase the realism of such simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Indeed, the addition of thermal conduction in cluster simulations has been shown to significantly affect the properties of the ICM and provides an additional source of gas heating (Voit 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Voit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Allowing heat transport in the ICM with the implementation of thermal conduction could help to decrease the dependence of numerical simulations on the feedback sub-grid model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' While it does not provide enough heating to offset cooling losses in clusters, recent studies have shown that thermal conduction has various impacts on AGN activity and ICM mixing (Yang & Reynolds 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Kannan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Barnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Beckmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' In this context, the properties of the intra-cluster gas intimately depend on the physical processes modelled in simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Quantifying such dependencies in simulations is therefore fundamental to providing robust GC scaling relations that are focused on the (hot) intra-cluster gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' In this paper, we perform cosmological magneto-hydrodynamic simulations of massive galaxy clusters (𝑀vir = 1015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='1 M⊙) that include anisotropic conduction, radiative cooling, stellar and AGN feedback to study their effects on cluster scaling relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' In Sec- tion 2, we introduce our Rhapsody-C sample of zoom-in simulations and the numerical methods and models that we employ for this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' We focus in Section 3 on our super-massive black hole (SMBH) mod- elling, presenting a new ‘tidal friction’ model that efficiently controls their orbits and studying different AGN feedback models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' We show the impact of anisotropic thermal conduction (ATC) on the cluster stellar and gaseous properties in Sections 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' In Section 6, we study the impact of AGN feedback models and ATC on the evolution of the simulated clusters along several mass-observable scaling rela- tions relevant to cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Finally, we summarise our results and conclude in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Additionally, we show in Appendix A the im- pact of anisotropic thermal conduction on the ICM in idealised con- figurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' We also provide in Appendix B and C the bias on cluster scaling relations induced by the choice of the X-ray temperature es- timates in the simulations and the impact of core inclusion/exclusion on X-ray observables, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2 METHODS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='1 The Rhapsody-C sample and initial conditions This paper presents the Rhapsody-C project1, a suite of high- resolution zoom-in magneto-hydrodynamical (MHD) simulations of nine haloes in the 𝑀vir = 1015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='1M⊙ range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' The haloes were selected using the cosmICweb database from the Rhapsody-New simulation2 (Buehlmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2023, in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' We list the properties of these haloes in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Sharing similar masses at 𝑧 = 0, our haloes have different assembly histories and probe extreme as well as median cluster properties: two haloes have extreme concentrations (𝑐vir > 8), high and low number of subhaloes (𝑁sub > 120 and 𝑁sub < 85 respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' On average, our haloes share the same number of substructures and concentrations as the Rhapsody-G sample (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Hahn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Martizzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' We use the ΛCDM cosmology of the Rhapsody-New simulation with density parameters Ωb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='049 for baryons, Ωm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='309 for total matter and ΩΛ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='691 for the cosmological constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' The pri- mordial spectral index, the amplitude normalisation and the Hubble constant are 𝑛𝑠 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='9667, 𝜎8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='8159 and 𝐻0 = 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='74 km/s/Mpc respectively (Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' In this new cosmol- ogy, we have a lower baryon fraction of 𝑓b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='1586 compared to the Rhapsody-G’s value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' The updated value is also much closer to more recent constraints from Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' (2021) of 𝑓b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='1564.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' We generated the initial conditions using Music (Hahn & Abel 2011) for our nine clusters at 𝑧 = 49 from the minimum bound- ing ellipsoid matrix retrieved from the cosmICweb database using a traceback-radius of 2𝑅vir centered in a 1 ℎ−1Gpc box with an effective resolution of 81923 particles3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' All initial conditions were performed 1 where the ‘C’ denotes for the inclusion of anisotropic thermal conduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2 See https://cosmicweb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='univie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='at for more details 3 We share the same resolution as the Rhapsody-G 8K run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' MNRAS 000, 1–30 (2022) The Rhapsody-C simulations 3 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Description of the Rhapsody-C runs presented in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' In the top part of the table we list the minimum cell size (Δ𝑥), the initial mass per hydro cell (𝑚gas), the dark matter (𝑚dm) and minimum stellar particle mass (𝑚∗,min).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' The middle part describes the physical models used in the simulations studied in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' In the bottom part, we list the properties of each Rhapsody-C haloes: the internal halo ID in the Rhapsody-New simulation, the number of substructures (𝑁sub), the virial mass (𝑀vir), radius (𝑅vir) and concentration (𝑐vir) as well as the radius enclosing 500 times the critical density of the Universe (𝑅500) and the total mass within (𝑀500).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Summary of the Rhapsody-C simulations Δ𝑥 [ kpc] 𝑚dm [M⊙] 𝑚gas [M⊙] 𝑚∗,min [M⊙] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='82 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='54 × 108 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='68 × 107 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='58 × 106 Sub-grid modelling and baryonic processes Cooling, AGN energy AGN energy Anisotropic Label SF, deposition accumulation thermal SN scheme threshold conduction NR – – – – VW ✓ volume-weighted 107 K – MW ✓ mass-weighted 107 K – MC ✓ mass-weighted 107 K ✓ MW6 ✓ mass-weighted 106 K – MW8 ✓ mass-weighted 108 K – Halo Properties ID 𝑁sub 𝑐vir 𝑀vir 𝑅vir 𝑀500 𝑅500 [1015M⊙] [Mpc] [1014M⊙] [Mpc] 174742934 111 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='22 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='82 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='80 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='62 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='73 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='36 174743229 83 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='98 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='62 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='24 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='33 173587157 84 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='86 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='51 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='19 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='25 using second-order Lagrangian perturbation theory (LPT) with dark matter and baryon perturbations at 𝑧 = 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Compared to the original Rhapsody-G simulations, we do not use the local Lagrangian ap- proximation for the construction of the baryon density field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Baryons and dark matter did not co-move prior to recombination and sub- percent effects are expected at cluster scales (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Angulo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Hahn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Khoraminezhad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' However, for the simulations used here, we assume that baryons fully trace cold dark matter perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='2 Numerical approach For our cluster zoom simulations, we use the Eulerian adaptive mesh refinement Ramses code (Teyssier 2002) to follow the non-linear evolution of the initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Gas dynamics are computed using a second-order unsplit Godunov scheme for the ideal MHD equations (Fromang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Teyssier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2006) while collisionless dark matter particles as well as stars and sink particles are evolved using a particle-mesh solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Our simulations use the method introduced by Dubois & Commerçon (2016) for solving the anisotropic diffusion of heat using an implicit finite-volume method which is independent of the Courant time step constraint of the MHD scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' We employ a Lagrangian overdensity-based refinement strategy that splits cells if they reach an overdensity of eight: the refinement of the base grid by 𝑛 additional levels requires a density of 8𝑛 ¯𝜌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Our simulation boxes of 1 ℎ−1Gpc on a side, reach a maximum refinement level by maintaining a smallest cell size of physical Δ𝑥 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='8 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' The dark matter 𝑁-body particle mass is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='54×108M⊙ and initial mass per hydro cell is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='68 × 107M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' The high-resolution Lagrangian ellipsoid patch, from which the 2𝑅vir sphere centred on each cluster will form, is tagged using a passive scalar colour field that is advected with the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Dynamic refinement is restricted to the regions where this colour field is non-zero and no refinement is allowed outside the zoom region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' We thus focus most of the computational resources on the forming cluster and its immediate environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' The rest of this section details the various physical ingredients used in our high-resolution zoom-in simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' See Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='1 for gas cooling and heating as well as the star formation and stellar feedback, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='2 for the subgrid modelling of SMBH forma- tion, evolution and AGN feedback ang finally in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='3 we describe the magnetic field evolution with the anisotropic thermal conduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' The reader can skip to Section 3 and Section 4 for the scientific results regarding the impact of the the various BH-related sub-grid models and anisotropic thermal conduction respectively on the stellar and gaseous content of a GC, or directly to Sections 5 and 6 for properties of cluster galaxies and ICM as well as the the evolution of our clusters along various scaling relations respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='1 Radiative gas cooling, metallicity and stellar evolution Radiative gas cooling is calculated according to the tabulated rates of Sutherland & Dopita (1993) for Hydrogen, Helium and metal line cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' The total gas metallicity is not evolved separately but treated as a single species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' It is advected with the MHD equations as a passive scalar and is sourced by the supernovae feedback model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' We consider an UV background radiation according to the Haardt & Madau (1996) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' An instantaneous reionisation takes place at 𝑧 = 10 to take into account an earlier reionisation in the particularly overdense proto-cluster regions that we simulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' The unresolved cold and dense gas that will constitute the inter-stellar medium (ISM) of galaxies is approximated using a temperature floor given by a polytropic equation of state, 𝑇floor = 𝑇∗ � 𝑛H 𝑛∗ �𝛾∗−1 , (1) with 𝑛H the Hydrogen number density of the gas, 𝑛∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='1 cm−3 and 𝑇∗ = 104 K being respectively the star formation density threshold and the ISM polytropic temperature with 𝛾∗ = 5/3 being the ISM polytropic index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' In practice, gas can be heated above the temperature floor, but cannot cool below it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Star formation occurs when the gas density exceeds 𝑛∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' A portion of the gas in a cell is converted into a star particle that decouples from the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' We have a minimum stellar particle mass of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='6 × 106M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' The star particles are randomly drawn from Poisson process (Rasera & Teyssier 2006) following a Schmidt law �𝜌∗ = 𝜖∗ 𝜌gas / 𝑡ff, (2) with 𝜖∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='01 and 𝑡ff = �3𝜋/32G𝜌gas �−1/2, the local free-fall time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Stellar feedback is included using the model of Dubois & Teyssier (2008) in which each newly formed star that traces a continuous stellar mass distribution following the Salpeter (1955) initial mass function and releases, after 20 Myr, a fraction 𝜂 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='1 of its mass and metals with a yield of 𝑦 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Therefore 𝑦𝜂 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='01 of the time- integrated SFR is returned as metals in the ISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' In addition, each MNRAS 000, 1–30 (2022) 4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Pellissier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' SN feedback event injects a thermal energy of 1051 erg into the sur- rounding ISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Compared to the original Rhapsody-G simulations, we chose to enable the delayed cooling of the SN heated gas with a dissipation time scale of 20 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' This additional sub-grid model mimics the effect of non-thermal processes, such as turbulence or CRs (Rodríguez Montero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2022), which can dissipate energy on longer time scales before being radiated away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' The calibration of the free parameters of the SN feedback listed above is able to reproduce stellar masses consistent with abundance-matching results at masses lower than 1012M⊙ for resolved haloes with at least 1000 particles (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='2 Black holes and active galactic nuclei Ramses uses collisionless sink particles to model black hole growth and evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' The SMBH formation and evolution follow the model of Biernacki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' (2017), itself based on the precedent models of Dubois et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' (2010) and Teyssier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' (2011), and build on a sink particle implementation developed within the context of star-forming molecular clouds (Bleuler & Teyssier 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Super-massive black hole seeding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' The Phew clump finder (Bleuler & Teyssier 2014), directly implemented in Ramses, de- termines potential sites for SMBH sink particle formation by identi- fying relevant peaks in the density field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' We will briefly discuss the main steps and free parameters of the sink seeding model that we use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' First, all density peaks above a threshold 𝜌peak are identified as well as their connecting saddle points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' To keep only relevant density peaks, we merge all peaks that have a peak-to-saddle ratio lower than 3 to the neighbouring peak with which it shares the highest density saddle point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' This merging process, or noise removal, is halted when a saddle density falls below the 𝜌saddle threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' In short, a noise removal is performed on the density field to select only the relevant peaks above a density 𝜌peak which are later divided by the saddle den- sity threshold 𝜌saddle into clumps to finally yield the sink formation sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' The gas in the spherical region of radius equal to 4 (highest) resolution elements Δ𝑥, defining the sink sphere, is investigated to make sure that the gravitational field is compressive, strong enough to overcome internal gas support and not only accelerated toward the sink sphere centre but that this gas is contracting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' As a proximity check, we forbid the gas that is infalling to an already existing sink to create another sink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' While the choice for the initial seed mass is arbitrary, we set it to be the same as our 𝑁-body dark matter particle mass with 𝑚BH,seed = 108M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Gas accretion and black hole dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Once SMBHs are formed, they grow in mass at the (un-boosted) Bondi-Hoyle accretion rate (Hoyle & Lyttleton 1939;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Bondi & Hoyle 1944;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Bondi 1952) capped by the Eddington rate : �𝑀acc = min � �𝑀Edd , �𝑀Bondi � , (3) with : �𝑀Bondi = 4𝜋𝜌∞𝑟2 Bondi𝑣Bondi, (4) �𝑀Edd = 4𝜋G𝑀BH𝑚 𝑝 𝜖𝑟 𝜎𝑇 𝑐 = 𝑀BH 𝑡𝑆 , (5) where 𝜎𝑇 is the Thomson cross-section, 𝐺 the gravitational constant, 𝑀BH and 𝑚 𝑝, sink and proton mass respectively, 𝜖𝑟 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='1 is the Shakura & Sunyaev (1973) radiative efficiency for a SMBH and 𝑡𝑆 ∼ 45 Myr is the Salpeter time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' We also have 𝜌∞ = ¯𝜌/𝛼(𝑥sink) with 𝛼 is the dimensionless density profile of the Bondi self-similar solution (see Biernacki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2017), ¯𝜌 the mean density inside the sink sphere, 𝑥sink = 𝑟sink/𝑟Bondi and the sink radius and velocity defined as follows : 𝑟Bondi = G𝑀BH 𝑣2 Bondi , (6) 𝑣Bondi = √︃ 𝑐2𝑠 + 𝑣2 rel, (7) with 𝑣rel the relative velocity of the sink to the average gas velocity inside the sink sphere and 𝑐𝑠, the local sound speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' While we use MHD, we generically find high plasma beta values in our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Therefore, in the Bondi formula, the magneto-sonic speed effectively reduces to the adiabatic sound speed .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' In addition to gas accretion, SMBHs can also grow via mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' In this work, we do not check if two sinks form a bound system but directly merge if they are less than one accretion radius apart, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 4Δ𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' The dynamics of a single SMBH cannot be resolved in cosmolog- ical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' This can lead to spurious oscillations of the SMBH in the potential well of its host halo, due to external perturbations and the finite resolution effects, particularly during merger events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Bier- nacki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' (2017) implemented in Ramses a physically motivated model based on Eddington-limited accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Their main assump- tion is that the gas accretion rate onto the accretion disc is set by the Bondi formula ( �𝑀Bondi) which corresponds to a large scale accretion flow, while the accretion onto the SMBH is set by the Eddington rate ( �𝑀Edd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' The difference between the two rates therefore gives the amount of gas not being accreted by the central SMBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Instead, it should be pushed away from the accretion disc by the Eddington radi- ation pressure at a rate �𝑀dec = �𝑀Bondi − �𝑀acc, which we however do not model explicitly in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' We also stress that we are not using radiation hydrodynamics in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' This process of gas accretion and ejection leads to an additional momentum exchange between the gas and the sink particle, hence an additional drag force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' This addi- tional drag force is modelled by requiring a fixed center of mass of the joint gas+sink system during the accretion and a conserved total momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' We implemented in Ramses a further modification to the model of Biernacki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' (2017) to move the SMBHs towards the potential minimun (described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Active galactic nucleus feedback The accretion rate of gas onto the SMBH sink particle is always computed from the cells in the sink sphere (of radius 4Δ𝑥) using mass-weighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Following Booth & Schaye (2009), we do not inject the thermal AGN energy at each time-step but store the rest-mass energy of the accreted gas until it would be enough to raise the gas temperature inside the sink sphere by Δ𝑇 = 107 K (unless specified otherwise, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' studies of Teyssier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2011 and Le Brun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2014 using this Δ𝑇 threshold strategy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' In other words, we inject this accumulated AGN energy when 𝐸AGN > 3 2𝑚gaskB Δ𝑇 (8) in every gas cell of the sink sphere in a mass- or volume-weighted way (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='3) with a maximum allowed temperature of the AGN feedback set to 𝑇AGN = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='5 × 1011 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' The rate at which this thermal energy is released to the ambient gas is given by : �𝐸AGN = 𝜖𝑐𝜖𝑟 �𝑀accc2, (9) where 𝜖𝑐 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='15 is the coupling efficiency (Dubois et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2012), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' the fraction of radiated energy that couples to the surrounding gas, and is calibrated on the local 𝑀BH − 𝑀∗ relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' MNRAS 000, 1–30 (2022) The Rhapsody-C simulations 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='3 Magnetic fields and anisotropic thermal conduction To solve the MHD equations, the Ramses code uses the second-order unsplit Godunov method based on the monotonic upstream-centred scheme for conservation laws (MUSCL-Hancock method, van Leer 1977;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Evans & Hawley 1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' The constrained transport approach is used to evolve the induction equation 𝜕𝑩 𝜕𝑡 = ∇ × 𝒖 × 𝑩, (10) where 𝒖 is the gas velocity and 𝑩 the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' The scheme satisfies the solenoidal constraint ∇ · 𝑩 = 0 to machine precision (Teyssier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' The 2D Riemann problem at the cell edges is solved using the approximate Harten-Lax-van Leer-Discontinuities (HLLD) solver from Miyoshi & Kusano (2005) to compute time averaged electromotive forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' In the ideal MHD limit, the generation of magnetic fields from a previously unmagnetised fluid is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Therefore, the magnetic fields must be seeded in our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' For simplicity, we seed a uniform magnetic field along the box 𝑧 axis with a comoving magnitude of 𝐵0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='56 × 10−12 G, which ensures a divergence-free initial field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' In the presence of a magnetic field, the conduction of heat in a plasma becomes anisotropic since the motion of charged particles perpendicular to the field lines is restricted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' We use the implementa- tion of Dubois & Commerçon (2016) using an implicit finite-volume method for solving the anisotropic diffusion of heat through electrons (Braginskii 1965) 𝜕𝜌𝜖e 𝜕𝑡 = −∇ · Qcond, (11) with 𝜖e the specific internal energy of electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' The conductive heat flux, Qcond, can saturate once the characteristic length scale of the electron temperature gradient ℓ𝑇e is comparable to or less than the mean free path of electron 𝜆e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Hence, following Sarazin (1986), we introduce an effective conductivity which interpolates between the unsaturated (Spitzer conductivity) and saturated regime by Qcond,sat = − 𝑓sat 𝜅Sp∇𝑇e, = − 𝑓sat � −𝜅∥b (b · ∇) 𝑇e � − 𝑓sat (−𝜅iso∇𝑇e) , (12) with 𝑓sat = �1 + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='2𝜆e/ℓ𝑇e �−1, b = B/|B| the unit vector in the direction of the local magnetic field, 𝑇e the electronic temperature, , 𝜅iso and 𝜅∥ the isotropic and parallel conduction coefficient (with respect to the magnetic field lines) respectively with 𝜅∥ = 𝜅Sp − 𝜅iso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' In many astrophysical cases, 𝜅iso/𝜅∥ ≪ 1 since the Larmor radius, 𝜆L, is much smaller than the mean-free-path of electrons, 𝜆e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' For instance, in a hot intra-cluster plasma with 𝑇e = 3 keV, 𝑛e = 10−2 cm−3 and 𝐵 = 1𝜇G, we have 𝜆L = 108 cm and 𝜆e = 1021 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Here, we set a perpendicular conductivity coefficient of 1 per cent to ensure numerical stability (Dubois & Commerçon 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' The electron energy is tracked separately from that of the ions as described in Dubois & Commerçon (2016) and the rate of energy transfer between the electron and ion temperatures is given by 𝑄e↔i = 𝑇i − 𝑇e 𝜏eq,ei 𝑛e𝑘B 𝛾 − 1, (13) with the equilibrium timescale 𝜏eq,ei = 3𝑚e𝑚p 8 √ 2𝜋𝑛i𝑞4e ln Λ � 𝑘B𝑇e 𝑚e � 3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' (14) Both ion and electron adiabatic indexes are equal to 𝛾 = 5/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' By modelling the anisotropic transport using Braginskii MHD (equation 11), we follow the Spitzer ansatz that assumes a high degree of electron-ion collisionality, which is a good assumption in cluster cores (in which we are particularly interested here), but would need to be corrected in cluster outskirts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Additionally, we do not take into account the suppression of thermal conduction by the ion mirror instability (Komarov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2016) caused by magnetic trapping of electrons by magnetic field strength fluctuations, or the Whistler instability (Levinson & Eichler 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Pistinner & Eichler 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Roberg-Clark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2016, 2018) where electron-whistler scat- terings can significantly alter conduction at very sharp temperature gradients such as in cold fronts (Komarov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2018) or at tem- perature scale lengths below the critical value 𝛽e𝜆e (Drake et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2021, with typical values of 𝛽e ∼ 100 and 𝜆e ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='1 kpc in cool-cores to 1 kpc in cluster outskirts)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' In the recent idealised simulations of Berlok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' (2021) and Beckmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' (2022), it was shown that whistler-based suppression of thermal conduction has only a small impact on the ICM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' In this work, we hence model the upper limit of anisotropic thermal conduction within the ICM, which is sufficient for our purposes to study the potential impact on cosmological observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 3 THE MODELLING OF SUPER-MASSIVE BLACK HOLES The key ingredients of our SMBH formation and evolution models are: (a) the conditions for the formation of the SMBH and the SMBH seed mass,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' (b) the SMBH dynamics with a possible inclusion of a dynamical friction model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' (c) the SMBH growth by mass accretion at the Bondi-Hoyle-Lyttleton rate limited to the Eddington rate and finally (d) the induced AGN feedback which affects the surrounding gas which couples back to all the previous model ingredients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Ramses uses the so-called sink particle technique (Bate et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 1995) to model SMBH formation and evolution, which is a point mass which can move through the fluid accretion and interact with it by the ejection of mass, energy and momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Motivated by the low efficiency of the AGN feedback model in the Rhapsody-G simulations, our sub-grid models for the SMBH formation, evolution and AGN feedback need to be revised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' In this section we test how the free parameters in the model influence the cluster evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Respectively, for (a) we investigate in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='1 the effect of different SMBH seeding scenarii on the gaseous and stellar content on one of our proto-clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Regarding (b), we will present in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='2 our new ‘tidal friction’ model which allow to control SMBH orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Lastly, we study for (d) different AGN feedback models in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='1 which impact completely differently cluster evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' These analyses are all carried out on a fiducial halo (173917492).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' In the simulations discussed in this section, we do not implement yet the anisotropic thermal conduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='1 Seeding of SMBHs The specifics of SMBH seeding in simulations is an important aspect of controlling the effect of AGN feedback in simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Different models for black hole (BH) seeding are used in cosmological simu- lations such as placing a BH particle in the centre of every massive halo (Schaye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Weinberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' McCarthy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2017) or models that use thresholds of local gas properties such as 4 where 𝛽e = 8𝜋𝑛e𝑇e/𝐵2 is the electron plasma beta, the ratio of electron thermal to magnetic field pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' MNRAS 000, 1–30 (2022) 6 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Pellissier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' metallicity, density, temperature and velocity (Dubois et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Tremmel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Habouzit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Dubois et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' In this work, we generally adopt the same procedure as in Rhapsody-G for black hole seeding, albeit with modified param- eters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Following Biernacki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' (2017), we use the ‘minimal’ Jeans mass corresponding to the highest refinement level of our simula- tion to define the initial SMBH sink particle mass 𝑀seed = 108M⊙ which also correspond to our dark matter particle mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Sink parti- cle formation sites are identified on-the-fly using the Phew clump finder algorithm which identifies density peaks with a given contrast relative to the next saddle-point (see Bleuler & Teyssier 2014, for a detailed description) which is directly implemented in the Ramses code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' The Phew parameters adopted for the original Rhapsody-G simulations favoured the seeding of a sink particle in fewer but larger patches of gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Due to the stochastic nature of star formation and supernova feedback that impact the local gas properties (hence the SMBH seeding), we observed a large variability in the efficiency of AGN feedback in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' For the new suite of simulations dis- cussed in this paper, we followed a more systematic investigation into the impact of seeding on the proto-cluster region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' In particular we studied the following scenarios where we varied peak density and saddle thresholds but kept all other parameters fixed: 𝜌peak = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='5 ¯𝜌, 𝜌saddle = 2 ¯𝜌: Phew parameters as the original Rhapsody-G setup, with ¯𝜌 = Ω𝑚𝜌𝑐 the mean matter density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 𝜌peak = 8 ¯𝜌, 𝜌saddle = 20 ¯𝜌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' With a higher 𝜌peak value, only the highest density regions in the simulation are probed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' In that case, smaller gas patches are selected but spatially more frequent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Thus, it allows to seed more SMBHs in the simulation compared to the original Rhapsody-G configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 𝜌peak = 8 ¯𝜌, 𝜌saddle = 200 ¯𝜌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' We increase the saddle density threshold by a factor of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' As the result, a much lesser number of peaks are merged which results in an increased number of SMBH seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 𝜌peak = 8 ¯𝜌, 𝜌saddle = 15 ¯𝜌, with a lower saddle threshold which induce more peak merging hence a lowered number of SMBH seeds in the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' We show the impact of these choices on the enclosed total stel- lar mass in the proto-cluster region in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 1 at 𝑧 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' At that time, we find 21, 102, 168, 113 SMBHs of mean masses 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='5 × 108 M⊙, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='8×108 M⊙, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='6×108 M⊙ and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content='1×108 M⊙ inside the virial radius respectively in the above-mentioned simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' It demonstrates the tight connection of the 𝜌peak parameter with the total number of created sinks and the mean mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' The Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' 1 clearly indicates the resulting effect on the star formation suppression in the proto-cluster environment: The simulations hosting a higher number of SMBHs (being also spatially more frequent), shows a greater amount of AGN feedback energy injected in haloes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' As a result, this more profuse AGN heating will reduce the gas cooling in haloes which decrease the accretion of cold gas onto the central SMBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' The resulting mass accretion rates are seen to be inversely proportional to the number of SMBHs in our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' In consequence, the total stellar mass in the proto-cluster is consistently reduced with an increasing number of SMBHs in the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' We see that the total stellar mass for the simulation using 𝜌peak = 8 ¯𝜌, 𝜌saddle = 200 ¯𝜌 is reduced by a factor of 5 while the number of SMBHs is increased by the same factor approximately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' The total stellar mass in the proto-cluster can be directly controlled by the number of SMBHs seeded in the simu- lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE0T4oBgHgl3EQf0QK5/content/2301.02684v1.pdf'} +page_content=' Galaxy masses at 𝑧 = 2 were found to be in agreement with abun- dance matching results with the use of 𝜌peak = 8 ¯𝜌, 𝜌saddle = 20 ¯𝜌 101 102 103 r[kpc] 1011 1012 M*( 0 represents the upper bound of +allowed perturbations. The loss function L(f(x + δ), t) could be +specified as the cross entropy loss in untargeted attack where t +indicates the ground-truth label of x, while as the negative cross +entropy in targeted attack where t indicates a target label that is +different from the ground-truth label of x. Many gradient-based +optimization methods have been utilized to solve the above problem, +including the fast gradient sign method (FSGM) [1], iterative fast +gradient sign method (I-FGSM) [3], C&W method [4], functional +adversarial attack [5], adversarial camouflage [6], etc. Moreover, +several works [7], [8], [9] focus on the adversarial attack in the +physical world, where many types of environmental distortions may +weaken the effectiveness of the generated adversarial perturbations. +2.2 +Black-box Adversarial Attack +The formulation of Problem (1) is also applicable to the black- +box attack. However, the parameters of the target model f(·) is + +JOURNAL OF LATEX CLASS FILES, VOL. , NO. , +3 +unknown, while only the objective value f(x + δ) for each query +x + δ is provided. Consequently, the gradient-based optimization +methods cannot be directly utilized. According to the type of query +feedback f(x + δ), black-box attacks can be further partitioned +to two sub-categories, including score-based and decision-based +attacks. +2.2.1 +Score-based Black-box Adversarial Attack +In the scenario of score-based attack, the feedback f(x + δ) +is continuous, such as the posterior probability in the image +classification task. Existing methods for solving this problem can +be generally partitioned into three categories, including query- +based and combination-based methods. 1) Query-based methods +iteratively adjust the perturbation only based on queries to the target +model. Many black-box optimization approaches have been utilized, +mainly including random search (e.g., [10], Square, SignHunter +[11]), evolution strategies (e.g., NES [12], Bandit [13]), and +gradient-estimation approaches (e.g., ZOO [14], AutoZoom [15], +ZO-signSGD [16]). Compared to transfer-based methods, query- +based methods often achieve a higher attack success rate, but with +the cost of many queries to the target model. 2) Combination- +based methods aim to achieve a high attack access rate and +high query efficiency simultaneously, by taking advantage of both +queries to the target model and the transferred item from the +surrogate model. Existing methods proposed different kinds of +transferring strategies, such as transferring the perturbation magni- +tude (e.g., Square [17]), perturbation gradient (e.g., [18], [19] and +Meta attack [20]), perturbation distribution (e.g., N ATTACK [21], +AdvFlow [22]), the projection to a low-dimensional space (e.g., +TREMBA [23]). Combination-based methods have shown superior +attack performance to the other two categories, and our method also +belongs to this category. However, most existing methods (only +except Meta attack [20]) only utilized the model-level adversarial +transferability, while our method captures both example-level and +model-level transferability to improve the query efficiency further. +2.2.2 +Decision-based Black-box Adversarial Attack +In the scenario of decision-based attack, the feedback f(x + δ) +is discrete, such as the class label in the image classification +task. Existing methods for solving this problem can be generally +partitioned into random search based and gradient-estimation- +based methods. Random search based methods aim to find the +best perturbation around the invisible decision boundary, such as +sampling from a normal distribution in Boundary method [24] +or from a learnable Gaussian distribution in Evolutionary method +[25], searching along the estimated normal direction of the decision +boundary in GeoDA [26], or searching on the surface of the allowed +perturbation region in the vicinity of the benign example in SFA +[27] and Rays [28]. Gradient-estimation-based methods propose +different estimation approaches of gradient, such as utilizing the +neighboring points around the current solution in NES [12] and +qFool [29], Monte Carlo estimation in HopSkipJumpAttack [30], +or estimating the gradient in a low-dimensional subspace for +acceleration in QEBA [31]. OPT [32] and Sign-OPT [33] were +developed based on a continuous formulation that alternatively +optimizes the magnitude and direction of perturbation, such that +any gradient-estimation approaches can be utilized. +2.3 +Transfer-based Adversarial Attack +We list the transfer-based methods as another category, as each +transfer-based method could be applied for the white-box attack or +the black-box attack. For example, (MI-FGSM) [34] and Nesterov +iterative fast gradient sign method (NI-FGSM) [35] can be used as +white-box attacks, since they are extensions of the classic white- +box attacks FSGM [1] and I-FGSM [3]. However, as claimed in +the manuscripts of [34] and [35], their goals are to generate more +transferable perturbations to improve the attack success rate in the +black-box settings, and their experiments contain both white-box +and black-box settings. Thus, if one method mainly aims to improve +the adversarial transferability, we partition it to the transfer-based +category, rather than white-box or black-box. According to the +transfer’s objective, we further present example-level and model- +level transferability, respectively. +2.3.1 +Example-level adversarial transferability +Although without explicit illustration, some works have actually +studied the example-level transferability. For example, universal +adversarial perturbations [2] revealed that it is possible to find a +perturbation to fool multiple benign examples simultaneously, with +respect to the same attacked model. It reveals that different benign +examples may have some common fragile directions, following +which adversarial perturbations can be found easily. Another +example is generation-based adversarial attacks [23], [22], where +a generative model is trained to directly generate an adversarial +perturbation or adversarial example for each benign example. Its +default assumption is that adversarial perturbations around different +benign examples may follow the same distribution or mapping. +However, the example-level adversarial transferability has been +rarely utilized to boost the attack performance, especially in the +scenario of black-box attack. The only attempt we have found +is called Meta attack [20], where a meta attacker is trained to +generate gradient, which is then used as the update direction in +gradient-estimation-based black-box attack methods. In contrast, +our proposed meta-learning framework aims at capturing the +distribution of adversarial perturbations, thus can be naturally +combined with any kinds of query-based attack methods. +2.3.2 +Model-level adversarial transferability +The model-level adversarial transferability has been observed +and studied in many existing works [36], [37], [23], [38], [39], +[40], [41]. Two important issues are mainly explored about the +model-level transferability, including the intrinsic reason, and how +to enhance the transferability across models, especially in the +black-box scenario. 1) The intrinsic reason of the model-level +transferability. Tram`er et al. [40] found that different models +share a large fraction of adversarial subspace, which consists of +orthogonal basis vectors that are highly aligned with the gradient +of the loss function. Consequently, perturbations within this shared +subspace are likely to be transferable across models. A geometric +perspective provided by [37] showed that the transferability is +partially due to the fact that decision boundaries of different models +align well with each other. Demontis et al. [42] demonstrated that +the model-level transferability is closely related to the intrinsic +vulnerability of the target model, and the complexity of the +surrogate model. The theoretical analysis provided in [41] derives +two lower bounds for transferability based on data distribution +similarity and model gradient similarity, as well as the upper bound +for transferability based on gradient orthogonality and smoothness. +Wang et al. [43] illustrated that the model-level transferability is +negatively correlated with the interaction inside adversarial pertur- +bations. 2) Enhancing the model-level transferability. Compared +to FGSM [1], its extensions, including momentum iterative fast + +JOURNAL OF LATEX CLASS FILES, VOL. , NO. , +4 +gradient sign method (MI-FGSM) [34] and Nesterov iterative fast +gradient sign method (NI-FGSM) [35], showed better model-level +transferability. In [44], MI-FGSM and NI-FGSM were further +extended by reducing the variance of the iterative update directions, +such that the update direction is stabilized to escape from poor local +optima, leading to the transferability improvement even when there +are defenses for the target model. The ensemble attack method [37] +generated adversarial perturbations based on multiple models +to enhance the transferability for both untargeted and targeted +attacks. Intermediate level attack (ILA) [45] proposed to generate +adversarial perturbations based on intermediate layers of surrogate +models to avoid overfitting, such that the transferability to the target +model can be enhanced. Feature distribution attack (FDA) [46] +focused on improving the transferability of a targeted attack by +maximizing the activation of one intermediate layer between the +benign example and the perturbed example. It is extended in +[38] from one intermediate layer to multiple intermediate layers, +such that the transferability of targeted attack is further enhanced. +Some works employed the meta-learning ideology to enhance +transferability either. MSM [47] obtained a Meta-Surrogate Model +via optimizing a differentiable attacker. The Meta-surrogate model +gained prior from one or a set of surrogate models and was able to +generate adversarial examples with eximious transferability. Meta +Gradient [48] randomly sampled multiple models from a model zoo +to compose different tasks and iteratively simulated a white-box +attack and a black-box attack in each task, which narrowed the +gap between the gradient directions in white-box and black-box +attacks. +3 +THE PROPOSED APPROACH +3.1 +Problem Formulation +We denote the classification model as fθ : X → Y, where the +model parameters θ are unknown in the black-box attack setting, +and X and Y are the input and output spaces, respectively. Given +an input example x, the index of its ground-truth label is denoted as +y; fθ(x, i) ∈ [0, 1] indicates the posterior probability w.r.t. the i-th +class, and fθ(x, i) denotes the corresponding logit. Our attack goal +is to generate an adversarial perturbation δ for the benign example +x to fool the model fθ(·), given that the model parameters θ are +unknown, dubbed score-based black-box adversarial attack. It can +be generally formulated as the following optimization problem: +min +δ∈Bϵ(x) Ladv(δ, x, y) = +� +max(0, △ut), +if untargeted attack +max(0, △t), +if targeted attack +(2) +where △ut = fθ(x + δ, y) − maxj̸=y fθ(x + δ, j), and △t = +maxj̸=y fθ(x + δ, j) − fθ(x + δ, t), with t ∈ Y being the target +label. Note that Ladv is non-negative, and if 0 is achieved, then the +corresponding δ is a successful adversarial perturbation. +3.2 +Meta Conditional Generator for Black-Box Attack +Conditional Perturbation Generator. Unlike most previous +combination-based methods that use a deterministic network to +predict the initial perturbation for a benign image, our generator +captures a conditional distribution of perturbation conditioned +on the benign image, i.e., p(δ|x; ϕ), where δ = Gϕ(z; x). G +is the generator with ϕ as its parameters. δ is the perturbation +and z is a random vector that follows a simple distribution, e.g., +Gaussian distribution. In the conditional distribution, effective +… +Tasks +𝒯1 +𝒯2 +𝒯𝑁 +Meta Train +Meta Generator +Meta Test +New Task +𝑵(𝝁, 𝚺) +Adaptation +Prior Learning +Perturbation +Generator +Target +Logit +Inconsistency +𝑵(𝝁, 𝚺) +Gaussian +Model +Sampling +Fig. 2. Task definition in meta-learning. Given an image and a target +model, the goal is to learn a generator that can generate an effective +adversarial example to attack the target model. +… +Tasks +𝒯1 +𝒯2 +𝒯𝑁 +Meta Train +Meta Generator +Meta Test +New Task +𝑵(𝝁, 𝚺) +Adaptation +Prior Learning +Perturbation +Generator +Target +Logit +Inconsistency +𝑵(𝝁, 𝚺) +Gaussian +Model +Fig. 3. Overview of meta-learning. During the meta-train phase, a meta +generator can be obtained by training on a large set of tasks, which +contains the generic prior of how to generate effective perturbations for +different images to attack the target model. During the meta-test phase, +the meta generator can be quickly adapted to new tasks with only a few +steps of fine-tuning. +perturbations are supposed to have a high probability of being +sampled. When attacking the target model, we can sample a +perturbation δ ∼ p(δ|x; ϕ) and add it to the benign image to +construct an adversarial example to fool the target model. +Modeling Conditional Distribution. The generator captures a +conditional distribution of perturbation, which can be realized with +using a simple distribution and a complex non-linear function that +maps the simple distribution to a complex one with an image as +the condition. In this work, we use a conditional generative flow +(c-Glow) [49] as the generator due to its superior property that the +mapping between a random vector and the output perturbation is +invertible. By using c-Glow, we have δ = gϕ(z; x) and the inverse +version z = g−1 +ϕ (δ; x). The random vector z follows a Gaussian +distribution, i.e., z ∼ N(µ, Σ). Since c-Glow consists of a set of +invertible functions/layers, the parameters ϕ can be decomposed +into several individual parts, i.e., g = gx,ϕ1 ◦· · ·◦gx,ϕM . M is the +number of layers and ϕi represents the parameters of the i-th layer. +With the change of variables [50], we can write the conditional +likelihood as +log p(δ|x; ϕ) = log p(z) + +M +� +i=1 +log +�����det(∂g−1 +ϕi (ri−1; x) +∂ri−1 +) +����� , +(3) +where ri = gϕi(ri−1; x), r0 = x, and rM = z. +Perspective of Meta Learning. Attacking on one benign image +can be viewed as an individual process where the generator can be +optimized by sampling several perturbations from the conditional +distribution and obtaining their corresponding feedback scores +from the target model as supervision. However, when attacking a +black-box model, the budget of query is always limited, i.e., we +can only sample a few perturbations. Hence, the learning of the +generator in each attacking process can be formulated as a few-shot +learning problem, i.e., given {δi, fθ(xi), yi}K +i=1, the goal is to +learn the generator Gϕ(z; x). K is the number of shots. +As mentioned in Sec. 1, adversarial perturbations around +different benign images may share certain common properties, i.e., + +JOURNAL OF LATEX CLASS FILES, VOL. , NO. , +5 +𝒙 +Perturbation +Generator +Logit +𝑳𝐚𝐝𝐯 +𝑵(𝝁, 𝚺) +Gaussian +Surrogate +Model +Target +Inner Loop Update +Outer Loop +Update +𝑵(𝝁, 𝚺) +Gaussian +Meta Generator +𝐺(𝒛; 𝒙, 𝝋) +𝑔𝒘(𝒙) +… +A Batch of Tasks +𝒯1 +𝒯2 +𝒯𝑛 +𝜹 +ෝ𝒚 +𝒚 +𝒙 + 𝜹 +Sampling +Sampling Task 𝓣𝒊 +Fig. 4. The pipeline of meta training. The batch version of REPTILE is +exploited for meta training. We sample a batch of tasks and perform the +inner loop update of task-specific parameters for each task. Then the +task-specific parameters of all tasks in the batch are aggregated to do +the outer loop update, i.e., updating the meta parameters. +example-level adversarial transferability. Inspired by the concept +of “learning to learn” in meta-learning, we can solve the few-shot +learning problem from the perspective of meta-learning, and learn +a meta generator to capture the common properties of perturbations +by performing a large set of attacking tasks. The prior of how +to generate a conditional distribution of perturbation around a +benign image is learned across various and diverse tasks. Since the +perturbation distributions of different benign images are generated +through the same generator, the prior is implicitly encoded in the +parameters of the generator. +Task Definition. Task is the atom in meta-learning. In the scenario +of adversarial attack, “task” is defined as: “given a benign image +and a target model, the goal is to learn a conditional generative +model from which a sampled perturbation could successfully fool +the target model. ” As shown in Fig. 2, given a benign image, we +can sample a perturbation from the generative model. The addition +of the benign image and the perturbation yields an adversarial +example which is then fed into the target network for attack. The +adversarial example is effective if its the predicted label is not +consistent with the ground truth label (i.e., untargeted attack) or the +predicted label of the adversarial example is the same as a specified +label (i.e., targeted attack). Hence, the parameters of the generator +are updated by maximizing the difference between the prediction +and the ground truth or minimizing the difference between the +prediction and the specified label. +Since the parameters and architecture of the target model are un- +known, inspired by the transferability of adversarial example [36], +we use a surrogate model to replace the target model in the meta- +train phase. As shown in Fig. 3, we can achieve a meta generator +by performing a large set of tasks, which captures the prior of +generating adversarial perturbations and is adaptive to different +tasks during the meta-test phase. +3.3 +Meta Training with A Surrogate Model +As described in Sec. 3.2, in a task T , given a benign image x and a +target model fθ(x, ·), the goal is to learn a conditional perturbation +generator Gϕ(z; x) that can generate an effective perturbation δ to +fool the target model, e.g., fθ(x + δ, y) < maxj̸=y fθ(x + δ, j) +for untargeted attack and fθ(x+δ, t) > maxj̸=t fθ(x+δ, j) for +targeted attack, where δ = Gϕ(z; x). The meta training process +is illustrated in Fig. 4. In this work, the target black-box model is +the same for different tasks. +In the scenario of black-box attack, we have no access to the +parameters and the architecture of the target model. Hence, we have +to make many queries for each benign image to generate successful +perturbations and then use them to learn the meta generator, which +is costly for query and hard to realize in real-world scenarios. +Based on the model-level adversarial transferability, in the meta- +train phase, the target model is replaced by a surrogate model +gw(x) of which the architecture and parameters are available. +Therefore, the gradients can backpropagate through the surrogate +model to update the generator. Since the meta-train phase does not +involve any target model, once the meta training is over, the meta +generator can be applied to any target model in the meta-test phase. +To evaluate the effectiveness of the generated perturbation, the +adversarial loss function is similar to that in Eq. (2). The difference +is that ˜△ut and ˜△t are computed by using the surrogate model +rather than the target model, i.e., +˜△ut = gw(x + δ, y) − max +j̸=y gw(x + δ, j), +˜△t = max +j̸=y gw(x + δ, j) − gw(x + δ, t), +(4) +where t is the specified target class in targeted attack. +Given a set of tasks {Ti}N +i=1, we follow the batch version of +REPTILE [51] to perform meta-learning. We sample n tasks to +form a batch and update the task-specific parameters k times +using Adam [52] for each task. The objective of the inner +loop optimization is φTi = arg minφTi Ladv +Ti . The optimization +procedure can be represented as +φTi = Adam(Ladv +Ti , ϕ, k, α), +(5) +where φTi is the final task-specific parameters of the generator +after k steps of performing Adam, starting from ϕ. At each of +the k steps, a perturbation is sampled from the current conditional +distribution. Ladv +Ti is the adversarial loss of the i-th task. α is the +learning rate of the inner loop. +Then, for the outer loop optimization, we can update the +meta parameters of the generator with the resulted task-specific +parameters in a mini-batch, i.e., +ϕ ← ϕ + β 1 +n +n +� +i=1 +(φTi − ϕ) , +(6) +where β is the learning rate of the outer loop. +3.4 +Meta Test Using Historical Attack Experience +The standard meta-test process is not applicable in black-box attack +because the target model is unknown and the gradient cannot be +backpropagated to fine-tune the meta generator through it. To solve +this issue in the meta-test, we propose to transfer the information +of the black-box target model to a surrogate model by using the +feedback of previous attacks to fine-tune the surrogate model. The +usage of the historical attack experience could make the surrogate +imitate the behaviour of the target network, which provides more +accurate and effective information to fine-tune the meta generator +than directly using the surrogate model. The pipeline of meta-test is +illustrated in Fig. 5. Given a new benign image, it consists of three +steps to conduct the attack to the target model, i.e., fine-tuning the +surrogate model for introducing the information of the target model, +fine-tuning the meta generator for adaptation to the new benign +image, and boosting off-the-shelf black-box attack methods. The +attacking ability of the whole framework is gradually improved in +the process of circulation. + +JOURNAL OF LATEX CLASS FILES, VOL. , NO. , +6 +𝑷𝝋′(𝜹|𝒙) +Distribution +Previous +Tasks +History Attack Info +Perturbation +Target +Logit +𝑵(𝝁, 𝚺) +Gaussian +𝑳𝐚𝐝𝐯 +Surrogate +Model +Surrogate +Model +New Tasks +𝑳𝐂𝐄 +Adapted Generator +Meta Generator +Black-Box +Optimization +/Search +Perturbation +Target Black- +Box Model +𝑓𝜽(𝒙) +Logit +Target +Optimized +Perturbation +(b) Fine-tune Surrogate Model +(a) Fine-tune Meta Generator +(c) Attack Target Black-Box Model +Attack +Info +𝑵(𝝁, 𝚺) +Gaussian +Fig. 5. The pipeline of meta test. (a) Fine-tune the meta generator. Given the image of the new task and the updated surrogate model, the meta +generator is fine-tuned by performing the inner optimization with Ladv. (b) Fine-tune the surrogate model. In order to transfer the information of the +target black-box model to the surrogate model, the historical information (i.e., logits of adversarial examples and the target labels) of previous tasks +as well as the current task are used to make the surrogate model mimic the behaviour of the target model. (c) Attack the target black-box model. +The adapted generator is combined with the off-the-shelf black-box attack methods. The generator can provide an initial distribution of perturbation +or a sampled perturbation according to the given image. The initialization is leveraged by the attack methods as the starting state to get a refined +perturbation. Then, the generated adversarial example is used to attack the target model. The logits of the attack are recorded to fine-tune the +surrogate model in stage (b). +Fine-tuning the Surrogate Model. In Fig. 5(b), to use historical +attack experience, the predicted logits of previous benign images, +their adversarial examples, and the current benign image by the +target model are collected to provide supervision for the fine-tuning +of the surrogate model. The loss function is defined as +LCE = CE(gw(x + δ), fθ(x + δ)) ++ CE(gw(x), fθ(x)) +(7) +where CE(·) is the cross entropy loss function. The two terms +represent the losses of the adversarial example and the benign +image, respectively. For the benign image in the current task, only +the second term is used. The optimization of the parameters of the +surrogate model can be represented as +w′ = Adam( +m +� +i +LCE +Ti , w, s, λ), +(8) +where w′ represents the parameters of the updated surrogate model. +λ is the learning rate and m is the number of tasks in a mini-batch. +LCE +Ti is the loss for the i-th task. s is the number of update steps. +Fine-tuning the Meta Generator. Fig. 5(a) presents the fine- +tuning of the meta generator with using the updated surrogate +model gw′(x) for the benign image of the new task. The fine- +tuning procedure in the meta-test phase is similar to the inner +optimization in the meta-train phase. We use the loss in Eq. (4) and +Eq. (5) to update the parameters of the meta generator, i.e., +φT = Adam(Ladv +T , ϕ, k, α), +(9) +where φT represents the adapted parameters for the new task. +Different from Eq. (5), the updated surrogate model is used +in Eq. (9) to yield out the loigts for the adversarial examples, +i.e., gw′(x + δ). As the updated surrogate model contains the +transferred information from the target model, it can provide more +accurate and effective supervision to fine-tune the meta generator +than the original surrogate model. Hence, the generated perturbation +is more specific to the target model. +Boosting Off-the-shelf Black-Box Attack Methods. Our adapted +generator can provide an initial distribution of perturbation or a +perturbation conditioned on the given benign image, enabling it +be combined with other off-the-shelf black-box attack methods to +boost their original performance. Fig. 5 (c) shows the process of +attacking the target model. The initial distribution or perturbation +is leveraged by a black-box attack method as the starting state. The +further optimized perturbation is then added to the benign image +to generate an adversarial example. The output logits from the +target model are recorded as historical attack information, which +is then used to fine-tune the surrogate model. When combined +with sampling-based methods [12], [53], the adapted generator +served as a distribution. When combined with random-search- +based methods [10], [11], [17], a perturbation can be a sample +from the generator and serves as an initial state. +4 +EXPERIMENTS +4.1 +Experimental Settings +Datasets. To demonstrate the effectiveness of the proposed method, +we conduct comprehensive experiments on two commonly used +benchmark databases, i.e., CIFAR-10 [54] and ImageNet [55]. +Following the setting in [53], for the CIFAR-10 dataset, we +randomly select 1, 000 images from the testing set for evaluation +which cover all classes evenly. The images are resized to 32 × 32. +For the ImageNet dataset, we first randomly select 10 classes + +JOURNAL OF LATEX CLASS FILES, VOL. , NO. , +7 +from the 1, 000 classes and then use the 500 images of each class +from the validation set for evaluation. The images are resized to +224 × 224. The target and surrogate models are trained on the +training set of the corresponding dataset. On CIFAR-10, we use +the full training set for meta-learning to learn the meta generator. +On ImageNet, the training set of the 10 chosen classes are used for +meta-learning. +Evaluation. We select l∞-based attacks and set the maximal +distortion as ϵ = 0.031 for CIFAR-10 and ϵ = 0.05 for ImageNet +with image pixel values re-scaled to [0, 1]. We set the maximal +query budget to 10,000 times in all experiments. If the attacker +cannot successfully fool the target m odel within the query limit, we +consider it a failure case. Following the prior work [53], we adopt +the attack success rate (ASR), the mean query number (Mean), +and the median query number (Median) of successful attacks to +evaluate the attack performance. +Target and Surrogate Models. On CIFAR-10, we consider four +target models: ResNet-Preact-110 [56], DenseNet-121 [57], VGG- +19 [58], and PyramidNet-110 [59]. We follow the standard training +process of image classification to obtain the checkpoints of these +target models. The top-1 error rates of these four target models +are 6.29%, 6.17%, 7.28%, and 7.51% on the standard testing +set, respectively. On ImageNet, we also evaluate our method on +four target models: ResNet-18 [56], VGG-16-BN [58], WRN- +50 [60], and InceptionV3 [61]. We use the official implementation +of these methods and download their pre-trained checkpoints +from torchvision. The top-1 error rates of these target models +are 28.41%, 30.24%, 21.53%, and 22.71% on the validation set +of ImageNet, respectively. In all experiments, ResNet-18 [56] and +ResNet-50 [56] are used as the surrogate models on CIFAR-10 +and ImageNet, respectively. The corresponding top-1 error rates +of the surrogate models are 6.37% for ResNet-18 and 23.97% for +ResNet-50. +To further verify the performance of our framework, we also +conduct experiments of attacking black-box adversarial defense +models on ImageNet, including JPEG-Compression-WideResNet- +50 [62], Small-Noise-Defense-WideResNet-50 [63], FreeAdv- +ResNet-50 [64], and FastAdv-ResNet-50 [65]. +Competing Methods. As our framework can provide a good +initialization of perturbation, it can be treated as a plug-and-play +component that can be combined with other black-box attack +methods to boost their performance. In order to verify the versatility +of our model, we combine our framework with 6 query-based +black-box attack methods, including NES [12], CG-Attack [53], +SimBA [10], SignHunter [11], Square [17], and MetaAttack [20]. +For search-based methods such as SignHunter, Square, SimBA, +and MetaAttack, a sampled perturbation from the generator is the +initialization. For sampling-based methods such as NES and CG- +Attack, a distribution of perturbation represented by the generator +is the initialization. +We also compare with several transfer-based attack methods +to verify the transferability of the proposed method, including +PGD [66], MI [34], TIMI [67], and DI [68], under the transfer +attack setting where only the information of the surrogate model is +accessible. Moreover, we finally compare with several combination- +based methods under the query attack setting, including Ad- +vFlow [22] and TREMBA [23]. They exploit both the queries to +the target model and the transferred item from the surrogate model. +To ensure the fairness of comparison, we retrain the combination- +based methods with the same surrogate model as ours. All the +experiments are implemented with the source code provided by +their authors under the same setting. +Implementation Details. Following [49], in all the experiments, +we adopt the same architecture for the generator c-Glow with 3 +blocks composed of 8 flow steps. Each block starts with a squeeze +operation followed by 8 flow steps and ends with a split operation. +To improve the efficiency, we adopt the discrete cosine transform +(DCT) and inverse DCT for dimension reduction by downsampling +the size of images in ImageNet to 1 +8 × 1 +8 lower frequency subspace +before feeding them into the generator. On CIFAR-10, we use the +original shape of images as they are in a small size. +Before meta training, we pre-train the c-Glow to provide an +initial state of modelling the distribution of perturbation. We +use the surrogate model to generate a large set of perturbations +through PGD attack with the perturbation strength of ℓinf = 0.05, +the step size of 0.01, and the number of iterations of 50. The +parameters are optimized by maximizing the log-likelihood, i.e., +maxϕ log p(δ|x; ϕ). +For the pre-training of the generator, the learning rate is set to +0.001. The batch size is m = 16. The generator is trained for 10 +epochs. For meta training, we sample 16 tasks every batch. The +update stepsize of the inner optimization is set to k = 4. The +learning rates of the inner and outer loops are set to α = 0.0003 +and β = 0.0006, respectively. For meta-test, when fine-tuning the +surrogate model, we freeze the parameters of all layers except the +last three layers. The surrogate model is fine-tuned with both benign +images and their adversarial examples. The learning rate of fine- +tuning is set to λ = 0.0003. The number of benign images is set to +4 in a batch. When fine-tuning meta generator, the process is similar +to the inner loop optimization of the task-specific parameters. +4.2 +Experiments in Closed-set Attack Scenario +In the closed-set attack scenario, the surrogate and target models +are trained on the same training set, i.e., both the training images +and the categories are the same. Experiments includes boosting the +off-the-shelf black-box attack methods, attacking defended models, +comparisons with transfer-based methods, and comparisons with +combination-based methods. +Performance on CIFAR-10. The specification of the surrogate +model and the target models is presented in Sec. 4.1. Table 1 +illustrates the results of several off-the-shelf black-box attack +methods and those of the combinations with our proposed MCG. +The MCG can boost the attack efficiency of all the black- +box attack methods without ASR drop under both untargeted and +targeted attacks. Specifically, under the untargeted attack setting, +the median query numbers of these methods are decreased to +1s for all the target models by using the initialization from our +MCG, which means that we can fool the target models with the +initially generated perturbations for over 50% images. Meanwhile, +the ASRs are improved to nearly 100% for almost all the cases. +Besides, the mean query numbers are also improved significantly. +For example, for the state-of-the-art Square attack, our MCG can +further improve its query efficiency in terms of the mean query +number by a factor of 5 for all the four target models. We further +plot the tendency curves of ASR w.r.t. the query number in Fig. 6. +We can observe that the MCG can boost the attack performance +under all values of query number for all the competing attack +methods, especially for small query numbers. +Under the targeted attack setting, our proposed MCG can also +boost the attack performance. The targeted attack is generally + +JOURNAL OF LATEX CLASS FILES, VOL. , NO. , +8 +TABLE 1 +Closed-set evaluation on the CIFAR-10 dataset. +Target model → +ResNet-PreAct-110 +DenseNet-121 +VGG-19 +PyramidNet-110 +Attack Method ↓ +ASR +Mean +Median +ASR +Mean +Median +ASR +Mean +Median +ASR +Mean +Median +Untargeted Attack +NES [12] +100.0% +285.2 +211.0 +99.4% +430.8 +274.0 +99.1% +822.8 +421.0 +100.0% +287.9 +190.0 +MCG + NES +100.0% +124.1 +1.0 +100.0% +133.2 +1.0 +99.9% +371.7 +1.0 +100.0% +90.4 +1.0 +CG-Attack [53] +100.0% +230.0 +81.0 +100.0% +222.7 +21.0 +100.0% +386.4 +101.0 +100.0% +93.7 +1.0 +MCG + CG-Attack +100.0% +112.3 +1.0 +100.0% +113.4 +1.0 +100.0% +262.7 +1.0 +100.0% +74.1 +1.0 +SimBA-DCT [10] +100.0% +428.0 +359.5 +100.0% +449.9 +352.0 +99.7% +557.6 +426.0 +100.0% +341.1 +273.5 +MCG + SimBA-DCT +100.0% +199.3 +1.0 +100.0% +127.0 +1.0 +99.7% +234.8 +1.0 +100.0% +114.6 +1.0 +Signhunter [11] +100.0% +167.0 +83.0 +100.0% +196.6 +87.0 +100.0% +238.3 +99.0 +100.0% +128.4 +63.5 +MCG + Signhunter +100.0% +83.3 +1.0 +100.0% +61.3 +1.0 +100.0% +157.9 +1.0 +100.0% +26.9 +1.0 +Square [17] +100.0% +227.3 +144.5 +100.0% +260.3 +159.0 +100.0% +342.0 +175.5 +100.0% +165.5 +100.5 +MCG + Square +100.0% +39.7 +1.0 +100.0% +47.6 +1.0 +100.0% +57.9 +1.0 +100.0% +29.6 +1.0 +MetaAttack [20] +100.0% +642.6 +632.0 +99.8% +759.4 +635.0 +100.0% +686.4 +633.0 +100.0% +601.8 +430.0 +MCG + MetaAttack +100.0% +204.6 +1.0 +100.0% +81.4 +1.0 +100.0% +188.6 +1.0 +100.0% +99.3 +1.0 +Targeted Attack +NES [12] +100.0% +774.3 +610.0 +99.8% +1205.2 +946.0 +92.4% +2738.8 +1954.0 +100.0% +889.7 +694.0 +MCG + NES +100.0% +591.8 +443.0 +100.0% +1009.2 +800.0 +98.5% +2688.6 +1472.0 +100.0% +657.8 +506.0 +CG-Attack [53] +100.0% +884.8 +741.0 +99.8% +859.7 +681.0 +96.8% +1476.9 +901.0 +100.0% +567.3 +501.0 +MCG + CG-Attack +100.0% +430.8 +181.0 +99.8% +608.5 +241.0 +97.4% +944.1 +381.0 +100.0% +290.6 +141.0 +SimBA-DCT [10] +99.6% +836.3 +729.0 +99.7% +944.8 +842.0 +98.6% +1170.3 +986.0 +99.8% +735.7 +661.0 +MCG + SimBA-DCT +99.8% +664.2 +623.5 +99.8% +845.5 +768.00 +98.9% +983.3 +861.0 +99.9% +601.9 +543.5 +Signhunter [11] +100.0% +386.1 +272.0 +100.0% +465.1 +323.0 +100.0% +556.3 +385.5 +100.0% +320.9 +232.0 +MCG + Signhunter +100.0% +267.3 +124.5 +100.0% +321.0 +181.0 +100.0% +399.1 +210.0 +100.0% +167.9 +81.0 +Square [17] +99.6% +504.7 +369.0 +100.0% +624.8 +471.0 +100.0% +827.2 +593.5 +100.0% +400.1 +301.0 +MCG + Square +100.0% +92.3 +20.0 +100.0% +133.1 +33.0 +100.0% +141.7 +22.0 +100.0% +55.0 +17.0 +MetaAttack [20] +100.0% +1174.7 +899.0 +100.0% +1294.5 +1106.0 +99.0% +1721.4 +1106.0 +100.0% +1065.9 +890.0 +MCG + MetaAttack +100.0% +757.3 +669.0 +100.0% +992.9 +887.0 +100.0% +1180.7 +882.0 +100.0% +718.3 +669.0 +TABLE 2 +Closed-set evaluation on the ImageNet dataset. +Target Model → +ResNet-18 +VGG-16 +WRN-50 +Inception-V3 +Attack Method ↓ +ASR +Mean +Median +ASR +Mean +Median +ASR +Mean +Median +ASR +Mean +Median +Untargeted Attack +NES [12] +99.0% +2085.5 +1597.0 +98.2% +1554.0 +1072.0 +96.5% +2366.0 +1681.0 +95.8% +1111.6 +526.0 +MCG + NES +99.7% +1057.5 +1.0 +99.1% +596.5 +1.0 +98.4% +1385.1 +1.0 +96.7% +889.9 +390.5 +CG-Attack [53] +96.3% +272.0 +21.0 +96.5% +178.3 +21.0 +89.6% +323.0 +21.0 +94.1% +251.1 +21.0 +MCG + CG-Attack +100.0% +95.4 +21.0 +99.7% +69.3 +1.0 +99.2% +247.3 +21.0 +97.0% +271.1 +21.0 +SimBA-DCT [10] +100.0% +520.6 +381 +98.3% +486.1 +350.0 +94.9% +810.6 +593.0 +81.9% +496.9 +290.5 +MCG + SimBA-DCT +98.1% +68.7 +1.0 +98.8% +73.6 +1.0 +94.7% +159.6 +1.0 +95.0% +129.2 +9.5 +Signhunter [11] +100.0% +60.2 +23.0 +100.0% +69.8 +34.0 +100.0% +116.0 +38.5 +99.4% +178.3 +49.0 +MCG + Signhunter +100.0% +29.6 +1.0 +100.0% +19.0 +1.0 +100.0% +62.8 +1.0 +100.0% +91.8 +8.0 +Square [17] +100.0% +72.8 +26.0 +100.0% +82.6 +28.0 +100.0% +136.1 +68.0 +99.4% +210.8 +64.0 +MCG + Square +100.0% +31.7 +1.0 +100.0% +24.8 +1.0 +100.0% +59.9 +1.0 +100.0% +123.8 +24.0 +MetaAttack [20] +92.2% +3302.6 +2641.0 +95.4% +3075.2 +2213.0 +87.4% +3824.2 +3309.0 +94.7% +2634.8 +1772.0 +MCG + MetaAttack +94.1% +1692.3 +1.0 +96.8% +1025.9 +1.0 +92.5% +2076.8 +1.0 +95.3% +2107.1 +1324.0 +Targeted Attack +NES [12] +67.9% +5734.8 +5860.0 +79.4% +4944.1 +4621.0 +33.0% +6137.9 +6259.0 +63.3% +4921.1 +4726.0 +MCG + NES +75.3% +5499.6 +5294.0 +81.0% +4720.9 +4559.0 +37.5% +5839.1 +5882.0 +66.2% +4687.7 +4370.0 +CG-Attack [53] +96.9% +2553.3 +1801.0 +92.9% +2447.7 +1731.0 +77.8% +2976.5 +2191.0 +91.0% +2260.3 +1481.0 +MCG + CG-Attack +98.0% +2501.8 +1781.0 +91.9% +2374.4 +1721.0 +79.8% +2960.2 +2101.0 +94.9% +2272.1 +1441.0 +SimBA-DCT [10] +56.0% +6927.2 +7196.0 +71.7% +6569.9 +6507.0 +41.0% +6795.4 +7028.0 +56.8% +6094.9 +6107.0 +MCG + SimBA-DCT +66.8% +6460.8 +6607.0 +79.3% +6023.3 +6038.0 +60.6% +5744.8 +5626.0 +72.8% +5576.9 +5564.0 +Signhunter [11] +100.0% +1332.7 +922.0 +100.0% +1115.8 +734.0 +99.4% +1786.8 +1064.0 +100.0% +1836.2 +1063.0 +MCG + Signhunter +100.0% +1043.8 +596.5 +100.0% +788.8 +446.0 +99.7% +1428.4 +782.0 +99.7% +1486.7 +865.0 +Square [17] +100.0% +1097.7 +819.0 +100.0% +1112.7 +819.0 +99.7% +1678.7 +1242.0 +99.0% +1789.0 +1120.0 +MCG + Square +100.0% +797.6 +518.5 +100.0% +784.4 +529.0 +100.0% +1148.9 +661.0 +99.7% +1455.2 +868.0 +MetaAttack [20] +33.8% +8202.6 +8041.5 +17.5% +8341.4 +8806.0 +10.3% +8408.4 +8585.0 +19.5% +7176.8 +8141.5 +MCG + MetaAttack +45.6% +7315.1 +8033.0 +38.1% +6425.6 +7721.0 +20.6% +7029.9 +7596.5 +19.5% +6946.5 +7706.0 + +JOURNAL OF LATEX CLASS FILES, VOL. , NO. , +9 +1 +2 +3 +3.7 +log10(Queries) +0 +20 +40 +60 +80 +100 +ASR (%) +PreactResNet +Square +MCG+Square +SignHunter +MCG+SignHunter +CG-Attack +MCG+CG-Attack +NES +MCG+NES +SimBA +MCG+SimBA +1 +2 +3 +3.7 +log10(Queries) +0 +20 +40 +60 +80 +100 +VGG +1 +2 +3 +3.7 +log10(Queries) +0 +20 +40 +60 +80 +100 +ASR (%) +DenseNet +1 +2 +3 +3.7 +log10(Queries) +0 +20 +40 +60 +80 +100 +PyramidNet +Fig. 6. Attack success rate (ASR%) w.r.t. the query number of untargeted attack on the CIFAR-10 dataset. +TABLE 3 +Untargeted Attack against adversarial defended models on the ImageNet dataset. +Target Model → +JPEG-Compress-WRN-50 +SND-WRN-50 +FastAdv-ResNet-50 +FreeAdv-ResNet-50 +Attack Method ↓ +ASR +Mean +Median +ASR +Mean +Median +ASR +Mean +Median +ASR +Mean +Median +NES [12] +13.2% +5682.1 +3243.0 +76.6% +3879.8 +3308.5 +23.5% +7501.5 +7986.5 +15.7% +7188.5 +6322.0 +MCG + NES +98.9% +1412.1 +1.0 +81.7% +1795.7 +1.0 +23.5% +2415.4 +715.0 +22.4% +2104.1 +685.0 +CG-Attack [53] +39.0% +2227.5 +1161.0 +79.8% +540.5 +81.0 +58.5% +1789.7 +871.0 +61.3% +2374.0 +1121.0 +MCG + CG-Attack +91.7% +180.9 +1.0 +79.8% +331.5 +1.0 +64.3% +1634.4 +801.0 +64.7% +2305.1 +1081.0 +SimBA-DCT [10] +14.3% +4688.3 +4545.0 +10.5% +608.3 +22.0 +45.5% +702.5 +72.5 +52.7% +792.5 +245.0 +MCG + SimBA-DCT +67.1% +543.9 +1.0 +60.4% +498.3 +1.0 +45.5% +634.4 +34.0 +52.7% +821.9 +77.0 +Signhunter [11] +100.0% +120.7 +39.0 +89.4% +133.4 +31.0 +89.4% +1101.5 +37.5 +88.2% +931.3 +42.0 +MCG + Signhunter +100.0% +69.8 +1.0 +89.1% +65.6 +1.0 +89.4% +1083.5 +36.5 +86.5% +1058.3 +40.0 +Square [17] +100.0% +140.9 +71.0 +88.9% +1487.8 +154.0 +92.4% +940.5 +182.0 +91.4% +871.3 +170.0 +MCG + Square +100.0% +57.5 +1.0 +90.7% +307.4 +1.0 +92.4% +864.9 +137.0 +91.4% +724.9 +123.5 +MetaAttack [20] +8.0% +3515.9 +1557.0 +11.7% +2449.1 +1777.0 +53.8% +4832.2 +5063.5 +55.3% +4614.9 +4631.0 +MCG + MetaAttack +50.9% +117.2 +1.0 +53.1% +535.7 +1.0 +55.7% +4119.9 +3969.5 +56.3% +3580.2 +3094.0 +TABLE 4 +Comparison with combination-based methods on the ImageNet dataset +Target Model → +ResNet-18 +VGG-16 +WRN-50 +Inception-V3 +Attack Method ↓ +ASR +Mean +Median +ASR +Mean +Median +ASR +Mean +Median +ASR +Mean +Median +Untargeted Attack +TREMBA [23] +100.0% +664.5 +169.0 +99.1% +160.2 +1.0 +99.0% +697.6 +148.0 +96.9% +1232.7 +211.0 +AdvFlow [22] +100.0% +578.8 +400.0 +100.0% +693.3 +420.0 +100.0% +937.9 +400.0 +97.2% +716.3 +200.0 +MCG + CG-Attack +100.0% +95.4 +21.0 +99.7% +69.3 +1.0 +99.2% +247.3 +21.0 +97.0% +271.1 +21.0 +Targeted Attack +TREMBA [23] +81.4% +3197.2 +1997.5 +91.1% +2493.5 +1759.0 +60.2% +5140.5 +3529.0 +13.3% +9462.1 +9433.0 +AdvFlow [22] +83.8% +4650.0 +4400.0 +81.6% +4407.8 +4200.0 +61.4% +5210.6 +4980.0 +81.8% +3734.1 +3200.0 +MCG + CG-Attack +98.0% +2501.8 +1781.0 +91.9% +2374.4 +1721.0 +79.8% +2960.2 +2101.0 +94.9% +2272.1 +1441.0 +harder to achieve than the untargeted attack, yet our MCG still +obtains a satisfactory ASR attacking all the four target models. The +best attack performance is achieved by MCG+Square. Compared +to the original Square attack, the MCG+Square achieves the ASR +of 100% with a significantly fewer number of queries. For example, +the mean and median query numbers of MCG are over 4.6 times +and 14.3 times less than those of the original Square attack. +Performance on ImageNet. ImageNet is a much larger dataset +than CIFAR-10. The performance of both untargeted and targeted +attacks on ImageNet is reported in Table 2. +We have a similar observation that the proposed MCG obtains +consistent improvements when combined with different attack +methods. Under the untargeted attack setting, the improvement of +ASR in several cases can be apparently observed. For example, +when attacking WRN-50 and VGG-16 with CG-Attack, the ASRs +are 89.6% and 96.5%, respectively. Our MCG further improves +CG-Attack by about 10% for attacking WRN-50 and 3% for +attacking VGG-16. Besides, the efficiency improvements are also +noticeable, especially for NES and SimBA. Under the targeted +attack setting, the MCG brings in different degrees of improvements +in almost all cases. The increase in ASR is more evident than +the untargeted attack setting. Although when combined with +SignHunter against InceptionV3, the ASR has been slightly reduced +(0.3% lower) while the attack cost is saved near 20%. +All these results demonstrate that our MCG can provide an +effective initial perturbation or a distribution of perturbation for +various off-the-shelf black-box attack methods to boost their +performance. Please note that the meta training procedure of the +meta generator does not involve any target model or attack method. +Hence, the meta generator only needs to be trained once, and it +can be combined with different black-box attack methods to attack +different black-box models without re-training, which corroborates + +JOURNAL OF LATEX CLASS FILES, VOL. , NO. , +10 +TABLE 5 +Comparison with Transfer-based methods on the ImageNet dataset. +Target model → +ResNet-18 +VGG-16 +WRN-50 +Inception-V3 +Attack Method ↓ +ASR +ASR +ASR +ASR +PGD [66] +35.5% +29.6% +36.6% +15.7% +PGD + MCG w/o surrogate +56.8% +70.0% +49.5% +26.4% +PGD + MCG w/ fixed surrogate +57.9% +70.4% +50.1% +26.4% +MI [34] +56.1% +62.0% +66.8% +26.4% +MI + MCG w/o surrogate +62.6% +71.6% +67.6% +45.1% +MI + MCG w/ fixed surrogate +63.2% +73.6% +69.3% +45.4% +TIMI [67] +56.7% +63.4% +62.3% +42.5% +TIMI + MCG w/o surrogate +66.4% +62.1% +58.3% +42.7% +TIMI + MCG w/ fixed surrogate +66.7% +63.4% +59.4% +45.4% +DI [68] +44.2% +42.0% +43.3% +20.1% +DI + MCG w/o surrogate +68.5% +80.0% +67.1% +49.6% +DI + MCG w/ fixed surrogate +69.2% +80.0% +68.7% +50.7% +the flexibility and generalization ability of the proposed framework. +Attacking Defended Models. To verify the effectiveness of the +proposed method against adversarial defense models, we perform +experiments of attacking various defended models trained with +different defense strategies, including JPEG-Compression [62], +random noise perturbation [63], and adversarial training [64]. +JPEG-Compression defense (i.e., JPEG-Compress-WRN-50) +attempts to remove the influence of adversarial examples through +the compression process. Small-Noise-Defense (i.e., SND-WRN- +50) is specially designed for query-based attack via introducing +additional random noise to hinder the attacker to estimate gradients +correctly. As shown in Table 3, when attacking JPEG-Compress- +WRN-50 and SND-WRN-50, the performance of NES, CG-Attack, +SimBA-DCT, and MetaAttack drops sharply. In contrast, the com- +bination with our framework greatly improves their performance in +all the three metrics. For example, when attacking JPEG-Compress- +WRN-50 with CG-Attack, the ASR is only 39% and the median +query number is 1161, which fails in most attacks. Our framework +boosts its ASR to 91.7% and reduces its median query number to +1. Besides, our method also reduces the mean and median query +numbers for Square and Signhunter. +Adversarial training enlarges the difference of the classification +boundaries between the robust model and the vanilla model and +greatly limits the model-level adversarial transferability. As shown +in Table 3, when attacking the models of adversarial training, +the attack performance of all methods drops a lot compared to +attacking the vanilla models. Our method can still improve the +attack performance in most cases, though the improvements are +not as significant as those of attacking the vanilla models. +Comparison with Transfer-based Methods. To verify the effec- +tiveness of example-level adversarial transferability boosted by +meta learning, we perform an untargeted attack experiment to +compare our meta generator with several transfer-based attack +methods in the transfer attack setting, i.e., no information from the +black-box target model is used for fine-tuning. +Our meta generator contains a pre-training stage of the c- +Glow and the pre-training is performed by using an attack method +to generate adversarial examples for training. The quality of +the adversarial examples affects the performance a lot. In other +experiments, the attack method is PGD. Since the transfer-based +methods can generate better transferable adversarial examples than +PGD, to fairly compare with the transfer-based methods, here we +pre-train the meta generator with using the adversarial examples +generated by the three transfer-based methods, respectively. +The results are shown in Table 5. ‘X + MCG w/o surrogate’ +means that we use ‘X’ to pre-train the c-Glow and then directly +use the meta generator to produce adversarial example without +using the surrogate model. ‘X + MCG w/ fixed surrogate’ means +that after pre-training we use the surrogate model to fine-tune the +meta generator first and then use the meta generator to produce +adversarial examples. As shown in Table 5, directly using our meta +generator has better performance than the transfer-based methods in +most cases. Fine-tuning the meta generator with the surrogate model +can further improve the performance. Transfer-based methods may +overfit the surrogate model due to that the adversarial example +generation totally depends on the surrogate model. Differently, our +meta generator captures the example-level transferability that can +alleviate the overfitting issue. The generation is determined by both +the learned prior of the meta generator and the surrogate model. +Comparison with Combination-based Methods. As our method +can be treated as a combination of transfer-based and query-based +method, we compare with two combination-based black-box attack +methods, i.e., TREMBA [23] and AdvFlow [22]. Since they take +advantage of model-level adversarial transferability to improve +the performance and are often combined with evolution methods +to adjust their distribution mapping. For fairness, in the meta- +test phase, we integrate the distribution adjustment method CG- +Attack [53] into our framework for evaluation. +The results on ImageNet are shown in Table 4. Our method +achieves the best performance under almost all attack cases in all +the three metrics. Although other methods attempt to utilize the +transferability between different models, they appear to be unstable +in ASR when attacking different target models. For example, when +attacking ResNet-18 and VGG-16 under targeted attack, the ASRs +of TREMBA are 81.4% and 91.1%. But when attacking WRN-50 +and Inception-V3, the ASRs drop to 60.2% and 13.3%. The results +show that TREMBA cannot generalize well to attack different +target models. Differently, our method can perform better in the +generalization ability to attack different models. The experimental +results on CIFAR-10 are presented in the Appendix Sec. B.1. +4.3 +Experiments in Open-set Attack Scenario +In the closed-set attack scenario, the surrogate and target models +share the same training dataset, i.e., the training dataset of the target +model is visible to attackers, which is hard to achieve in the real +attack scenario. In real-world scenarios, the surrogate model will +share less common knowledge with the target model. This situation +strongly increases the difficulty of attack. Therefore, in this section, +we verify the effectiveness of our method in the open-set attack +scenario where the surrogate and target models are trained on +disjoint training datasets, and there is no overlap between the +output categories of the two models. In this open-set attack setting, +it is quite challenging for the attacker to transfer the limited prior +to the unknown areas. Specifically, in our experiments, we employ +two datasets and train the surrogate model on one dataset and the +target model on another. The training data of the meta generator +is the same as the data used for the surrogate model. The results +of training on CIFAR-10 and testing on CIFAR-100 are shown +in Table 6. Please note that other experiments on ImageNet and +OpenImage [69] are presented in the Appendix Sec. B.2. +Since the surrogate and target models are trained from different +training sets with different categories, the effectiveness of model- +level adversarial transferability is limited, which decreases the +performance compared with the results in Table 1. Nevertheless, + +JOURNAL OF LATEX CLASS FILES, VOL. , NO. , +11 +TABLE 6 +Untargeted Attack trained on the CIFAR-10 dataset and tested on the CIFAR-100 dataset in the open-set scenario. +Target model → +ResNet-18 +VGG-16 +WRN-50 +Inception-V3 +Attack Method ↓ +ASR +Mean +Median +ASR +Mean +Median +ASR +Mean +Median +ASR +Mean +Median +NES [12] +96.1% +2054.0 +1555.0 +84.3% +1691.0 +904.0 +89.5% +2229.6 +1534.0 +94.6% +2044.7 +1471.0 +MCG + NES +97.0% +773.1 +1.0 +88.5% +910.5 +1.0 +94.0% +833.7 +1.0 +95.7% +1025.2 +1.0 +CG-Attack [53] +99.4% +226.2 +1.0 +98.1% +294.7 +1.0 +98.4% +285.2 +1.0 +99.1% +330.8 +1.0 +MCG + CG-Attack +99.5% +135.1 +1.0 +97.8% +244.8 +1.0 +98.7% +163.5 +1.0 +99.6% +196.5 +1.0 +SimBA-DCT [10] +99.2% +222.8 +84.0 +96.8% +321.4 +66.0 +97.4% +268.1 +86.5 +98.5% +231.6 +76.0 +MCG + SimBA-DCT +99.2% +119.5 +1.0 +97.0% +218.3 +1.0 +98.0% +155.2 +1.0 +98.9% +184.1 +1.0 +SignHunter [11] +100.0% +88.2 +28.0 +99.6% +136.7 +22.0 +99.8% +134.5 +28.0 +100.0% +117.8 +32.0 +MCG + SignHunter +100.0% +57.8 +1.0 +99.6% +110.2 +1.0 +99.8% +94.1 +1.0 +100.0% +92.0 +1.0 +Square [17] +99.9% +103.8 +17.0 +99.5% +213.9 +24.0 +99.8% +178.1 +16.0 +99.8% +121.2 +15.0 +MCG + Square +99.9% +47.9 +1.0 +99.5% +92.7 +1.0 +99.8% +83.1 +1.0 +99.9% +75.6 +1.0 +MetaAttack [20] +100.0% +518.9 +443.0 +99.7% +591.4 +443.0 +99.9% +610.1 +447.0 +100.0% +523.6 +445.0 +MCG + MetaAttack +99.9% +227.4 +1.0 +99.9% +396.4 +1.0 +100.0% +303.6 +1.0 +100.0% +291.4 +1.0 +TABLE 7 +Untargeted Attack against Imagga tagging API. +Baseline +Combined with MCG +Method +ASR +Mean +Median +ASR +Mean +Median +NES [12] +44.0% +35.3 +21.0 +67.0% +8.8 +1.0 +CG-Attack [53] +74.0% +33.5 +21.0 +81.0% +21.4 +1.0 +SimBA-DCT [10] +45.0% +93.8 +56.0 +57.0% +45.5 +1.0 +SignHunter [11] +51.0% +45.3 +20.5 +82.0% +19.5 +1.0 +Square [17] +49.0% +50.8 +15.5 +69.0% +13.7 +1.0 +MetaAttack [20] +16.0% +230.1 +95.0 +61.0% +101.4 +1.0 +our framework still works in boosting the existing black-box attack +methods in the open-set setting. MCG can reduce the query cost of +attacks and improve the ASR in most cases, which demonstrates +that the meta generator can be fast-adapted to attack different target +models across different datasets. +4.4 +Experiments of Attacking Real-World API +To verify the effectiveness of our framework in real-world scenarios, +we perform an experiment of attacking the Imagga Tagging API1. +The model of Imagga is trained on an unknown dataset of over +3, 000 types of daily-life objects. Given a query image, the API +will return a list of possible labels as well as the corresponding +confidence scores. We randomly select 100 images from the +validation set of ImageNet and set the query limit to 500. We +define the goal of untargeted attack as removing the top-3 labels +of the benign images. As the images are from ImageNet, the pre- +trained surrogate model can be used to fine-tune the meta generator. +Ladv is set as the maximal score of the top-3 labels. The adapted +generator is then used to generate an initial perturbation for existing +black-box attack methods to attack the API. As shown in Table 7, +the performance of all methods is significantly improved through +the combination with our framework. Specifically, the median query +numbers are decreased to 1s for all methods and the mean query +numbers are also highly improved. These results demonstrate that +our framework is applicable in real-world scenarios. +4.5 +Ablation Study +To verify the effectiveness of the meta training and the fine-tuning +stages in the meta-test, we conduct experiments of untargeted +1. https://imagga.com/solutions/auto-tagging +TABLE 8 +Ablation study on the ImageNet dataset. All methods in the table are +combined with Square attack for untargeted attack. +Target Model → +ResNet-18 +VGG-16 +Attack Method ↓ +FASR +Mean +Median +FASR +Mean +Median +Flow +35.2% +48.6 +13.0 +46.1% +39.1 +4.0 +MCG w/ fixed surrogate +57.9% +35.3 +1.0 +70.4% +30.0 +1.0 +MCG w/ fine-tuned surrogate +60.1% +31.7 +1.0 +71.3% +24.8 +1.0 +attacks on ImageNet for ablation study. The results are shown in +Table 8. ‘Flow’ is the method that directly uses the perturbations +to learn a conditional glow (c-Glow) model, rather than using the +adversarial loss or the parameter update strategy in meta learning. +The perturbations are generated by applying Projected Gradient +Descent (PGD) [66] attack to the surrogate model. During testing, +no fine-tuning is conducted. ‘MCG w/ fixed surrogate’ means +that during testing we fine-tune the meta generator with a fixed +surrogate model. ‘MCG w/ fine-tuned surrogate’ means that during +testing we update the surrogate model first by exploiting the query +feedback from the target model, and then we use the updated +surrogate model to fine-tune the meta generator. All these methods +are combined with Square attack to query the target model. To +evaluate the effectiveness of the initial perturbations, we use the first +Attack Success Rate (FASR) as the metric instead of ASR. FASR +means the success rate of straightforwardly using the perturbation +generated by the generators to attack the target model. +As shown in Table 8, compared to Flow, MCG w/ fixed surro- +gate achieves much better performance in all the three metrics. The +FASRs and the median query numbers are significantly improved. +The results demonstrate the effectiveness of the meta training +formulation, which improves the example-level transferability by +capturing more effective generic prior of how to attack different +samples. The provided initial perturbation is better than that from +Flow. Moreover, compared to MCG w/ fixed surrogate, MCG +w/ fine-tuned surrogate gains further improvements in the FASR +and the attack efficiency, which corroborates the effectiveness of +the historical attack information, i.e., we can get a better-adapted +generator by transferring the information of the target model to the +surrogate model. +5 +CONCLUSION +We propose a novel framework for black-box attack by formulating +it as a meta-learning problem to improve the example-level + +JOURNAL OF LATEX CLASS FILES, VOL. , NO. , +12 +adversarial transferability as well as the efficiency of attack. As +the architecture and parameters of the black-box target model +are unknown, we propose to perform the meta training with a +surrogate by leveraging the model-level adversarial transferability. +Since the standard meta-test process cannot be applied to the black- +box attack, we propose a three-stage attack pipeline to fine-tune +the meta model, including fine-tuning the surrogate model with +historical attack information of the target model, fine-tuning the +meta generator for the benign image with the updated surrogate +model, and serving as the initialization to boost off-the-shelf black- +box attack methods. Comprehensive experiments, including the +closed-set and open-set scenarios as well as attacking online APIs, +demonstrate the effectiveness of the proposed model. +REFERENCES +[1] +I. J. Goodfellow, J. Shlens, and C. Szegedy, “Explaining and harnessing +adversarial examples,” in Proc. Int. Conf. Learn. Represent., 2015. +[2] +S.-M. Moosavi-Dezfooli, A. Fawzi, O. 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Conf. +Comput. Vis. Worksh., 2019, pp. 2045–2048. +[72] Y. Xiong and C.-J. Hsieh, “Improved adversarial training via learned +optimizer,” in Proc. Eur. Conf. Comput. Vis., 2020, pp. 85–100. +[73] D. Wu, Y. Wang, S.-T. Xia, J. Bailey, and X. Ma, “Skip connections +matter: On the transferability of adversarial examples generated with +resnets,” in Proc. Int. Conf. Learn. Represent., 2020. +Fei Yin is currently a master student in Tsinghua +Shenzhen International Graduate School, Ts- +inghua University. His current research interests +include multimedia and computer vision. +Yong Zhang received the Ph.D. degree in pat- +tern recognition and intelligent systems from the +Institute of Automation, Chinese Academy of +Sciences in 2018. From 2015 to 2017, he was a +Visiting Scholar with the Rensselaer Polytechnic +Institute. He is currently with the Tencent AI Lab. +His research interests include computer vision +and machine learning. +Baoyuan Wu is an Associate Professor of School +of Data Science, the Chinese University of Hong +Kong, Shenzhen (CUHK-Shenzhen). He is also +the director of the Secure Computing Lab of +Big Data, Shenzhen Research Institute of Big +Data (SBRID). He received the PhD degree from +the National Laboratory of Pattern Recognition, +Institute of Automation, Chinese Academy of +Sciences, on June 2014. From November 2016 +to August 2020, he was a Senior and Principal +Researcher at Tencent AI lab. His research inter- +ests are AI security and privacy, machine learning, computer vision and +optimization. +Yan Feng is currently a master student in Ts- +inghua Shenzhen International Graduate School, +Tsinghua University. His current research inter- +ests include computer vision and AI security. +Jingyi Zhang is currently a master student in +School of Computer Science and Engineering, +University of Electronic Science and Technology +of China. His current research interests include +multimedia and computer vision. +Yanbo Fan is currently a Senior Researcher at +Tencent AI Lab. He received his Ph.D. degree +from Institute of Automation, Chinese Academy +of Sciences (CASIA), Beijing, China, in 2018, +and his B.S. degree in Computer Science and +Technology from Hunan University in 2013. His +research interests are computer vision and ma- +chine learning. +Yujiu Yang Yujiu Yang (Member, IEEE) received +the Ph.D. degree from the Institute of Automation, +Chinese Academy of Sciences. He is an Asso- +ciate Professor with the Tsinghua Shenzhen In- +ternational Graduate School, Tsinghua University. +His research interests include natural language +processing and computer vision. + +JOURNAL OF LATEX CLASS FILES, VOL. , NO. , +14 +APPENDIX A +METHOD ANALYSIS +A.1 +Comparison with Methods Directly Using the Sur- +rogate Model +Our framework can combine the learned prior with different types +of off-the-shelf query-based black-box attack methods in the meta- +test phase and significantly boost their performance in terms of +attack efficiency as well ASR. +Surrogate +Model +Attack +Achieved +Score +Surrogate +Model +Surrogate +Model +Meta +Generator +Achieved +Score +Attack +Target Black- +Box Model +Surrogate +Model +Achieved +Score +Attack +Meta +Generator +Achieved +Score +Attack +Target Black- +Box Model +Model to generate +perturbations: +Target model: +Performance: +Model to generate +perturbations: +Target model: +Performance: +Case 1: +Case 2: +(a) +(b) +(c) +(d) +Fig. 7. Two cases for transfer attack. +For a fair comparison with the transfer-based methods directly +using the surrogate model, we only use the trained generator +for evaluation. Figure 7 presents the comparison between the +surrogate model and our meta generator in two cases in the +scenario of transfer attack, i.e., no query of the unknown target +model. In Case 1, the surrogate model is treated as the target +model. Perturbations are generated by the surrogate model and +our meta generator, respectively. In this case, using the surrogate +model to generate perturbation achieves much better performance +than our meta generator (e.g., ASR: PGD-100 100% v.s. Ours +74%. Surrogate model: ResNet-50, Target model: ResNet-50). +Because the unknown target model is the same as the surrogate +model. Though our meta model is learned using the gradient +from the surrogate, it does not try to learn mapping exactly from a +sample to the gradient but learns the sample-dependent perturbation +distribution by attacking a large set of samples, i.e., it captures +some common properties among samples. Given a sample, our +meta generator can provide a perturbation distribution that tells +the probability of a sampled perturbation to be effective. It cannot +predict the exact perturbation as the surrogate model. Therefore, in +Case 1, the surrogate model wins. +In Case 2, the unknown target model is different from the +surrogate model. Given a sample, its generated perturbation is +totally determined by the sample and the model itself. The +perturbation seems to ‘overfit’ the surrogate model as the gradient +exactly comes from the surrogate model. Differently, our meta +model generates the perturbation according to the sample and the +learned prior (i.e., common properties among samples). Hence, +the perturbation is generated by considering not one sample but +a set of samples. It always generalizes better than the surrogate +model in this case (e.g., ASR: PGD-100 35.5% v.s. Ours 56.8%. +Surrogate model: ResNet-50, Target model: ResNet-18). +A.2 +Working Principle of MCG +We provide an illustration of how our method works for better +understanding in Figure 8. Figure 8 (a) presents the attack by using +the surrogate model via ‘PGD’. The perturbation is generated +according to the surrogate model and the clean example itself. +Figure 8 (b) shows that our meta generator captures the sample- +dependent conditional distribution by performing a large set of +attacking tasks involving a number of samples. The perturbation +distribution is denoted by the black dash circle. Figure 8 (c) shows +that the meta generator is directly used to attack the unknown +target model by sampling a perturbation with a high probability. +It is a transfer attack. Figure 8 (d) shows the meta update of the +meta generator. The meta generator is combined with a query- +based attack method. The query feedback is used to update the +meta generator. Hence, the perturbation distribution is shifted +from the black dash circle to the black dash circle. The black +one is better as the update uses the feedback information about +the target model. Figure 8 (e) shows the usage of the updated +meta generator. When combining with a query-based method +that requires a perturbation as initialization, we can sample a +perturbation with a high probability as the initialization. When +combining with a query-based method that requires a distribution +as initialization, then we can use the distribution represented by +the generator as the initialization. +According to Figure 8, our method learns the prior knowledge +in the meta-train phase and updates the generator in the meta-test +phase. It can be combined with off-the-shelf query-based methods. +The meta update involves the query feedback that improves the +meta generator. Our framework can boost the query-based methods +in attack efficiency as well as ASR. +A.3 +Advantages of Using c-Glow as Meta Generator +c-Glow can model the exact log-likelihood of the underlying +distribution, making it feasible to directly minimize the KL diver- +gence between the approximated and real conditional adversarial +distributions (CAD), rather than only optimizing the lower bound +as in VAE models or finding an approximate maximum point in +CAD as in learning-to-learn methods. +In learning-to-learn methods, CNN and RNN generators are +optimized based on the gradients of the classifiers, which means +attackers can only pre-train these generators on surrogate classifiers, +and then fully transfer their parameters for black-box attacks. As +pointed out in [53], such a fully-transfer mechanism will introduce +the so-called surrogate bias (due to differences in architectures +and training datasets between surrogate and target models), which +inevitably harms black-box attack performance. In contrast, as +c-Glow consists of two parts of parameters, i.e., Gaussian and +mapping parameters, we can utilize the partial transfer mechanism +to alleviate the surrogate bias. +A.4 +Visualization of Adversarial Perturbations +We present some visualization examples of five attack methods +in Fig. 9. The experimental settings are as follows. We perform +untargeted black-box attack with ResNet-50 as the surrogate model +and ResNet-18 as the target model. Five attack methods are +evaluated, including MCG (pure transfer), CG-Attack, MCG + CG- +Attack, Square, and MCG + Square. The perturbation limit is set to +ℓ∞ ≤ 0.05. It is interesting to see that our proposed MCG is likely +to generate nearly symmetric and rhombus-like patterns (see the +top row in Fig. 1). Considering the extremely high transferability of +these perturbations, they may provide good instances to analyze the +characteristics of highly transferable perturbations. However, we +realize that these perturbations will vary across different clean +images and different models. It requires more comprehensive + +JOURNAL OF LATEX CLASS FILES, VOL. , NO. , +15 +PGD Attack +Meta Update: +Finetuning Meta Generator +Sampling after Meta Update +Meta Train: +Distribution Modeling +Sampling & Attack +Surrogate Boundary +Target Boundary +Gradient Contour +Clean Example +Adversarial Example +Neighboring Example +Distribution +Finetuned Distribution +Fig. 8. Illustration of how the proposed method works. +evaluations and ingenious analysis tools/approaches to reveal some +general characteristics. It will be explored in our future work. +APPENDIX B +ADDITIONAL EXPERIMENT RESULTS +B.1 +Comparison with Combination-based Methods on +the CIFAR-10 dataset +In Table 4 of the manuscript, we present the comparison with +combination-based methods on the ImageNet dataset. Here, we +provide the comparison results on the CIFAR-10 dataset. Compet- +ing methods are TREMBA [23] and AdvFlow [22]. The results are +shown in Table 9. On CIFAR-10, our method achieves the best +performance under all attack cases in all three metrics, which is +consistent with the results on ImageNet. +B.2 +Additional Experiments in Open-set Attack Sce- +nario +In real-world scenarios, the surrogate model will share less common +knowledge with the target model. This situation strongly increases +the difficulty of attacking. In Sec.4.3 of the manuscript, we +verify the adaptability of the proposed method across datasets via +training on CIFAR-10 and testing on CIFAR-100. Here, we provide +additional experiments on ImageNet [55] and OpenImage [69]. +Experiments on ImageNet. To simulate the open-set scenarios, +we first randomly select 10 classes from the 1, 000 classes of +ImageNet and split the 10 classes into two groups evenly. We train +the surrogate model on the training set of the one group of classes +and train the target model on the training set of the other group +of classes. The training data of the meta generator is the same +as the data used for the surrogate model. The testing data is the +validation set corresponding to the training categories of the target +model. The results of the untargeted attack on ImageNet are shown +in Table 10. The open-set attack on ImageNet is more challenging +than that on CIFAR-10 as the median query numbers of several +attack methods are relatively high, e.g., NES, SimBA-DCT, and +MetaAttack. In this setting, our method can still improve their +performance, especially the ASR for NES and SimBA-DCT. +Experiments of training on ImageNet and testing on OpenIm- +age. To further verify the adaptability across different datasets, we +perform another untargeted attack experiment by training the meta +generator on the ImageNet dataset and testing it on the OpenImage +dataset. Target models (i.e., ResNet-18, VGG-16, WRN-50, and +Inception-V3) are trained with the training data of 10 randomly +selected classes of OpenImage. The meta generator is trained with +the training data of ImageNet guided by the surrogate model. The +results are shown in Table 11. Our MCG can reduce the query cost +of attacks and improve the ASR in most cases, which demonstrates +that the meta generator can be fast-adapted to different target +models across different datasets. +B.3 +Comparison with CNN and RNN-based generators. +CNN and RNN-based generators can also be fine-tuned to mitigate +the bias in our framework. In our framework the generator is used +to capture the prior distribution of adversarial examples, which is a +replaceable component. Other types of generators can be flexibly +incorporated into the framework to replace the c-Glow. To compare +the influence of different generators, we perform experiments by +integrating the CNN or RNN-based generator into our framework +to replace c-Glow. +Implementation details of the CNN-based generator. We follow +[70] and [71] to re-implement the CNN-based generator. The +backbone includes a feature extractor f, a generator network +G, and a discriminator network D. We concatenate the feature +f(x) of image x and a noise vector sampled from learnable +mean parameters z. Then we feed the concatenation to the +generator G. The generator G predicts an adversary perturbation +xadv corresponding to x. The discriminator D distinguishes the +output distribution of the generator with the real distribution. The +adversarial loss of the surrogate model (Eq. 4 of the manuscript) is +also used. To bound the magnitude of perturbation, we minimize +Linf bound norm of adversary perturbation. The loss function is +defined as: +L = LGAN + αLadv + βLinf, +(10) +where +LGAN = Ex[log D(x)+Ex log(1−D(x+G(z, f(x)))], (11) + +JOURNAL OF LATEX CLASS FILES, VOL. , NO. , +16 +Query Number: 1 +Query Number: 1 +Query Number: 1 +Query Number: 1 +Query Number: 1 +Query Number: 1 +Query Number: 1 +Query Number: 1 +Query Number: 1 +Query Number: 1 +Query Number: 1 +Query Number: 1 +a) MCG +Query Number: 261 +Query Number: 41 +Query Number: 1 +Query Number: 141 +Query Number: 1 +Query Number: 81 +Query Number: 301 +Query Number: 5441 +Query Number: 21 +Query Number: 41 +Query Number: 1021 +Query Number: 281 +b) CG-Attack +Query Number: 1 +Query Number: 41 +Query Number: 1 +Query Number: 281 +Query Number: 1 +Query Number: 1 +Query Number: 121 +Query Number: 3841 +Query Number: 1 +Query Number: 81 +Query Number: 641 +Query Number: 281 +Query Number: 1 +Query Number: 69 +Query Number: 274 +Query Number: 439 +Query Number: 328 +Query Number: 30 +Query Number: 150 +Query Number: 160 +Query Number: 1 +Query Number: 4 +Query Number: 1 +Query Number: 26 +c) MCG + CG-Attack +Query Number: 82 +Query Number: 54 +Query Number: 101 +Query Number: 53 +d) Square Attack +Query Number: 1 +Query Number: 88 +Query Number: 204 +Query Number: 65 +Query Number: 1 +Query Number: 1 +Query Number: 81 +Query Number: 192 +e) MCG + Square Attack +Fig. 9. Visualization of the generated perturbations of the ImageNet dataset. In each triple block, each column represents the benign image, the +adversarial example, and the perturbation, respectively. The origin range of the perturbation is [−0.05, 0.05] and we scale it to the range [0, 255] for +better visualization. + +Irish +Cove +ZSOOLIrish +CoveIrish +CoveIrish +Cove +1002rish +Cove +OOLIrish +Cove +SOOLJOURNAL OF LATEX CLASS FILES, VOL. , NO. , +17 +TABLE 9 +Comparison with combination-based methods on the CIFAR-10 dataset +Target model → +ResNet-PreAct-110 +DenseNet-121 +VGG-19 +PyramidNet-110 +Attack Method ↓ +ASR +Mean +Median +ASR +Mean +Median +ASR +Mean +Median +ASR +Mean +Median +Untargeted Attack +TREMBA [23] +90.9% +120.7 +64.0 +97.8% +126.4 +66.0 +97.7% +125.5 +63.0 +97.9% +82.3 +39.0 +AdvFlow [22] +97.2% +841.4 +600.0 +100.0% +1025.3 +740.0 +98.2% +1079.1 +860.0 +99.7% +857.5 +560.0 +MCG + CG-Attack +100.0% +41.8 +1.0 +100.0% +21.3 +1.0 +100.0% +38.8 +1.0 +100.0% +12.8 +1.0 +Targeted Attack +TREMBA [23] +91.2 +1125.3 +868.0 +92.3 +1123.4 +879.0 +96.5 +1331.5 +1142.0 +98.1 +1082.4 +759.0 +AdvFlow [22] +98.6 +911.7 +820.0 +96.3 +1021.5 +860.0 +97.4 +1144.1 +940.0 +100.0 +908.1 +820.0 +MCG + CG-Attack +100.0 +430.8 +181.0 +99.8 +608.5 +241.0 +97.4 +944.1 +381.0 +100.0 +290.6 +141.0 +TABLE 10 +Untargeted attack on the ImageNet dataset in the open-set scenario. +Target Model → +ResNet-18 +VGG-16 +WRN-50 +Inception-V3 +Attack Method ↓ +ASR +Mean +Median +ASR +Mean +Median +ASR +Mean +Median +ASR +Mean +Median +NES [12] +66.7% +4684.9 +4516.0 +68.5% +4903.7 +4778.5 +74.0% +4199.8 +3760.0 +94.9% +1989.0 +1439.5 +MCG + NES +77.5% +4113.1 +3845.0 +77.4% +4392.6 +4517.0 +78.9% +3503.1 +3089.0 +96.8% +1773.5 +1251.5 +CG-Attack [53] +50.2% +524.8 +61.0 +36.3% +476.3 +41.0 +55.1% +615.6 +141.0 +64.4% +497.3 +41.0 +MCG + CG-Attack +57.7% +446.2 +21.0 +41.2% +468.6 +61.0 +55.2% +591.3 +61.0 +65.0% +521.4 +41.0 +SimBA-DCT [10] +20.9% +2123.4 +2002.0 +21.4% +2782.8 +3202.0 +33.3% +2947.2 +3187.0 +53.3% +2485.2 +2283.0 +MCG + SimBA-DCT +42.9% +1182.9 +829.0 +35.2% +1006.6 +669.0 +52.4% +1293.4 +1335.0 +60.0% +1402.7 +1343.0 +Signhunter [11] +100.0% +142.2 +60.0 +98.5% +391.6 +145.0 +100.0% +157.7 +49.0 +100.0% +129.5 +45.0 +MCG + Signhunter +100.0% +161.0 +64.0 +99.2% +364.6 +137.5 +100.0% +153.0 +44.0 +100.0% +121.9 +46.0 +Square [17] +100.0% +288.1 +196.0 +100.0% +632.6 +336.0 +100.0% +272.5 +156.0 +100.0% +207.1 +126.0 +MCG + Square +100.0% +220.1 +119.0 +100.0% +572.3 +271.0 +100.0% +220.3 +112.0 +100.0% +175.8 +97.0 +MetaAttack [20] +78.8% +5646.7 +5951.0 +75.7% +5485.2 +5293.0 +76.2% +5346.7 +5505.5 +93.6% +4168.6 +4198.5 +MCG + MetaAttack +79.2% +4360.4 +4416.0 +76.6% +5078.9 +5619.0 +82.4% +4555.6 +4412.0 +94.5% +3896.9 +3974.0 +TABLE 11 +Untargeted attack evaluation on the OpenImage dataset. +Target Model → +ResNet-18 +VGG-16 +WRN-50 +Inception-V3 +Attack Method ↓ +ASR +Mean +Median +ASR +Mean +Median +ASR +Mean +Median +ASR +Mean +Median +NES [12] +28.3% +4915.9 +5041.0 +40.4% +3894.7 +3718.0 +61.1% +4035.6 +3823.0 +86.9% +2671.2 +1954.0 +MCG + NES +52.6% +1306.7 +281.0 +69.9% +764.3 +1.0 +74.0% +1327.8 +1.0 +92.9% +1712.8 +23.0 +CG-Attack [53] +62.3% +2033.8 +21.0 +66.7% +2044.4 +21.0 +66.9% +1948.1 +81.0 +61.5% +2273.4 +41.0 +MCG + CG-Attack +72.7% +1337.4 +1.0 +82.5% +1419.3 +1.0 +78.7% +1201.2 +101.0 +75.5% +1039.1 +1.0 +SimBA-DCT [10] +90.4% +374.201 +17 +94.9% +254.7 +10.0 +92.5% +301.4 +12.0 +83.2% +320.180 +11.0 +MCG + SimBA-DCT +90.8% +348.5 +7.0 +94.7% +226.2 +1.0 +92.5% +250.7 +1.0 +83.9% +249.6 +1.0 +Signhunter [11] +99.7% +331.8 +11.0 +100.0% +149.2 +7.0 +100.0% +233.3 +13.0 +100.0% +259.9 +8.0 +MCG + Signhunter +98.6% +267.9 +6.0 +100.0% +100.9 +1.0 +100.0% +159.9 +1.0 +100.0% +247.7 +4.0 +Square [17] +99.9% +292.1 +69.0 +100.0% +247.0 +68.5 +100.0% +156.9 +61.5 +100.0% +178.9 +56.0 +MCG + Square +99.7% +230.7 +24.0 +100.0% +180.9 +1.0 +100.0% +93.9 +1.0 +100.0% +123.9 +8.5 +MetaAttack [20] +69.7% +4366.9 +3964.0 +79.9% +4029.4 +3635.5 +72.2% +4364.2 +3754.0 +81.4% +3689.8 +3093.0 +MCG + MetaAttack +75.1% +3033.8 +1670.0 +84.7% +1637.9 +1.0 +76.9% +1722.0 +1.0 +86.9% +2351.1 +41.0 +Ladv = Ex[gw(x + G(z, f(x)), t)], +(12) +Linf = Ex∥G(z, f(x))∥∞. +(13) +t is the target class and gw denotes the surrogate classifier. +Implementation details of the RNN-based generator. We follow +[72] to re-implement the RNN-based model. We flatten the input +benign images into one-dimension feature vectors and directly +feed the features into the RNN network G. The initial hidden +state h for the input sequence is sampled from the learnable mean +parameters z. We reshape the output sequence from G back to +the corresponding spatial dimension and achieve the adversarial +perturbation. Similarly, the adversarial loss Ladv and the bound +loss Linf are used to optimize the generator. The loss function is +defined as: +L = Ladv + λLinf. +(14) +For the rest training and testing strategy, we keep the settings the +same as in our manuscript. +Experimental results. The experimental results are shown in +Tab. 12. We use Square [17] as the baseline method. It can be +observed that both CNN-based and RNN-based generators can +improve the performance of the baseline method considerably. +The results demonstrate the scalablility of our framework, i.e., the +c-Glow generator can be replaced by other types of generators. +Comparing the three types of generator, Flow-based MCG achieves +better performance than the other two. Moreover, overall the + +JOURNAL OF LATEX CLASS FILES, VOL. , NO. , +18 +TABLE 12 +Untargeted Attack comparison with CNN and RNN-based generators on the CIFAR-10 dataset. +Target model → +ResNet-PreAct-110 +DenseNet-121 +VGG-19 +PyramidNet-110 +Attack Method ↓ +ASR +Mean +Median +ASR +Mean +Median +ASR +Mean +Median +ASR +Mean +Median +Square +100.0% +227.3 +144.5 +100.0% +260.3 +159.0 +100.0% +342.0 +175.5 +100.0% +165.5 +100.5 +CNN MCG + Square +100.0% +67.4 +1.0 +100.0% +58.7 +1.0 +100.0% +120.8 +24.0 +100.0% +39.3 +1.0 +RNN MCG + Square +100.0% +49.9 +1.0 +100.0% +69.4 +6.0 +100.0% +97.9 +20.0 +100.0% +33.1 +1.0 +c-Glow MCG + Square +100.0% +39.7 +1.0 +100.0% +47.6 +1.0 +100.0% +57.9 +1.0 +100.0% +29.6 +1.0 +TABLE 13 +Untargeted Attack comparison with MAML-based meta-learning strategy on the CIFAR-10 dataset. +Target model → +ResNet-PreAct-110 +DenseNet-121 +VGG-19 +PyramidNet-110 +Attack Method ↓ +ASR +Mean +Median +ASR +Mean +Median +ASR +Mean +Median +ASR +Mean +Median +Square [17] +100.0% +227.3 +144.5 +100.0% +260.3 +159.0 +100.0% +342.0 +175.5 +100.0% +165.5 +100.5 +MCG-MAML + Square +100.0% +42.2 +1.0 +100.0% +45.7 +1.0 +100.0% +107.1 +18.0 +100.0% +27.2 +1.0 +MCG-REPTILE + Square +100.0% +39.7 +1.0 +100.0% +47.6 +1.0 +100.0% +57.9 +1.0 +100.0% +29.6 +1.0 +TABLE 14 +Validation of the extension with SGM. Untargeted attack evaluation on the ImageNet dataset. +Target model → +ResNet-18 +VGG-16 +WRN-50 +Inception-V3 +Attack Method ↓ +FASR +Mean +Median +FASR +Mean +Median +FASR +Mean +Median +FASR +Mean +Median +MCG + Square +60.1% +31.7 +1.0 +71.3% +24.8 +1.0 +51.3% +59.9 +1.0 +30.0% +123.8 +24.0 +SGM + MCG + Square +67.9% +22.9 +1.0 +76.2% +22.6 +1.0 +65.5% +41.0 +1.0 +43.9% +88.2 +6.0 +RNN-based generator performs slightly better than the CNN-based +generator. +B.4 +Comparison with MAML-based Meta-learning Meth- +ods. +Both MAML and REPTILE are powerful meta-learning algorithms +aiming at optimizing for an initial representation that can be +effectively fine-tuned. MAML unrolls and differentiates through the +computation graph of the gradient descent algorithm, while Reptile +simply performs stochastic gradient descent on each task, which +makes Reptile take less computation and memory than MAML. +We perform an experiment to compare REPTILE-based MCG +with MAML-based MCG in the untargeted attack scenario on the +CIFAR-10 dataset. The baseline method is Square [17]. Results +are shown in Tab. 13. It can be observed that ‘MCG-MAML + +Square’ and ‘MCG-REPTILE + Square’ achieve close performance. +This comparison also demonstrates the flexibility of using different +meta-learning algorithms in the proposed framework. +B.5 +Extended Experiments with Skip Gradient Method +Skip Gradient Method (SGM) [73] is a transfer-based attack +method that excavates the internal gradient flow of skip-connection +branches to generate more transferable perturbation. Since we +utilize the model-level adversarial transferability through the +surrogate model, we can introduce the strategy of SGM into our +framework to boost the attack performance. Similar to SGM, we +change the backward gradient weights of skip-connection branches +of our surrogate model to train the generator. During the attacking +process, we apply the same strategy to the surrogate model to fine- +tune our meta generator. We perform an experiment on ImageNet +in the untargeted attack scenario with ResNet-50 as the surrogate +model. The results are shown in Table 14. ‘MCG + Square’ is our +original method. ‘SGM + MCG + Square’ means that we additional +introduce the strategy of SGM. The results show the SGM strategy +helps our model achieve improvements in both FASR and the query +number. + diff --git a/NtAyT4oBgHgl3EQfgviP/content/tmp_files/load_file.txt b/NtAyT4oBgHgl3EQfgviP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..db8707d7274e54997bba7d6a81da9f62b049c9b1 --- /dev/null +++ b/NtAyT4oBgHgl3EQfgviP/content/tmp_files/load_file.txt @@ -0,0 +1,3262 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf,len=3261 +page_content='JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' , NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' , 1 Generalizable Black-Box Adversarial Attack with Meta Learning Fei Yin∗ , Yong Zhang∗ , Baoyuan Wu∗† , Member, IEEE, Yan Feng, Jingyi Zhang, Yanbo Fan , Yujiu Yang† , Member, IEEE Abstract—In the scenario of black-box adversarial attack, the target model’s parameters are unknown, and the attacker aims to find a successful adversarial perturbation based on query feedback under a query budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Due to the limited feedback information, existing query-based black-box attack methods often require many queries for attacking each benign example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' To reduce query cost, we propose to utilize the feedback information across historical attacks, dubbed example-level adversarial transferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Specifically, by treating the attack on each benign example as one task, we develop a meta-learning framework by training a meta generator to produce perturbations conditioned on benign examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' When attacking a new benign example, the meta generator can be quickly fine-tuned based on the feedback information of the new task as well as a few historical attacks to produce effective perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Moreover, since the meta-train procedure consumes many queries to learn a generalizable generator, we utilize model-level adversarial transferability to train the meta generator on a white-box surrogate model, then transfer it to help the attack against the target model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The proposed framework with the two types of adversarial transferability can be naturally combined with any off-the-shelf query-based attack methods to boost their performance, which is verified by extensive experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The source code is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='com/SCLBD/MCG-Blackbox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Index Terms—Black-box Adversarial Attack, Meta Learning, Example-level and Model-level Adversarial Transferability, Conditional Distribution of Perturbation !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 1 INTRODUCTION D EEP neural networks (DNNs) have been shown to be vulnera- ble to adversarial examples [1], where stealthy and malicious perturbations are added onto benign examples to fool the DNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' According to the accessible information about the attacked DNN model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=', the target model), existing adversarial attacks can be generally categorized into two scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The first is white-box attack, which assumes that the attacker knows the parameters of the target model, such that the adversarial perturbation can be easily generated based on the gradient of the target model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The second is black-box attack, where the attacker does not know the parameters of the target model, while only the query feedback is accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Compared to the white-box scenario, the black-box scenario is Fei Yin, Yan Feng, and Yujiu Yang are with Tsinghua Shenzhen International Graduate School, Tsinghua University, Beijing 100190, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' E-mail: {yinf20, y-feng18}@mails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='cn, yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='yujiu@sz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Baoyuan Wu is with the School of Data Science, Shenzhen Research Institute of Big Data, Chinese University of Hong Kong, Shenzhen 518172, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' E-mail: wubaoyuan@cuhk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Yong Zhang and Yanbo Fan are with Tencent AI Lab, Shenzhen, Guangdong 518057, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' E-mail: {zhangyong201303, fanyanbo0124}@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Jingyi Zhang is with the Center for Future Media and the School of Computer Science and Engineering, University of Electronic Sci- ence and Technology of China, Chengdu 610056, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' E-mail: jingyi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='zhang1995@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Fei Yin, Yong Zhang and Baoyuan Wu are co-first authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Baoyuan Wu and Yujiu Yang are corresponding authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' This work has been accepted by T-PAMI 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' This work was supported in part by the National Natural Science Foundation of China under Grant U1903213 and in part by the Shenzhen Key Laboratory of Marine IntelliSense and Computation under Grant ZDSYS20200811142605016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The work of Baoyuan Wu is supported by the Natural Science Foundation of China under Grant 62076213, Shenzhen Science and Technology Program under Grants RCYX20210609103057050 and ZDSYS20211021111415025, and in part by the University Development fund of the Chinese University of Hong Kong, Shenzhen under Grant 01001810, and CCF-Tencent Open Fund.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' more practical and more challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Thus, this work focuses on the black-box scenario, especially on the score-based black-box attack, where the feedback is a continuous score (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=', the posterior probability in classification problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The general procedure of attacking one benign example in the score-based black-box attack scenario can be described as follows: given a query budget (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=', the allowed query number) and an allowed perturbation region in the vicinity of the attacked benign example, with the starting solution at the benign example, the attacker keeps searching for a successful perturbation that satisfies the attack goal for a black-box target model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' if such a successful perturbation is found within the allowed perturbation region under the query budget, then the attack is successful and stopped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Otherwise, the attack failed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' As image statistics differ in benign images, attacking a benign image can be viewed as an individual task where the feedback information from the target model serves as the supervision to guide the perturbation generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' This perspective inspires us to utilize the information across different tasks to learn generic prior knowledge that can be transferred to boost the performance of each individual task, as did in meta-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The intuition is that, when attacking more benign examples, the attacker is supposed to be more experienced in how to generate perturbation conditioned on a given benign image and also know the target model better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Consequently, compared to the fresh attacker, one experienced attacker is expected to find a successful perturbation with fewer queries when attacking a new benign example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The underlying rationale is that adversarial perturbations around different benign examples may have some similar properties, and we call it example- level adversarial transferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' One typical example is universal adversarial perturbations [2], where one perturbation may fool multiple benign examples simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' However, to the best of our knowledge, example-level adversarial transferability has not arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='00364v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='LG] 1 Jan 2023 JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' , NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 2 Adversarial Attack White-box Attack Black-box Attack Score-based Attack Decision-based Attack Combination-based [17],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' [18],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' [19],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' [20],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' [21],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' [22],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' [23] Query-based [10],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' [11],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' [12],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' [13],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' [14],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' [15],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' [16] [1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' [3],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' [4],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' [5],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' [6],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' [7],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' [8],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' [9] Full Available Partial Available Label with Confidence Hard Label [29],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' [30],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' [31],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' [32],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' [33] Gradient Estimation Random Search [24],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' [25],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' [26],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' [27],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' [28] Transfer-based Attack [34],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' [35],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' [36],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' [37],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' [38],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' [39],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' [40],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' [41],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' [42],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' [43],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' [44],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' [45],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' [46],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' [47],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' [48] Model-level Example-level [2],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' [20],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' [22],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' [23] Non Available Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Taxonomy structure of existing black-box adversarial attack methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' been proposed explicitly to boost the attack performance, especially in the scenario of black-box attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' To capture the above intuition, here we propose to utilize meta-learning to learn a meta generator across different attacking tasks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=', attacking different benign examples), dubbed Meta Conditional Generator (MCG), which encodes the prior into the parameters of the network and can produce adversarial perturbations based on the benign example accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' When attacking a new benign example, the meta generator can be quickly fine-tuned based on the information of the benign example as well as the feedback information of a few historical attacks to produce effective perturbations that are specific to the new benign example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' However, directly training the meta generator with the perturba- tions of successful attacks in the meta-train process may still require many queries to the target model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' To further reduce the number of queries, we resort to the widely used model-level adversarial transferability, which assumes that some shared terms can be transferred from white-box surrogate models to the target model to help the black-box attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Specifically, we propose to conduct meta training based on white-box surrogate models, then the learned generator is transferred to help the attack against the target model, which serves as the meta-test process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Besides, to mitigate the difference between the surrogate and target models, we also fine-tune the surrogate model during the meta-test process by minimizing the feedback difference between the surrogate and target models for the same query, which encourages the surrogate model to mimic the behaviour of the target model and makes the generated perturbation more specific to the target model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The updated surrogate model is then exploited to fine-tune the meta generator for adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' In short, inspired by the perspective of meta-learning, we propose a general meta-learning framework for black-box adversarial attacks, by utilizing two levels of adversarial transferability, including example-level and model-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' One prominent advantage of the proposed framework is that it can be naturally combined with any off-the-shelf black-box attack methods to boost their original performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' For example, for sampling-based methods, the meta generator can serve as the sampling distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' for random-search-based methods that gradually adjust the perturbation, the meta generator can provide a suitably initialized perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Extensive experiments on bench- mark datasets and against several state-of-the-art attack methods verify the superiority of the proposed attack framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The main contributions of this work are three-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 1) We propose to treat the black-box attack to each benign example as an individual task, which inspires us to utilize the information across different tasks to boost the attack performance of each task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 2) We develop a general meta-learning framework for the black-box attack scenario by utilizing both the example- and model-levels of adversarial transferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 3) Extensive experiments demonstrate that the proposed framework can be naturally combined with any existing query-based black-box attack methods and significantly boost their original performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 2 RELATED WORK As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 1, we partition existing adversarial attack methods into three categories at the first levels, including white-box methods which utilize the full information (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=', parameters) of the attacked model to generate perturbations, black-box methods which utilize the query feedback returned by the attacked model, and transfer-based methods which don’t utilize any information of the attacked model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=', target model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Their detailed reviews are presented in the following sub-sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='1 White-box Adversarial Attack The white-box attack problem is generally formulated as follows: max δ∈Bϵ(x) L(f(x + δ), t), (1) where δ represents the adversarial perturbation, Bϵ(x) = {x′ − x| ∥x′ − x∥p ≤ ϵ} denotes a neighboring region around x, and the attacker-specified scalar ϵ > 0 represents the upper bound of allowed perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The loss function L(f(x + δ), t) could be specified as the cross entropy loss in untargeted attack where t indicates the ground-truth label of x, while as the negative cross entropy in targeted attack where t indicates a target label that is different from the ground-truth label of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Many gradient-based optimization methods have been utilized to solve the above problem, including the fast gradient sign method (FSGM) [1], iterative fast gradient sign method (I-FGSM) [3], C&W method [4], functional adversarial attack [5], adversarial camouflage [6], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Moreover, several works [7], [8], [9] focus on the adversarial attack in the physical world, where many types of environmental distortions may weaken the effectiveness of the generated adversarial perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='2 Black-box Adversarial Attack The formulation of Problem (1) is also applicable to the black- box attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' However, the parameters of the target model f(·) is JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' , NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' , 3 unknown, while only the objective value f(x + δ) for each query x + δ is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Consequently, the gradient-based optimization methods cannot be directly utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' According to the type of query feedback f(x + δ), black-box attacks can be further partitioned to two sub-categories, including score-based and decision-based attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='1 Score-based Black-box Adversarial Attack In the scenario of score-based attack, the feedback f(x + δ) is continuous, such as the posterior probability in the image classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Existing methods for solving this problem can be generally partitioned into three categories, including query- based and combination-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 1) Query-based methods iteratively adjust the perturbation only based on queries to the target model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Many black-box optimization approaches have been utilized, mainly including random search (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=', [10], Square, SignHunter [11]), evolution strategies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=', NES [12], Bandit [13]), and gradient-estimation approaches (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=', ZOO [14], AutoZoom [15], ZO-signSGD [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Compared to transfer-based methods, query- based methods often achieve a higher attack success rate, but with the cost of many queries to the target model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 2) Combination- based methods aim to achieve a high attack access rate and high query efficiency simultaneously, by taking advantage of both queries to the target model and the transferred item from the surrogate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Existing methods proposed different kinds of transferring strategies, such as transferring the perturbation magni- tude (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=', Square [17]), perturbation gradient (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=', [18], [19] and Meta attack [20]), perturbation distribution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=', N ATTACK [21], AdvFlow [22]), the projection to a low-dimensional space (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=', TREMBA [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Combination-based methods have shown superior attack performance to the other two categories, and our method also belongs to this category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' However, most existing methods (only except Meta attack [20]) only utilized the model-level adversarial transferability, while our method captures both example-level and model-level transferability to improve the query efficiency further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='2 Decision-based Black-box Adversarial Attack In the scenario of decision-based attack, the feedback f(x + δ) is discrete, such as the class label in the image classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Existing methods for solving this problem can be generally partitioned into random search based and gradient-estimation- based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Random search based methods aim to find the best perturbation around the invisible decision boundary, such as sampling from a normal distribution in Boundary method [24] or from a learnable Gaussian distribution in Evolutionary method [25], searching along the estimated normal direction of the decision boundary in GeoDA [26], or searching on the surface of the allowed perturbation region in the vicinity of the benign example in SFA [27] and Rays [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Gradient-estimation-based methods propose different estimation approaches of gradient, such as utilizing the neighboring points around the current solution in NES [12] and qFool [29], Monte Carlo estimation in HopSkipJumpAttack [30], or estimating the gradient in a low-dimensional subspace for acceleration in QEBA [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' OPT [32] and Sign-OPT [33] were developed based on a continuous formulation that alternatively optimizes the magnitude and direction of perturbation, such that any gradient-estimation approaches can be utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='3 Transfer-based Adversarial Attack We list the transfer-based methods as another category, as each transfer-based method could be applied for the white-box attack or the black-box attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' For example, (MI-FGSM) [34] and Nesterov iterative fast gradient sign method (NI-FGSM) [35] can be used as white-box attacks, since they are extensions of the classic white- box attacks FSGM [1] and I-FGSM [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' However, as claimed in the manuscripts of [34] and [35], their goals are to generate more transferable perturbations to improve the attack success rate in the black-box settings, and their experiments contain both white-box and black-box settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Thus, if one method mainly aims to improve the adversarial transferability, we partition it to the transfer-based category, rather than white-box or black-box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' According to the transfer’s objective, we further present example-level and model- level transferability, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='1 Example-level adversarial transferability Although without explicit illustration, some works have actually studied the example-level transferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' For example, universal adversarial perturbations [2] revealed that it is possible to find a perturbation to fool multiple benign examples simultaneously, with respect to the same attacked model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' It reveals that different benign examples may have some common fragile directions, following which adversarial perturbations can be found easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Another example is generation-based adversarial attacks [23], [22], where a generative model is trained to directly generate an adversarial perturbation or adversarial example for each benign example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Its default assumption is that adversarial perturbations around different benign examples may follow the same distribution or mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' However, the example-level adversarial transferability has been rarely utilized to boost the attack performance, especially in the scenario of black-box attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The only attempt we have found is called Meta attack [20], where a meta attacker is trained to generate gradient, which is then used as the update direction in gradient-estimation-based black-box attack methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' In contrast, our proposed meta-learning framework aims at capturing the distribution of adversarial perturbations, thus can be naturally combined with any kinds of query-based attack methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='2 Model-level adversarial transferability The model-level adversarial transferability has been observed and studied in many existing works [36], [37], [23], [38], [39], [40], [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Two important issues are mainly explored about the model-level transferability, including the intrinsic reason, and how to enhance the transferability across models, especially in the black-box scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 1) The intrinsic reason of the model-level transferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Tram`er et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' [40] found that different models share a large fraction of adversarial subspace, which consists of orthogonal basis vectors that are highly aligned with the gradient of the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Consequently, perturbations within this shared subspace are likely to be transferable across models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' A geometric perspective provided by [37] showed that the transferability is partially due to the fact that decision boundaries of different models align well with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Demontis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' [42] demonstrated that the model-level transferability is closely related to the intrinsic vulnerability of the target model, and the complexity of the surrogate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The theoretical analysis provided in [41] derives two lower bounds for transferability based on data distribution similarity and model gradient similarity, as well as the upper bound for transferability based on gradient orthogonality and smoothness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' [43] illustrated that the model-level transferability is negatively correlated with the interaction inside adversarial pertur- bations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 2) Enhancing the model-level transferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Compared to FGSM [1], its extensions, including momentum iterative fast JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' , NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' , 4 gradient sign method (MI-FGSM) [34] and Nesterov iterative fast gradient sign method (NI-FGSM) [35], showed better model-level transferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' In [44], MI-FGSM and NI-FGSM were further extended by reducing the variance of the iterative update directions, such that the update direction is stabilized to escape from poor local optima, leading to the transferability improvement even when there are defenses for the target model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The ensemble attack method [37] generated adversarial perturbations based on multiple models to enhance the transferability for both untargeted and targeted attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Intermediate level attack (ILA) [45] proposed to generate adversarial perturbations based on intermediate layers of surrogate models to avoid overfitting, such that the transferability to the target model can be enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Feature distribution attack (FDA) [46] focused on improving the transferability of a targeted attack by maximizing the activation of one intermediate layer between the benign example and the perturbed example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' It is extended in [38] from one intermediate layer to multiple intermediate layers, such that the transferability of targeted attack is further enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Some works employed the meta-learning ideology to enhance transferability either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' MSM [47] obtained a Meta-Surrogate Model via optimizing a differentiable attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The Meta-surrogate model gained prior from one or a set of surrogate models and was able to generate adversarial examples with eximious transferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Meta Gradient [48] randomly sampled multiple models from a model zoo to compose different tasks and iteratively simulated a white-box attack and a black-box attack in each task, which narrowed the gap between the gradient directions in white-box and black-box attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 3 THE PROPOSED APPROACH 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='1 Problem Formulation We denote the classification model as fθ : X → Y, where the model parameters θ are unknown in the black-box attack setting, and X and Y are the input and output spaces, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Given an input example x, the index of its ground-truth label is denoted as y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' fθ(x, i) ∈ [0, 1] indicates the posterior probability w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' the i-th class, and fθ(x, i) denotes the corresponding logit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Our attack goal is to generate an adversarial perturbation δ for the benign example x to fool the model fθ(·), given that the model parameters θ are unknown, dubbed score-based black-box adversarial attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' It can be generally formulated as the following optimization problem: min δ∈Bϵ(x) Ladv(δ, x, y) = � max(0, △ut), if untargeted attack max(0, △t), if targeted attack (2) where △ut = fθ(x + δ, y) − maxj̸=y fθ(x + δ, j), and △t = maxj̸=y fθ(x + δ, j) − fθ(x + δ, t), with t ∈ Y being the target label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Note that Ladv is non-negative, and if 0 is achieved, then the corresponding δ is a successful adversarial perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='2 Meta Conditional Generator for Black-Box Attack Conditional Perturbation Generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Unlike most previous combination-based methods that use a deterministic network to predict the initial perturbation for a benign image, our generator captures a conditional distribution of perturbation conditioned on the benign image, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=', p(δ|x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' ϕ), where δ = Gϕ(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' G is the generator with ϕ as its parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' δ is the perturbation and z is a random vector that follows a simple distribution, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=', Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' In the conditional distribution, effective … Tasks 𝒯1 𝒯2 𝒯𝑁 Meta Train Meta Generator Meta Test New Task 𝑵(𝝁, 𝚺) Adaptation Prior Learning Perturbation Generator Target Logit Inconsistency 𝑵(𝝁, 𝚺) Gaussian Model Sampling Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Task definition in meta-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Given an image and a target model, the goal is to learn a generator that can generate an effective adversarial example to attack the target model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' … Tasks 𝒯1 𝒯2 𝒯𝑁 Meta Train Meta Generator Meta Test New Task 𝑵(𝝁, 𝚺) Adaptation Prior Learning Perturbation Generator Target Logit Inconsistency 𝑵(𝝁, 𝚺) Gaussian Model Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Overview of meta-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' During the meta-train phase, a meta generator can be obtained by training on a large set of tasks, which contains the generic prior of how to generate effective perturbations for different images to attack the target model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' During the meta-test phase, the meta generator can be quickly adapted to new tasks with only a few steps of fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' perturbations are supposed to have a high probability of being sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' When attacking the target model, we can sample a perturbation δ ∼ p(δ|x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' ϕ) and add it to the benign image to construct an adversarial example to fool the target model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Modeling Conditional Distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The generator captures a conditional distribution of perturbation, which can be realized with using a simple distribution and a complex non-linear function that maps the simple distribution to a complex one with an image as the condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' In this work, we use a conditional generative flow (c-Glow) [49] as the generator due to its superior property that the mapping between a random vector and the output perturbation is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' By using c-Glow, we have δ = gϕ(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' x) and the inverse version z = g−1 ϕ (δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The random vector z follows a Gaussian distribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=', z ∼ N(µ, Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Since c-Glow consists of a set of invertible functions/layers, the parameters ϕ can be decomposed into several individual parts, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=', g = gx,ϕ1 ◦· · ·◦gx,ϕM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' M is the number of layers and ϕi represents the parameters of the i-th layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' With the change of variables [50], we can write the conditional likelihood as log p(δ|x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' ϕ) = log p(z) + M � i=1 log �����det(∂g−1 ϕi (ri−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' x) ∂ri−1 ) ����� , (3) where ri = gϕi(ri−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' x), r0 = x, and rM = z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Perspective of Meta Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Attacking on one benign image can be viewed as an individual process where the generator can be optimized by sampling several perturbations from the conditional distribution and obtaining their corresponding feedback scores from the target model as supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' However, when attacking a black-box model, the budget of query is always limited, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=', we can only sample a few perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Hence, the learning of the generator in each attacking process can be formulated as a few-shot learning problem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=', given {δi, fθ(xi), yi}K i=1, the goal is to learn the generator Gϕ(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' K is the number of shots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' As mentioned in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 1, adversarial perturbations around different benign images may share certain common properties, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=', JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' , NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' , 5 𝒙 Perturbation Generator Logit 𝑳𝐚𝐝𝐯 𝑵(𝝁, 𝚺) Gaussian Surrogate Model Target Inner Loop Update Outer Loop Update 𝑵(𝝁, 𝚺) Gaussian Meta Generator 𝐺(𝒛;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 𝒙, 𝝋) 𝑔𝒘(𝒙) … A Batch of Tasks 𝒯1 𝒯2 𝒯𝑛 𝜹 ෝ𝒚 𝒚 𝒙 + 𝜹 Sampling Sampling Task 𝓣𝒊 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The pipeline of meta training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The batch version of REPTILE is exploited for meta training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' We sample a batch of tasks and perform the inner loop update of task-specific parameters for each task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Then the task-specific parameters of all tasks in the batch are aggregated to do the outer loop update, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=', updating the meta parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' example-level adversarial transferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Inspired by the concept of “learning to learn” in meta-learning, we can solve the few-shot learning problem from the perspective of meta-learning, and learn a meta generator to capture the common properties of perturbations by performing a large set of attacking tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The prior of how to generate a conditional distribution of perturbation around a benign image is learned across various and diverse tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Since the perturbation distributions of different benign images are generated through the same generator, the prior is implicitly encoded in the parameters of the generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Task Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Task is the atom in meta-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' In the scenario of adversarial attack, “task” is defined as: “given a benign image and a target model, the goal is to learn a conditional generative model from which a sampled perturbation could successfully fool the target model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' ” As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 2, given a benign image, we can sample a perturbation from the generative model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The addition of the benign image and the perturbation yields an adversarial example which is then fed into the target network for attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The adversarial example is effective if its the predicted label is not consistent with the ground truth label (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=', untargeted attack) or the predicted label of the adversarial example is the same as a specified label (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=', targeted attack).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Hence, the parameters of the generator are updated by maximizing the difference between the prediction and the ground truth or minimizing the difference between the prediction and the specified label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Since the parameters and architecture of the target model are un- known, inspired by the transferability of adversarial example [36], we use a surrogate model to replace the target model in the meta- train phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 3, we can achieve a meta generator by performing a large set of tasks, which captures the prior of generating adversarial perturbations and is adaptive to different tasks during the meta-test phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='3 Meta Training with A Surrogate Model As described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='2, in a task T , given a benign image x and a target model fθ(x, ·), the goal is to learn a conditional perturbation generator Gϕ(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' x) that can generate an effective perturbation δ to fool the target model, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=', fθ(x + δ, y) < maxj̸=y fθ(x + δ, j) for untargeted attack and fθ(x+δ, t) > maxj̸=t fθ(x+δ, j) for targeted attack, where δ = Gϕ(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The meta training process is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' In this work, the target black-box model is the same for different tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' In the scenario of black-box attack, we have no access to the parameters and the architecture of the target model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Hence, we have to make many queries for each benign image to generate successful perturbations and then use them to learn the meta generator, which is costly for query and hard to realize in real-world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Based on the model-level adversarial transferability, in the meta- train phase, the target model is replaced by a surrogate model gw(x) of which the architecture and parameters are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Therefore, the gradients can backpropagate through the surrogate model to update the generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Since the meta-train phase does not involve any target model, once the meta training is over, the meta generator can be applied to any target model in the meta-test phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' To evaluate the effectiveness of the generated perturbation, the adversarial loss function is similar to that in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The difference is that ˜△ut and ˜△t are computed by using the surrogate model rather than the target model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=', ˜△ut = gw(x + δ, y) − max j̸=y gw(x + δ, j), ˜△t = max j̸=y gw(x + δ, j) − gw(x + δ, t), (4) where t is the specified target class in targeted attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Given a set of tasks {Ti}N i=1, we follow the batch version of REPTILE [51] to perform meta-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' We sample n tasks to form a batch and update the task-specific parameters k times using Adam [52] for each task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The objective of the inner loop optimization is φTi = arg minφTi Ladv Ti .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The optimization procedure can be represented as φTi = Adam(Ladv Ti , ϕ, k, α), (5) where φTi is the final task-specific parameters of the generator after k steps of performing Adam, starting from ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' At each of the k steps, a perturbation is sampled from the current conditional distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Ladv Ti is the adversarial loss of the i-th task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' α is the learning rate of the inner loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Then, for the outer loop optimization, we can update the meta parameters of the generator with the resulted task-specific parameters in a mini-batch, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=', ϕ ← ϕ + β 1 n n � i=1 (φTi − ϕ) , (6) where β is the learning rate of the outer loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='4 Meta Test Using Historical Attack Experience The standard meta-test process is not applicable in black-box attack because the target model is unknown and the gradient cannot be backpropagated to fine-tune the meta generator through it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' To solve this issue in the meta-test, we propose to transfer the information of the black-box target model to a surrogate model by using the feedback of previous attacks to fine-tune the surrogate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The usage of the historical attack experience could make the surrogate imitate the behaviour of the target network, which provides more accurate and effective information to fine-tune the meta generator than directly using the surrogate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The pipeline of meta-test is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Given a new benign image, it consists of three steps to conduct the attack to the target model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=', fine-tuning the surrogate model for introducing the information of the target model, fine-tuning the meta generator for adaptation to the new benign image, and boosting off-the-shelf black-box attack methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The attacking ability of the whole framework is gradually improved in the process of circulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' , NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' , 6 𝑷𝝋′(𝜹|𝒙) Distribution Previous Tasks History Attack Info Perturbation Target Logit 𝑵(𝝁, 𝚺) Gaussian 𝑳𝐚𝐝𝐯 Surrogate Model Surrogate Model New Tasks 𝑳𝐂𝐄 Adapted Generator Meta Generator Black-Box Optimization /Search Perturbation Target Black- Box Model 𝑓𝜽(𝒙) Logit Target Optimized Perturbation (b) Fine-tune Surrogate Model (a) Fine-tune Meta Generator (c) Attack Target Black-Box Model Attack Info 𝑵(𝝁, 𝚺) Gaussian Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The pipeline of meta test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' (a) Fine-tune the meta generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Given the image of the new task and the updated surrogate model, the meta generator is fine-tuned by performing the inner optimization with Ladv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' (b) Fine-tune the surrogate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' In order to transfer the information of the target black-box model to the surrogate model, the historical information (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=', logits of adversarial examples and the target labels) of previous tasks as well as the current task are used to make the surrogate model mimic the behaviour of the target model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' (c) Attack the target black-box model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The adapted generator is combined with the off-the-shelf black-box attack methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The generator can provide an initial distribution of perturbation or a sampled perturbation according to the given image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The initialization is leveraged by the attack methods as the starting state to get a refined perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Then, the generated adversarial example is used to attack the target model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The logits of the attack are recorded to fine-tune the surrogate model in stage (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Fine-tuning the Surrogate Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 5(b), to use historical attack experience, the predicted logits of previous benign images, their adversarial examples, and the current benign image by the target model are collected to provide supervision for the fine-tuning of the surrogate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The loss function is defined as LCE = CE(gw(x + δ), fθ(x + δ)) + CE(gw(x), fθ(x)) (7) where CE(·) is the cross entropy loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The two terms represent the losses of the adversarial example and the benign image, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' For the benign image in the current task, only the second term is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The optimization of the parameters of the surrogate model can be represented as w′ = Adam( m � i LCE Ti , w, s, λ), (8) where w′ represents the parameters of the updated surrogate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' λ is the learning rate and m is the number of tasks in a mini-batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' LCE Ti is the loss for the i-th task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' s is the number of update steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Fine-tuning the Meta Generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 5(a) presents the fine- tuning of the meta generator with using the updated surrogate model gw′(x) for the benign image of the new task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The fine- tuning procedure in the meta-test phase is similar to the inner optimization in the meta-train phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' We use the loss in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' (4) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' (5) to update the parameters of the meta generator, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=', φT = Adam(Ladv T , ϕ, k, α), (9) where φT represents the adapted parameters for the new task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Different from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' (5), the updated surrogate model is used in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' (9) to yield out the loigts for the adversarial examples, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=', gw′(x + δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' As the updated surrogate model contains the transferred information from the target model, it can provide more accurate and effective supervision to fine-tune the meta generator than the original surrogate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Hence, the generated perturbation is more specific to the target model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Boosting Off-the-shelf Black-Box Attack Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Our adapted generator can provide an initial distribution of perturbation or a perturbation conditioned on the given benign image, enabling it be combined with other off-the-shelf black-box attack methods to boost their original performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 5 (c) shows the process of attacking the target model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The initial distribution or perturbation is leveraged by a black-box attack method as the starting state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The further optimized perturbation is then added to the benign image to generate an adversarial example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The output logits from the target model are recorded as historical attack information, which is then used to fine-tune the surrogate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' When combined with sampling-based methods [12], [53], the adapted generator served as a distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' When combined with random-search- based methods [10], [11], [17], a perturbation can be a sample from the generator and serves as an initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 4 EXPERIMENTS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='1 Experimental Settings Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' To demonstrate the effectiveness of the proposed method, we conduct comprehensive experiments on two commonly used benchmark databases, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=', CIFAR-10 [54] and ImageNet [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Following the setting in [53], for the CIFAR-10 dataset, we randomly select 1, 000 images from the testing set for evaluation which cover all classes evenly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The images are resized to 32 × 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' For the ImageNet dataset, we first randomly select 10 classes JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' , NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' , 7 from the 1, 000 classes and then use the 500 images of each class from the validation set for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The images are resized to 224 × 224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The target and surrogate models are trained on the training set of the corresponding dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' On CIFAR-10, we use the full training set for meta-learning to learn the meta generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' On ImageNet, the training set of the 10 chosen classes are used for meta-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' We select l∞-based attacks and set the maximal distortion as ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='031 for CIFAR-10 and ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='05 for ImageNet with image pixel values re-scaled to [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' We set the maximal query budget to 10,000 times in all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' If the attacker cannot successfully fool the target m odel within the query limit, we consider it a failure case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Following the prior work [53], we adopt the attack success rate (ASR), the mean query number (Mean), and the median query number (Median) of successful attacks to evaluate the attack performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Target and Surrogate Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' On CIFAR-10, we consider four target models: ResNet-Preact-110 [56], DenseNet-121 [57], VGG- 19 [58], and PyramidNet-110 [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' We follow the standard training process of image classification to obtain the checkpoints of these target models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The top-1 error rates of these four target models are 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='29%, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='17%, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='28%, and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='51% on the standard testing set, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' On ImageNet, we also evaluate our method on four target models: ResNet-18 [56], VGG-16-BN [58], WRN- 50 [60], and InceptionV3 [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' We use the official implementation of these methods and download their pre-trained checkpoints from torchvision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The top-1 error rates of these target models are 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='41%, 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='24%, 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='53%, and 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='71% on the validation set of ImageNet, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' In all experiments, ResNet-18 [56] and ResNet-50 [56] are used as the surrogate models on CIFAR-10 and ImageNet, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The corresponding top-1 error rates of the surrogate models are 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='37% for ResNet-18 and 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='97% for ResNet-50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' To further verify the performance of our framework, we also conduct experiments of attacking black-box adversarial defense models on ImageNet, including JPEG-Compression-WideResNet- 50 [62], Small-Noise-Defense-WideResNet-50 [63], FreeAdv- ResNet-50 [64], and FastAdv-ResNet-50 [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Competing Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' As our framework can provide a good initialization of perturbation, it can be treated as a plug-and-play component that can be combined with other black-box attack methods to boost their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' In order to verify the versatility of our model, we combine our framework with 6 query-based black-box attack methods, including NES [12], CG-Attack [53], SimBA [10], SignHunter [11], Square [17], and MetaAttack [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' For search-based methods such as SignHunter, Square, SimBA, and MetaAttack, a sampled perturbation from the generator is the initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' For sampling-based methods such as NES and CG- Attack, a distribution of perturbation represented by the generator is the initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' We also compare with several transfer-based attack methods to verify the transferability of the proposed method, including PGD [66], MI [34], TIMI [67], and DI [68], under the transfer attack setting where only the information of the surrogate model is accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Moreover, we finally compare with several combination- based methods under the query attack setting, including Ad- vFlow [22] and TREMBA [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' They exploit both the queries to the target model and the transferred item from the surrogate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' To ensure the fairness of comparison, we retrain the combination- based methods with the same surrogate model as ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' All the experiments are implemented with the source code provided by their authors under the same setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Implementation Details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Following [49], in all the experiments, we adopt the same architecture for the generator c-Glow with 3 blocks composed of 8 flow steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Each block starts with a squeeze operation followed by 8 flow steps and ends with a split operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' To improve the efficiency, we adopt the discrete cosine transform (DCT) and inverse DCT for dimension reduction by downsampling the size of images in ImageNet to 1 8 × 1 8 lower frequency subspace before feeding them into the generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' On CIFAR-10, we use the original shape of images as they are in a small size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Before meta training, we pre-train the c-Glow to provide an initial state of modelling the distribution of perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' We use the surrogate model to generate a large set of perturbations through PGD attack with the perturbation strength of ℓinf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='05, the step size of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='01, and the number of iterations of 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The parameters are optimized by maximizing the log-likelihood, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=', maxϕ log p(δ|x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' For the pre-training of the generator, the learning rate is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The batch size is m = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The generator is trained for 10 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' For meta training, we sample 16 tasks every batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The update stepsize of the inner optimization is set to k = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The learning rates of the inner and outer loops are set to α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0003 and β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0006, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' For meta-test, when fine-tuning the surrogate model, we freeze the parameters of all layers except the last three layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The surrogate model is fine-tuned with both benign images and their adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The learning rate of fine- tuning is set to λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The number of benign images is set to 4 in a batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' When fine-tuning meta generator, the process is similar to the inner loop optimization of the task-specific parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='2 Experiments in Closed-set Attack Scenario In the closed-set attack scenario, the surrogate and target models are trained on the same training set, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=', both the training images and the categories are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Experiments includes boosting the off-the-shelf black-box attack methods, attacking defended models, comparisons with transfer-based methods, and comparisons with combination-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Performance on CIFAR-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The specification of the surrogate model and the target models is presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Table 1 illustrates the results of several off-the-shelf black-box attack methods and those of the combinations with our proposed MCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The MCG can boost the attack efficiency of all the black- box attack methods without ASR drop under both untargeted and targeted attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Specifically, under the untargeted attack setting, the median query numbers of these methods are decreased to 1s for all the target models by using the initialization from our MCG, which means that we can fool the target models with the initially generated perturbations for over 50% images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Meanwhile, the ASRs are improved to nearly 100% for almost all the cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Besides, the mean query numbers are also improved significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' For example, for the state-of-the-art Square attack, our MCG can further improve its query efficiency in terms of the mean query number by a factor of 5 for all the four target models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' We further plot the tendency curves of ASR w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' the query number in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' We can observe that the MCG can boost the attack performance under all values of query number for all the competing attack methods, especially for small query numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Under the targeted attack setting, our proposed MCG can also boost the attack performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The targeted attack is generally JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' , NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' , 8 TABLE 1 Closed-set evaluation on the CIFAR-10 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Target model → ResNet-PreAct-110 DenseNet-121 VGG-19 PyramidNet-110 Attack Method ↓ ASR Mean Median ASR Mean Median ASR Mean Median ASR Mean Median Untargeted Attack NES [12] 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0% 285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='2 211.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='5 7706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' , NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' , 9 1 2 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='7 log10(Queries) 0 20 40 60 80 100 ASR (%) PreactResNet Square MCG+Square SignHunter MCG+SignHunter CG-Attack MCG+CG-Attack NES MCG+NES SimBA MCG+SimBA 1 2 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='7 log10(Queries) 0 20 40 60 80 100 VGG 1 2 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='7 log10(Queries) 0 20 40 60 80 100 ASR (%) DenseNet 1 2 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='7 log10(Queries) 0 20 40 60 80 100 PyramidNet Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Attack success rate (ASR%) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='r.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='1 3200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 MCG + CG-Attack 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0% 2501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='8 1781.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='9% 2374.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='4 1721.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='8% 2960.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='2 2101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='9% 2272.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='1 1441.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 harder to achieve than the untargeted attack, yet our MCG still obtains a satisfactory ASR attacking all the four target models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The best attack performance is achieved by MCG+Square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Compared to the original Square attack, the MCG+Square achieves the ASR of 100% with a significantly fewer number of queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' For example, the mean and median query numbers of MCG are over 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='6 times and 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='3 times less than those of the original Square attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Performance on ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' ImageNet is a much larger dataset than CIFAR-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The performance of both untargeted and targeted attacks on ImageNet is reported in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' We have a similar observation that the proposed MCG obtains consistent improvements when combined with different attack methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Under the untargeted attack setting, the improvement of ASR in several cases can be apparently observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' For example, when attacking WRN-50 and VGG-16 with CG-Attack, the ASRs are 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='6% and 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='5%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Our MCG further improves CG-Attack by about 10% for attacking WRN-50 and 3% for attacking VGG-16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Besides, the efficiency improvements are also noticeable, especially for NES and SimBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Under the targeted attack setting, the MCG brings in different degrees of improvements in almost all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The increase in ASR is more evident than the untargeted attack setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Although when combined with SignHunter against InceptionV3, the ASR has been slightly reduced (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='3% lower) while the attack cost is saved near 20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' All these results demonstrate that our MCG can provide an effective initial perturbation or a distribution of perturbation for various off-the-shelf black-box attack methods to boost their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Please note that the meta training procedure of the meta generator does not involve any target model or attack method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Hence, the meta generator only needs to be trained once, and it can be combined with different black-box attack methods to attack different black-box models without re-training, which corroborates JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' , NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' , 10 TABLE 5 Comparison with Transfer-based methods on the ImageNet dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Target model → ResNet-18 VGG-16 WRN-50 Inception-V3 Attack Method ↓ ASR ASR ASR ASR PGD [66] 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='5% 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='6% 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='6% 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='7% PGD + MCG w/o surrogate 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='8% 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0% 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='5% 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='4% PGD + MCG w/ fixed surrogate 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='9% 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='4% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='1% 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='4% MI [34] 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='1% 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0% 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='8% 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='4% MI + MCG w/o surrogate 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='6% 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='6% 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='6% 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='1% MI + MCG w/ fixed surrogate 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='2% 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='6% 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='3% 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='4% TIMI [67] 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='7% 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='4% 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='3% 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='5% TIMI + MCG w/o surrogate 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='4% 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='1% 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='3% 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='7% TIMI + MCG w/ fixed surrogate 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='7% 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='4% 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='4% 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='4% DI [68] 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='2% 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0% 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='3% 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='1% DI + MCG w/o surrogate 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='5% 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0% 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='1% 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='6% DI + MCG w/ fixed surrogate 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='2% 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0% 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='7% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='7% the flexibility and generalization ability of the proposed framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Attacking Defended Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' To verify the effectiveness of the proposed method against adversarial defense models, we perform experiments of attacking various defended models trained with different defense strategies, including JPEG-Compression [62], random noise perturbation [63], and adversarial training [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' JPEG-Compression defense (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=', JPEG-Compress-WRN-50) attempts to remove the influence of adversarial examples through the compression process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Small-Noise-Defense (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=', SND-WRN- 50) is specially designed for query-based attack via introducing additional random noise to hinder the attacker to estimate gradients correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' As shown in Table 3, when attacking JPEG-Compress- WRN-50 and SND-WRN-50, the performance of NES, CG-Attack, SimBA-DCT, and MetaAttack drops sharply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' In contrast, the com- bination with our framework greatly improves their performance in all the three metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' For example, when attacking JPEG-Compress- WRN-50 with CG-Attack, the ASR is only 39% and the median query number is 1161, which fails in most attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Our framework boosts its ASR to 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='7% and reduces its median query number to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Besides, our method also reduces the mean and median query numbers for Square and Signhunter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Adversarial training enlarges the difference of the classification boundaries between the robust model and the vanilla model and greatly limits the model-level adversarial transferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' As shown in Table 3, when attacking the models of adversarial training, the attack performance of all methods drops a lot compared to attacking the vanilla models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Our method can still improve the attack performance in most cases, though the improvements are not as significant as those of attacking the vanilla models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Comparison with Transfer-based Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' To verify the effec- tiveness of example-level adversarial transferability boosted by meta learning, we perform an untargeted attack experiment to compare our meta generator with several transfer-based attack methods in the transfer attack setting, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=', no information from the black-box target model is used for fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Our meta generator contains a pre-training stage of the c- Glow and the pre-training is performed by using an attack method to generate adversarial examples for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The quality of the adversarial examples affects the performance a lot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' In other experiments, the attack method is PGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Since the transfer-based methods can generate better transferable adversarial examples than PGD, to fairly compare with the transfer-based methods, here we pre-train the meta generator with using the adversarial examples generated by the three transfer-based methods, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The results are shown in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' ‘X + MCG w/o surrogate’ means that we use ‘X’ to pre-train the c-Glow and then directly use the meta generator to produce adversarial example without using the surrogate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' ‘X + MCG w/ fixed surrogate’ means that after pre-training we use the surrogate model to fine-tune the meta generator first and then use the meta generator to produce adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' As shown in Table 5, directly using our meta generator has better performance than the transfer-based methods in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Fine-tuning the meta generator with the surrogate model can further improve the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Transfer-based methods may overfit the surrogate model due to that the adversarial example generation totally depends on the surrogate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Differently, our meta generator captures the example-level transferability that can alleviate the overfitting issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The generation is determined by both the learned prior of the meta generator and the surrogate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Comparison with Combination-based Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' As our method can be treated as a combination of transfer-based and query-based method, we compare with two combination-based black-box attack methods, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=', TREMBA [23] and AdvFlow [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Since they take advantage of model-level adversarial transferability to improve the performance and are often combined with evolution methods to adjust their distribution mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' For fairness, in the meta- test phase, we integrate the distribution adjustment method CG- Attack [53] into our framework for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The results on ImageNet are shown in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Our method achieves the best performance under almost all attack cases in all the three metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Although other methods attempt to utilize the transferability between different models, they appear to be unstable in ASR when attacking different target models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' For example, when attacking ResNet-18 and VGG-16 under targeted attack, the ASRs of TREMBA are 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='4% and 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' But when attacking WRN-50 and Inception-V3, the ASRs drop to 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='2% and 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The results show that TREMBA cannot generalize well to attack different target models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Differently, our method can perform better in the generalization ability to attack different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The experimental results on CIFAR-10 are presented in the Appendix Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='3 Experiments in Open-set Attack Scenario In the closed-set attack scenario, the surrogate and target models share the same training dataset, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=', the training dataset of the target model is visible to attackers, which is hard to achieve in the real attack scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' In real-world scenarios, the surrogate model will share less common knowledge with the target model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' This situation strongly increases the difficulty of attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Therefore, in this section, we verify the effectiveness of our method in the open-set attack scenario where the surrogate and target models are trained on disjoint training datasets, and there is no overlap between the output categories of the two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' In this open-set attack setting, it is quite challenging for the attacker to transfer the limited prior to the unknown areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Specifically, in our experiments, we employ two datasets and train the surrogate model on one dataset and the target model on another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The training data of the meta generator is the same as the data used for the surrogate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The results of training on CIFAR-10 and testing on CIFAR-100 are shown in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Please note that other experiments on ImageNet and OpenImage [69] are presented in the Appendix Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Since the surrogate and target models are trained from different training sets with different categories, the effectiveness of model- level adversarial transferability is limited, which decreases the performance compared with the results in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Nevertheless, JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' , NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' , 11 TABLE 6 Untargeted Attack trained on the CIFAR-10 dataset and tested on the CIFAR-100 dataset in the open-set scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Target model → ResNet-18 VGG-16 WRN-50 Inception-V3 Attack Method ↓ ASR Mean Median ASR Mean Median ASR Mean Median ASR Mean Median NES [12] 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='1% 2054.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 1555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='3% 1691.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='5% 2229.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='6 1534.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='6% 2044.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='7 1471.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 MCG + NES 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0% 773.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 88.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 CG-Attack [53] 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='4% 226.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='1% 294.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='4% 285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='1% 330.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 MCG + CG-Attack 99.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 Square [17] 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='8 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='5 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0% 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 MetaAttack [20] 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0% 230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='1 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0% 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 our framework still works in boosting the existing black-box attack methods in the open-set setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' MCG can reduce the query cost of attacks and improve the ASR in most cases, which demonstrates that the meta generator can be fast-adapted to attack different target models across different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='4 Experiments of Attacking Real-World API To verify the effectiveness of our framework in real-world scenarios, we perform an experiment of attacking the Imagga Tagging API1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The model of Imagga is trained on an unknown dataset of over 3, 000 types of daily-life objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Given a query image, the API will return a list of possible labels as well as the corresponding confidence scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' We randomly select 100 images from the validation set of ImageNet and set the query limit to 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' We define the goal of untargeted attack as removing the top-3 labels of the benign images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' As the images are from ImageNet, the pre- trained surrogate model can be used to fine-tune the meta generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Ladv is set as the maximal score of the top-3 labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The adapted generator is then used to generate an initial perturbation for existing black-box attack methods to attack the API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' As shown in Table 7, the performance of all methods is significantly improved through the combination with our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Specifically, the median query numbers are decreased to 1s for all methods and the mean query numbers are also highly improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' These results demonstrate that our framework is applicable in real-world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='5 Ablation Study To verify the effectiveness of the meta training and the fine-tuning stages in the meta-test, we conduct experiments of untargeted 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' https://imagga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='com/solutions/auto-tagging TABLE 8 Ablation study on the ImageNet dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' All methods in the table are combined with Square attack for untargeted attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Target Model → ResNet-18 VGG-16 Attack Method ↓ FASR Mean Median FASR Mean Median Flow 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='2% 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='6 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='1% 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 MCG w/ fixed surrogate 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='9% 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='4% 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 MCG w/ fine-tuned surrogate 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='1% 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='3% 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 attacks on ImageNet for ablation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The results are shown in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' ‘Flow’ is the method that directly uses the perturbations to learn a conditional glow (c-Glow) model, rather than using the adversarial loss or the parameter update strategy in meta learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The perturbations are generated by applying Projected Gradient Descent (PGD) [66] attack to the surrogate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' During testing, no fine-tuning is conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' ‘MCG w/ fixed surrogate’ means that during testing we fine-tune the meta generator with a fixed surrogate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' ‘MCG w/ fine-tuned surrogate’ means that during testing we update the surrogate model first by exploiting the query feedback from the target model, and then we use the updated surrogate model to fine-tune the meta generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' All these methods are combined with Square attack to query the target model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' To evaluate the effectiveness of the initial perturbations, we use the first Attack Success Rate (FASR) as the metric instead of ASR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' FASR means the success rate of straightforwardly using the perturbation generated by the generators to attack the target model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' As shown in Table 8, compared to Flow, MCG w/ fixed surro- gate achieves much better performance in all the three metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The FASRs and the median query numbers are significantly improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The results demonstrate the effectiveness of the meta training formulation, which improves the example-level transferability by capturing more effective generic prior of how to attack different samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The provided initial perturbation is better than that from Flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Moreover, compared to MCG w/ fixed surrogate, MCG w/ fine-tuned surrogate gains further improvements in the FASR and the attack efficiency, which corroborates the effectiveness of the historical attack information, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=', we can get a better-adapted generator by transferring the information of the target model to the surrogate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 5 CONCLUSION We propose a novel framework for black-box attack by formulating it as a meta-learning problem to improve the example-level JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' , NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' , 12 adversarial transferability as well as the efficiency of attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' As the architecture and parameters of the black-box target model are unknown, we propose to perform the meta training with a surrogate by leveraging the model-level adversarial transferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Since the standard meta-test process cannot be applied to the black- box attack, we propose a three-stage attack pipeline to fine-tune the meta model, including fine-tuning the surrogate model with historical attack information of the target model, fine-tuning the meta generator for the benign image with the updated surrogate model, and serving as the initialization to boost off-the-shelf black- box attack methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Comprehensive experiments, including the closed-set and open-set scenarios as well as attacking online APIs, demonstrate the effectiveness of the proposed model.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Ma, “Skip connections matter: On the transferability of adversarial examples generated with resnets,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Represent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Fei Yin is currently a master student in Tsinghua Shenzhen International Graduate School, Ts- inghua University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' His current research interests include multimedia and computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Yong Zhang received the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' degree in pat- tern recognition and intelligent systems from the Institute of Automation, Chinese Academy of Sciences in 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' From 2015 to 2017, he was a Visiting Scholar with the Rensselaer Polytechnic Institute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' He is currently with the Tencent AI Lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' His research interests include computer vision and machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Baoyuan Wu is an Associate Professor of School of Data Science, the Chinese University of Hong Kong, Shenzhen (CUHK-Shenzhen).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' He is also the director of the Secure Computing Lab of Big Data, Shenzhen Research Institute of Big Data (SBRID).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' He received the PhD degree from the National Laboratory of Pattern Recognition, Institute of Automation, Chinese Academy of Sciences, on June 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' From November 2016 to August 2020, he was a Senior and Principal Researcher at Tencent AI lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' His research inter- ests are AI security and privacy, machine learning, computer vision and optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Yan Feng is currently a master student in Ts- inghua Shenzhen International Graduate School, Tsinghua University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' His current research inter- ests include computer vision and AI security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Jingyi Zhang is currently a master student in School of Computer Science and Engineering, University of Electronic Science and Technology of China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' His current research interests include multimedia and computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Yanbo Fan is currently a Senior Researcher at Tencent AI Lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' He received his Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' degree from Institute of Automation, Chinese Academy of Sciences (CASIA), Beijing, China, in 2018, and his B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' degree in Computer Science and Technology from Hunan University in 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' His research interests are computer vision and ma- chine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Yujiu Yang Yujiu Yang (Member, IEEE) received the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' degree from the Institute of Automation, Chinese Academy of Sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' He is an Asso- ciate Professor with the Tsinghua Shenzhen In- ternational Graduate School, Tsinghua University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' His research interests include natural language processing and computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' , NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' , 14 APPENDIX A METHOD ANALYSIS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='1 Comparison with Methods Directly Using the Sur- rogate Model Our framework can combine the learned prior with different types of off-the-shelf query-based black-box attack methods in the meta- test phase and significantly boost their performance in terms of attack efficiency as well ASR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Surrogate Model Attack Achieved Score Surrogate Model Surrogate Model Meta Generator Achieved Score Attack Target Black- Box Model Surrogate Model Achieved Score Attack Meta Generator Achieved Score Attack Target Black- Box Model Model to generate perturbations: Target model: Performance: Model to generate perturbations: Target model: Performance: Case 1: Case 2: (a) (b) (c) (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Two cases for transfer attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' For a fair comparison with the transfer-based methods directly using the surrogate model, we only use the trained generator for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Figure 7 presents the comparison between the surrogate model and our meta generator in two cases in the scenario of transfer attack, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=', no query of the unknown target model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' In Case 1, the surrogate model is treated as the target model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Perturbations are generated by the surrogate model and our meta generator, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' In this case, using the surrogate model to generate perturbation achieves much better performance than our meta generator (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=', ASR: PGD-100 100% v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Ours 74%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Surrogate model: ResNet-50, Target model: ResNet-50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Because the unknown target model is the same as the surrogate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Though our meta model is learned using the gradient from the surrogate, it does not try to learn mapping exactly from a sample to the gradient but learns the sample-dependent perturbation distribution by attacking a large set of samples, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=', it captures some common properties among samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Given a sample, our meta generator can provide a perturbation distribution that tells the probability of a sampled perturbation to be effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' It cannot predict the exact perturbation as the surrogate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Therefore, in Case 1, the surrogate model wins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' In Case 2, the unknown target model is different from the surrogate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Given a sample, its generated perturbation is totally determined by the sample and the model itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The perturbation seems to ‘overfit’ the surrogate model as the gradient exactly comes from the surrogate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Differently, our meta model generates the perturbation according to the sample and the learned prior (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=', common properties among samples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Hence, the perturbation is generated by considering not one sample but a set of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' It always generalizes better than the surrogate model in this case (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=', ASR: PGD-100 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='5% v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Ours 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='8%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Surrogate model: ResNet-50, Target model: ResNet-18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='2 Working Principle of MCG We provide an illustration of how our method works for better understanding in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Figure 8 (a) presents the attack by using the surrogate model via ‘PGD’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The perturbation is generated according to the surrogate model and the clean example itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Figure 8 (b) shows that our meta generator captures the sample- dependent conditional distribution by performing a large set of attacking tasks involving a number of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The perturbation distribution is denoted by the black dash circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Figure 8 (c) shows that the meta generator is directly used to attack the unknown target model by sampling a perturbation with a high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' It is a transfer attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Figure 8 (d) shows the meta update of the meta generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The meta generator is combined with a query- based attack method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The query feedback is used to update the meta generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Hence, the perturbation distribution is shifted from the black dash circle to the black dash circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The black one is better as the update uses the feedback information about the target model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Figure 8 (e) shows the usage of the updated meta generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' When combining with a query-based method that requires a perturbation as initialization, we can sample a perturbation with a high probability as the initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' When combining with a query-based method that requires a distribution as initialization, then we can use the distribution represented by the generator as the initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' According to Figure 8, our method learns the prior knowledge in the meta-train phase and updates the generator in the meta-test phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' It can be combined with off-the-shelf query-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The meta update involves the query feedback that improves the meta generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Our framework can boost the query-based methods in attack efficiency as well as ASR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='3 Advantages of Using c-Glow as Meta Generator c-Glow can model the exact log-likelihood of the underlying distribution, making it feasible to directly minimize the KL diver- gence between the approximated and real conditional adversarial distributions (CAD), rather than only optimizing the lower bound as in VAE models or finding an approximate maximum point in CAD as in learning-to-learn methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' In learning-to-learn methods, CNN and RNN generators are optimized based on the gradients of the classifiers, which means attackers can only pre-train these generators on surrogate classifiers, and then fully transfer their parameters for black-box attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' As pointed out in [53], such a fully-transfer mechanism will introduce the so-called surrogate bias (due to differences in architectures and training datasets between surrogate and target models), which inevitably harms black-box attack performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' In contrast, as c-Glow consists of two parts of parameters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=', Gaussian and mapping parameters, we can utilize the partial transfer mechanism to alleviate the surrogate bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='4 Visualization of Adversarial Perturbations We present some visualization examples of five attack methods in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The experimental settings are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' We perform untargeted black-box attack with ResNet-50 as the surrogate model and ResNet-18 as the target model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Five attack methods are evaluated, including MCG (pure transfer), CG-Attack, MCG + CG- Attack, Square, and MCG + Square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The perturbation limit is set to ℓ∞ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' It is interesting to see that our proposed MCG is likely to generate nearly symmetric and rhombus-like patterns (see the top row in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Considering the extremely high transferability of these perturbations, they may provide good instances to analyze the characteristics of highly transferable perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' However, we realize that these perturbations will vary across different clean images and different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' It requires more comprehensive JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' , NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' , 15 PGD Attack Meta Update: Finetuning Meta Generator Sampling after Meta Update Meta Train: Distribution Modeling Sampling & Attack Surrogate Boundary Target Boundary Gradient Contour Clean Example Adversarial Example Neighboring Example Distribution Finetuned Distribution Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Illustration of how the proposed method works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' evaluations and ingenious analysis tools/approaches to reveal some general characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' It will be explored in our future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' APPENDIX B ADDITIONAL EXPERIMENT RESULTS B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='1 Comparison with Combination-based Methods on the CIFAR-10 dataset In Table 4 of the manuscript, we present the comparison with combination-based methods on the ImageNet dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Here, we provide the comparison results on the CIFAR-10 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Compet- ing methods are TREMBA [23] and AdvFlow [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The results are shown in Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' On CIFAR-10, our method achieves the best performance under all attack cases in all three metrics, which is consistent with the results on ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='2 Additional Experiments in Open-set Attack Sce- nario In real-world scenarios, the surrogate model will share less common knowledge with the target model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' This situation strongly increases the difficulty of attacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='3 of the manuscript, we verify the adaptability of the proposed method across datasets via training on CIFAR-10 and testing on CIFAR-100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Here, we provide additional experiments on ImageNet [55] and OpenImage [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Experiments on ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' To simulate the open-set scenarios, we first randomly select 10 classes from the 1, 000 classes of ImageNet and split the 10 classes into two groups evenly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' We train the surrogate model on the training set of the one group of classes and train the target model on the training set of the other group of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The training data of the meta generator is the same as the data used for the surrogate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The testing data is the validation set corresponding to the training categories of the target model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The results of the untargeted attack on ImageNet are shown in Table 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The open-set attack on ImageNet is more challenging than that on CIFAR-10 as the median query numbers of several attack methods are relatively high, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=', NES, SimBA-DCT, and MetaAttack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' In this setting, our method can still improve their performance, especially the ASR for NES and SimBA-DCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Experiments of training on ImageNet and testing on OpenIm- age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' To further verify the adaptability across different datasets, we perform another untargeted attack experiment by training the meta generator on the ImageNet dataset and testing it on the OpenImage dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Target models (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=', ResNet-18, VGG-16, WRN-50, and Inception-V3) are trained with the training data of 10 randomly selected classes of OpenImage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The meta generator is trained with the training data of ImageNet guided by the surrogate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The results are shown in Table 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Our MCG can reduce the query cost of attacks and improve the ASR in most cases, which demonstrates that the meta generator can be fast-adapted to different target models across different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='3 Comparison with CNN and RNN-based generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' CNN and RNN-based generators can also be fine-tuned to mitigate the bias in our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' In our framework the generator is used to capture the prior distribution of adversarial examples, which is a replaceable component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Other types of generators can be flexibly incorporated into the framework to replace the c-Glow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' To compare the influence of different generators, we perform experiments by integrating the CNN or RNN-based generator into our framework to replace c-Glow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Implementation details of the CNN-based generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' We follow [70] and [71] to re-implement the CNN-based generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The backbone includes a feature extractor f, a generator network G, and a discriminator network D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' We concatenate the feature f(x) of image x and a noise vector sampled from learnable mean parameters z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Then we feed the concatenation to the generator G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The generator G predicts an adversary perturbation xadv corresponding to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The discriminator D distinguishes the output distribution of the generator with the real distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The adversarial loss of the surrogate model (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 4 of the manuscript) is also used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' To bound the magnitude of perturbation, we minimize Linf bound norm of adversary perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The loss function is defined as: L = LGAN + αLadv + βLinf, (10) where LGAN = Ex[log D(x)+Ex log(1−D(x+G(z, f(x)))], (11) JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' , NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='a) MCG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 261 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 41 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 141 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 81 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 301 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 5441 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 41 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 1021 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 281 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='b) CG-Attack ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 41 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 281 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 121 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 3841 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 81 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 641 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 160 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='c) MCG + CG-Attack ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 82 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 54 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 53 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='d) Square Attack ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 88 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 204 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 65 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 81 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Query Number: 192 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='e) MCG + Square Attack ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Visualization of the generated perturbations of the ImageNet dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' In each triple block, each column represents the benign image, the adversarial example, and the perturbation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The origin range of the perturbation is [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='05] and we scale it to the range [0, 255] for better visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Irish Cove ZSOOLIrish CoveIrish CoveIrish Cove 1002rish Cove OOLIrish Cove SOOLJOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' , NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' , 17 TABLE 9 Comparison with combination-based methods on the CIFAR-10 dataset Target model → ResNet-PreAct-110 DenseNet-121 VGG-19 PyramidNet-110 Attack Method ↓ ASR Mean Median ASR Mean Median ASR Mean Median ASR Mean Median Untargeted Attack TREMBA [23] 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='9% 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='7 64.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0% 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0% 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 Targeted Attack TREMBA [23] 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='2 1125.' 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dataset in the open-set scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Target Model → ResNet-18 VGG-16 WRN-50 Inception-V3 Attack Method ↓ ASR Mean Median ASR Mean Median ASR Mean Median ASR Mean Median NES [12] 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='7% 4684.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='9 4516.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='5% 4903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='4% 4555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='6 4412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='5% 3896.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='9 3974.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 TABLE 11 Untargeted attack evaluation on the OpenImage dataset.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='9% 2351.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='1 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 Ladv = Ex[gw(x + G(z, f(x)), t)], (12) Linf = Ex∥G(z, f(x))∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' (13) t is the target class and gw denotes the surrogate classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Implementation details of the RNN-based generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' We follow [72] to re-implement the RNN-based model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' We flatten the input benign images into one-dimension feature vectors and directly feed the features into the RNN network G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The initial hidden state h for the input sequence is sampled from the learnable mean parameters z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' We reshape the output sequence from G back to the corresponding spatial dimension and achieve the adversarial perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Similarly, the adversarial loss Ladv and the bound loss Linf are used to optimize the generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The loss function is defined as: L = Ladv + λLinf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' (14) For the rest training and testing strategy, we keep the settings the same as in our manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The experimental results are shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' We use Square [17] as the baseline method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' It can be observed that both CNN-based and RNN-based generators can improve the performance of the baseline method considerably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The results demonstrate the scalablility of our framework, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=', the c-Glow generator can be replaced by other types of generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Comparing the three types of generator, Flow-based MCG achieves better performance than the other two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Moreover, overall the JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' , NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' , 18 TABLE 12 Untargeted Attack comparison with CNN and RNN-based generators on the CIFAR-10 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Target model → ResNet-PreAct-110 DenseNet-121 VGG-19 PyramidNet-110 Attack Method ↓ ASR Mean Median ASR Mean Median ASR Mean Median ASR Mean Median Square 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0% 227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='3 144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='5 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0% 260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='3 159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0% 342.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='5 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0% 165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='5 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='5 CNN MCG + Square 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0% 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0% 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0% 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='8 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0% 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 RNN MCG + Square 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0% 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0% 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0% 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='9 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0% 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 c-Glow MCG + Square 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0% 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0% 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0% 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0% 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 TABLE 13 Untargeted Attack comparison with MAML-based meta-learning strategy on the CIFAR-10 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Target model → ResNet-PreAct-110 DenseNet-121 VGG-19 PyramidNet-110 Attack Method ↓ ASR Mean Median ASR Mean Median ASR Mean Median ASR Mean Median Square [17] 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0% 227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='3 144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='5 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0% 260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='3 159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0% 342.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='5 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0% 165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='5 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='5 MCG-MAML + Square 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0% 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0% 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0% 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='1 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0% 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 MCG-REPTILE + Square 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0% 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0% 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0% 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0% 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 TABLE 14 Validation of the extension with SGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Untargeted attack evaluation on the ImageNet dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Target model → ResNet-18 VGG-16 WRN-50 Inception-V3 Attack Method ↓ FASR Mean Median FASR Mean Median FASR Mean Median FASR Mean Median MCG + Square 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='1% 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='3% 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='3% 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0% 123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='8 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 SGM + MCG + Square 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='9% 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='2% 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='5% 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='9% 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='0 RNN-based generator performs slightly better than the CNN-based generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='4 Comparison with MAML-based Meta-learning Meth- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Both MAML and REPTILE are powerful meta-learning algorithms aiming at optimizing for an initial representation that can be effectively fine-tuned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' MAML unrolls and differentiates through the computation graph of the gradient descent algorithm, while Reptile simply performs stochastic gradient descent on each task, which makes Reptile take less computation and memory than MAML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' We perform an experiment to compare REPTILE-based MCG with MAML-based MCG in the untargeted attack scenario on the CIFAR-10 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The baseline method is Square [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Results are shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' It can be observed that ‘MCG-MAML + Square’ and ‘MCG-REPTILE + Square’ achieve close performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' This comparison also demonstrates the flexibility of using different meta-learning algorithms in the proposed framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content='5 Extended Experiments with Skip Gradient Method Skip Gradient Method (SGM) [73] is a transfer-based attack method that excavates the internal gradient flow of skip-connection branches to generate more transferable perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Since we utilize the model-level adversarial transferability through the surrogate model, we can introduce the strategy of SGM into our framework to boost the attack performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' Similar to SGM, we change the backward gradient weights of skip-connection branches of our surrogate model to train the generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' During the attacking process, we apply the same strategy to the surrogate model to fine- tune our meta generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' We perform an experiment on ImageNet in the untargeted attack scenario with ResNet-50 as the surrogate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The results are shown in Table 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' ‘MCG + Square’ is our original method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' ‘SGM + MCG + Square’ means that we additional introduce the strategy of SGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} +page_content=' The results show the SGM strategy helps our model achieve improvements in both FASR and the query number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQfgviP/content/2301.00364v1.pdf'} diff --git a/NtE0T4oBgHgl3EQfTQBT/content/tmp_files/2301.02233v1.pdf.txt b/NtE0T4oBgHgl3EQfTQBT/content/tmp_files/2301.02233v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..baab200c6257fe556be6b4bbf52173c3fa9d722f --- /dev/null +++ b/NtE0T4oBgHgl3EQfTQBT/content/tmp_files/2301.02233v1.pdf.txt @@ -0,0 +1,2256 @@ +arXiv:2301.02233v1 [math.OA] 5 Jan 2023 +THE STABLE EXOTIC CUNTZ ALGEBRAS ARE HIGHER-RANK +GRAPH ALGEBRAS +JEFFREY L. BOERSEMA AND SARAH L. BROWNE AND ELIZABETH GILLASPY +Abstract. For each odd integer n ≥ 3, we construct a rank-3 graph Λn with involution γn +whose real C∗-algebra C∗ +R (Λn, γn) is stably isomorphic to the exotic Cuntz algebra E +R +n. This +construction is optimal, as we prove that a rank-2 graph with involution (Λ, γ) can never +satisfy C∗ +R (Λ, γ) ∼ME E +R +n, and the first author reached the same conclusion for rank-1 graphs +(directed graphs) in [Boe17, Corollary 4.3]. Our construction relies on a rank-1 graph with +involution (Λ, γ) whose real C∗-algebra C∗ +R (Λ, γ) is stably isomorphic to the suspension SR. +In the Appendix, we show that the i-fold suspension SiR is stably isomorphic to a graph +algebra iff −2 ≤ i ≤ 1. +1. Introduction +For every odd integer n ≥ 3, the (complex) Cuntz algebra On has two real forms: the real +Cuntz algebra O +R +n and the exotic Cuntz algebra En. While the existence of En follows from the +classification of simple purely infinite real C∗-algebras [Boe06, BRS11], the non-constructive +nature of the existence portion of this classification theorem [Boe06, Theorem 1] means that +we know very little about En beyond its K-theory. In particular, until now there has been +no construction or representation of En in terms of familiar C∗-algebraic objects. +In this paper, we give an explicit realization of the stabilized exotic Cuntz algebras KR ⊗R +En as higher-rank graph algebras associated to rank-3 graphs with involution. Given the +extensive literature on the properties of higher-rank graph C∗-algebras, we anticipate that +this concrete description will facilitate an improved understanding of these elusive algebras. +Higher-rank graphs, or k-graphs, are a k-dimensional generalization of directed graphs +which were introduced by Kumjian and Pask in [KP00]. Many of the properties of (complex) +directed graph C∗-algebras, such as their K-theory [RS04] and their ideal structure [BHRS02, +HS04], are visible from the graph. +While the structure of k-graph C∗-algebras is more +intricate than that of graph C∗-algebras, k-graph C∗-algebras also encompass a broader +range of examples. Indeed [RSS15], every complex UCT Kirchberg algebra is a direct limit +of 2-graph C∗-algebras. The real C∗-algebra C∗ +R(Λ, γ) of a higher-rank graph with involution +(Λ, γ) was recently introduced by the first and third authors in [BG22]. In that paper, the +authors also generalized the work of [Eva08] and [Boe17] to describe a spectral sequence +which converges to the CR K-theory of these real C∗-algebras. +The main result (Theorem 4.3) of the present paper, that the exotic Cuntz algebra is stably +isomorphic to the C∗-algebra of a 3-graph with involution, is the best possible in terms of +the rank of Λ. In [Boe17], the first author made an extensive analysis of the K-theory of the +real C∗-algebra C∗ +R(Λ, γ) of a rank-1 graph (directed graph) with involution. 1 In particular, +[Boe17, Corollary 4.3] establishes that the exotic Cuntz algebra cannot be isomorphic or +stably isomorphic to the real C∗-algebra C∗ +R(Λ) of a directed graph Λ, or to the real C∗- +algebra C∗ +R(Λ, γ) of a graph with involution, since KO7(En) = Z2 but KO7(C∗ +R(Λ, γ)) is +always torsion-free. Theorem 3.1 below uses the K-theory spectral sequence for real higher- +rank graph C∗-algebras ([BG22, Section 3]) to show that En ̸∼ME C∗ +R(Λ, γ)) for any rank-2 +1This class of C∗-algebras includes the real C∗-algebras C∗ +R (Λ) of a directed graph, introduced in [Boe14], +as C∗ +R (Λ) ∼= C∗ +R (Λ, γtriv). +1 + +2 +JEFFREY L. BOERSEMA AND SARAH L. BROWNE AND ELIZABETH GILLASPY +graph with involution (Λ, γ). +However, we construct in Theorem 4.3 a family of rank-3 +graphs with involution (Λn, γn) such that C∗ +R(Λn, γn) ∼= En ⊗R KR. +Our construction combines the directed graphs with involution (En, γn) of [Boe17, Exam- +ple 6.2], which satisfy C∗ +R(En, γn) ∼ME S6En, with a directed graph with involution (Λ, γ) +such that (Proposition 4.1) C∗ +R(Λ, γ) is a real Kirchberg algebra which is KK-equivalent +to the suspension algebra SR ∼= C0((0, 1)). +To be precise, the 3-graph Λn which gives +C∗ +R(Λn, γn) ∼= En ⊗R KR is a product graph, Λn = En × Λ × Λ. +Prompted by the graph with involution of Proposition 4.1, we consider in Section 5 the +question of which suspensions SiR are KK-equivalent to the real C∗-algebra of a graph with +involution. For −2 ≤ i ≤ 1 we exhibit an example of a graph with involution (Λ, γ) such that +C∗ +R(Λ, γ) is KK-equivalent to SiR, and we show in Proposition 5.2 that SiR ̸∼KK C∗ +R(Λ, γ) +if 2 ≤ i ≤ 5. (However, we can realize these suspensions as 2-graph or 3-graph algebras, by +taking products of the graphs which do realize suspensions of R.) +Many key questions remain open for further investigation about the class of real C∗- +algebras that can be obtained using higher-rank graphs. For example, it is still unknown +whether or not En itself can be realized as a rank-k graph-with-involution algebra. Similarly, +it remains unknown which real Kirchberg algebras can be realized by higher-rank graphs +with involution (as opposed to inductive limits of such objects); we would particularly like +to find a K-theoretic characterization of such algebras. +Acknowledgments: E.G. was partially supported by NSF grant 1800749. +2. Preliminaries +2.1. Higher-rank graphs. +Definition 2.1. [KP00, Definition 1.1] A higher-rank graph of rank k, or a k-graph, is +a countable small category Λ equipped with a degree functor d: Λ → Nk such that, if a +morphism λ ∈ Λ satisfies d(λ) = m + n, then there exist unique morphisms µ, ν ∈ Λ such +that λ = µν, d(µ) = m and d(ν) = n. +Write ei for the standard ith basis vector of Nk. +The morphisms of degree ei can be +advantageously viewed as the “edges of color i” in Λ. In this perspective, if e is an edge of +color i and f is an edge of color j, their composition ef ∈ Λ satisfies +d(ef) = ei + ej = ej + ei, +so we must be able to rewrite ef = f ′e′ for some morphisms e′, f ′ ∈ Λ with d(f ′) = ej and +d(e′) = ei. +Indeed, by [HRSW13, Theorems 4.4 and 4.5], a k-graph can be equivalently thought of +as arising from a directed graph G, with k colors of edges and with a factorization rule on +multicolored paths. That is, given any two colors (“red” and “blue”) and any two vertices +v, w in G, the factorization rule identifies each red-blue path ef from v to w with an equivalent +blue-red path f ′e′ from v to w. +We would like the quotient of the space G∗ of directed paths in G by the equivalence +relation ∼ generated by the factorization rule to be a k-graph. +For this to occur, the +factorization rule must also satisfy certain consistency conditions which ensure that, for each +path in G∗, its equivalence class under ∼ corresponds to a k-dimensional hyper-rectangle; +see [EFG+21, Theorem 2.3] for more details. As our work in this paper does not depend on +these consistency conditions, we will not reproduce them here. That said, we remark that in + +THE STABLE EXOTIC CUNTZ ALGEBRAS ARE HIGHER-RANK GRAPH ALGEBRAS +3 +a rank-1 graph, the factorization rule is nonexistent, and so a 1-graph is precisely the space +of paths of a directed graph. +Let Λ be a k-graph. Given n ∈ Nk and objects v, w ∈ Λ, we write +(1) +Λn = {λ ∈ Λ : d(λ) = n}. +By the factorization rule, for every λ ∈ Λ, there are unique v, w ∈ Λ0 with vλ = λw = +vλw = λ. That is, we can identify Λ0 with the objects of Λ. If λ = vλw, we write v = r(λ) +and w = s(λ). Thus, expanding on Equation (1), we have +vΛn = {λ ∈ Λ : r(λ) = v and d(λ) = n} +Λnw = {λ ∈ Λ : s(λ) = w and d(λ) = n}, +(2) +as well as the obvious variations such as vΛnw. +A k-graph Λ has k adjacency matrices Mi ∈ MΛ0(N), which are given by +(3) +Mi(v, w) = #vΛeiw. +In the graphical picture, λ ∈ Λ(n1,...,nk) means that λ represents the ∼-equivalence class of +a path with ni edges of color i, for each 1 ≤ i ≤ k. That is, Λ0 consists of the length-0 paths, +ie, the vertices. Then Mi(v, w) is the number of edges of color i from vertex w to vertex v. +If Λ1 is a k1-graph and Λ2 is a k2-graph, then [KP00, Proposition 1.8] their (Cartesian) +product Λ1 ×Λ2 is a (k1 +k2)-graph; the degree functor is given by d(λ1, λ2) = (d(λ1), d(λ2). +We have (Λ1 × Λ2)0 = Λ0 +1 × Λ0 +2 and s(λ1 × λ2) = (s(λ1), s(λ2)). +In this paper we will focus on k-graphs which are row-finite and source-free (or have no +sources). We say a k-graph Λ is row-finite if |vΛn| < ∞ for all n ∈ Nk and v ∈ Λ0. The +k-graph has no sources if vΛn ̸= ∅ for all v, n. It is straightforward to check that if Λ1, Λ2 +are row-finite and source-free, then so is Λ1 × Λ2. +For a row-finite source-free k-graph Λ, its (complex) C∗-algebra is the universal C∗-algebra +generated by a Cuntz–Krieger Λ-family. +Definition 2.2. [KP00, Definition 1.5] Given a row-finite source-free k-graph Λ, a Cuntz–Krieger +Λ-family is a collection {tλ}λ∈Λ of partial isometries in a C∗-algebra A which satisfy the fol- +lowing conditions: +(CK1) For each v ∈ Λ0, tv is a projection, and tvtw = δv,wtv. +(CK2) For each λ ∈ Λ, t∗ +λtλ = ts(λ). +(CK3) For each λ, µ ∈ Λ, tλtµ = tλµ. +(CK4) For each v ∈ Λ0 and each n ∈ Nk, tv = +� +λ∈vΛn +tλt∗ +λ. +We define C∗(Λ) to be the universal (complex) C∗-algebra generated by a Cuntz–Krieger +family, in the sense that for any Cuntz–Krieger Λ-family {tλ}λ∈Λ, there is a surjective ∗- +homomorphism C∗(Λ) → C∗({tλ}λ). +We write {sλ}λ∈Λ for the generators of C∗(Λ). By using the Cuntz–Krieger relations, +one can compute that C∗(Λ) = span{sλs∗ +µ : s(λ) = s(µ)}. Corollary 3.5(iv) of [KP00] also +establishes that C∗(Λ1 × Λ2) ∼= C∗(Λ1) ⊗ C∗(Λ2); the isomorphism takes s(λ1,λ2) to sλ1 ⊗ sλ2. +We will use one more ingredient – an involution – to construct the real C∗-algebras asso- +ciated to higher-rank graphs. +Definition 2.3. An involution γ on a k-graph Λ is a degree-preserving functor γ : Λ → Λ +which satisfies γ ◦ γ = id Λ. + +4 +JEFFREY L. BOERSEMA AND SARAH L. BROWNE AND ELIZABETH GILLASPY +As established in [BG22, Lemma 2.4], the real C∗-algebra associated to a k-graph Λ and +an involution γ : Λ → Λ is +(4) +C∗ +R(Λ, γ) = spanR{zsλs∗ +µ + zsγ(λ)s∗ +γ(µ) | z ∈ C, λ, µ ∈ Λ}. +Equivalently, we have C∗ +R(Λ, γ) = {a ∈ C∗(Λ) | �γ(a) = a∗}, where �γ is the antimultiplicative +C∗-involution uniquely determined by �γ(sλ) = s∗ +γ(λ). For any involution γ on Λ, we have +C∗(Λ) ∼= C ⊗R C∗ +R(Λ, γ). +2.2. CRT K-theory. In our work, we will use the full united K-theory K +CRT(A) (introduced +in [Boe02]) as well as the abbreviated variation K +CR(A) which contains just the real and +complex parts. Theorem 10.2 of [BRS11] shows that the category of real purely infinite +simple C∗-algebras, whose complexifications are simple and in the UCT class, is classified up +to isomorphism by either of these invariants. We tend to use K +CR(A) since it is simpler and +usually sufficient, but we will also need to use K +CRT(A) on occasion since that is the context +in which we have the K¨unneth formula. Specifically, recall that for a real C∗-algebra A, +K +CR(A) = {KO∗(A), KU∗(A)} +K +CRT(A) = {KO∗(A), KU∗(A), KT∗(A)} +where KO∗(A) is the standard 8-periodic real K-theory for a real C∗-algebra and KU∗(A) = +K∗(C ⊗C A) is the 2-periodic K-theory of the complexification of A. Meanwhile KT∗(A) is +the 4-periodic self-conjugate K-theory. These invariants also include the additional CR and +CRT -module structure. In particular for K +CR(A) there are natural transformations +ri : KUi(A) → KOi(A) +induced by the standard inclusion C → M2(R) +ci : KOi(A) → KUi(A) +induced by the standard inclusion R → C +ψi : KUi(A) → KUi(A) +induced by conjugation C → C +ηi : KOi(A) → KOi+1(A) +induced by multiplication by η ∈ KO1(R) = Z2 +ξi : KOi(A) → KOi+1(A) +induced by multiplication by ξ ∈ KO4(R) = Z. +This additional structure tends to aid in the computations of KO∗(A) because the natural +transformations satisfy the relations +rc = 2 +cr = 1 + ψ +2η = 0 +rψ = r +ψ2 = id +η3 = 0 +ψc = c +ψβU = −βUψ +ξ = rβ2 +Uc +and they fit into a long exact sequence +(5) +· · · +rβ−1 +U +−−−→ KOi(A) +η−→ KOi+1(A) +c−→ KUi+1(A) +rβ−1 +U +−−−→ KOi−1(A) +η−→ · · · +These two invariants K +CR(A) and K +CRT(A) contain the same information by results of +[Hew96] (summarized, for example, in [BRS11, Proposition 2.5]). +2.3. K-theory for higher-rank graphs. For the real C∗-algebra C∗ +R(Λ, γ) of a higher- +rank graph with involution, [BG22, Theorem 3.10] establishes the existence of a spectral +sequence {Er, dr} of CR-modules that converges to K +CR(C∗ +R(Λ, γ)). The complex part of this +spectral sequence (Er +p,q) +U coincides with the Evans spectral sequence [Eva08] and converges +to KU∗(C∗ +R(Λ, γ)) = K∗(C∗(Λ)). The real part of this spectral sequence (Er +p,q) +O converges to +KO∗(C∗ +R(Λ, γ)). + +THE STABLE EXOTIC CUNTZ ALGEBRAS ARE HIGHER-RANK GRAPH ALGEBRAS +5 +The E2 page of the spectral sequence arises from the homology of a certain chain complex C +based on the combinatorial information of Λ and γ. We will use the spectral sequence only in +the rank-1 and rank-2 cases, where these chain complexes have the following straightforward +descriptions which we recall from [BG22, Theorems 3.14 and 3.15]. First let Λ0 +f be the set +of vertices fixed by the involution γ and let Λ0 +g ⊔ Λ0 +h be any partition of Λ0\Λ0 +f satisfying +γ(Λ0 +g) = Λ0 +h and γ(Λ0 +h) = Λ0 +g. Then let A = K +CR(R)Λ0 +f ⊕ K +CR(C)Λ0 +g. +For a rank-1 graph with involution the chain complex C is given by +0 → A +∂1 +−→ A → 0, +where ∂1 = ρ1. For a rank-2 graph with involution the chain complex C is given by +0 → A +∂2 +−→ A2 ∂1 +−→ A → 0, +where ∂1 = +� +ρ1 +ρ2� +and ∂2 = +� +−ρ2 +ρ1 +� +. Here the maps ρi are determined by the adjacency +structure of Λ as follows. For 1 ≤ i ≤ k, the complex part (ρi) +U +0 : ZΛ0 → ZΛ0 is represented +by the matrix Bi = I − Mt +i , where Mi is the adjacency matrix of the graph Λ for the edges +of degree ei, and (ρi) +U +1 = 0 for all i. The real parts of this map (ρi) +O +j , for 0 ≤ j ≤ 7, can +similarly be determined for each i by some variations of Bi as shown in Table 3 of [BG22] +and Theorem 4.4 of [Boe17], which we also reproduce here as Table 1. +complex part +0 + + +B11 +B12 +B12 +B21 +B22 +B23 +B21 +B23 +B22 + + +Z|Λ0 +f | ⊕ Z|Λ0 +g| ⊕ Z|Λ0 +h → Z|Λ0 +f| ⊕ Z|Λ0 +g| ⊕ Z|Λ0 +h +1 +0 +0 +real part +0 +� +B11 +2B12 +B21 +B22 + B23 +� +Z|Λ0 +f| ⊕ Z|Λ0 +g| → Z|Λ0 +f| ⊕ Z|Λ0 +g| +1 +B11 +Z +|Λ0 +f | +2 +→ Z +|Λ0 +f| +2 +2 +� +B11 +B12 +0 +B22 − B23 +� +Z +|Λf| +2 +⊕ Z|Λ0 +g| → Z +|Λf| +2 +⊕ Z|Λ0 +g| +3 +0 +0 +4 +� +B11 +B12 +2B21 +B22 + B23 +� +Z|Λ0 +f| ⊕ Z|Λ0 +g| → Z|Λ0 +f| ⊕ Z|Λ0 +g| +5 +0 +0 +6 +B22 − B23 +Z|Λ0 +g| → Z|Λ0 +g| +7 +0 +0 +Table 1. Chain complex maps for real K-theory +For reference, the groups of K +CR(R) and K +CR(C) are shown below. The natural trans- +formations η, c, r, and ψ that are part of the structure of united K-theory are uniquely +determined from these groups and the long exact sequence (5); they are also shown in Ta- +bles 1 and 2 of [BG22]. In particular we note that for KO∗(R), the map ηi is non-trivial +exactly for i = 0, 1. + +6 +JEFFREY L. BOERSEMA AND SARAH L. BROWNE AND ELIZABETH GILLASPY +0 +1 +2 +3 +4 +5 +6 +7 +KO∗(R) +Z +Z2 +Z2 +0 +Z +0 +0 +0 +KU∗(R) +Z +0 +Z +0 +Z +0 +Z +0 +0 +1 +2 +3 +4 +5 +6 +7 +KO∗(C) +Z +0 +Z +0 +Z +0 +Z +0 +KU∗(C) +Z2 +0 +Z2 +0 +Z2 +0 +Z2 +0 +3. Non-Existence of a Rank-2 Graph with Involution +Theorem 3.1. Let n be an odd integer, n ≥ 3. There does not exist a (row-finite, source- +free) rank-2 graph with involution (Λ, γ) such that K +CR(C∗ +R(Λ, γ)) ∼= K +CR(En). +Proof. Suppose that (Λ, γ) is a row-finite, source-free rank-2 graph with involution and that +K +CR(C∗ +R(Λ, γ)) ∼= K +CR(En). For reference we reproduce the groups of K +CR(En) here: +0 +1 +2 +3 +4 +5 +6 +7 +KO∗(En) +Z2(n−1) +Z2 +Z2 +0 +Z(n−1)/2 +0 +Z2 +Z2 +KU∗(En) +Zn−1 +0 +Zn−1 +0 +Zn−1 +0 +Zn−1 +0 +Note that since KU7(En) = 0 and since im η6 = ker c7 from the long exact sequence (5) relat- +ing KO∗(A) and KU∗(A), we see immediately that η6: Z2 → Z2 must be an isomorphism. +Now, we consider only the real part of the spectral sequence from [BG22, Theorem 3.15]. +This is a spectral sequence converging to KO∗(C∗ +R(Λ, γ)), where the E2-page consists of the +homology of the chain complex +(6) +0 → A +∂2 +−−→ A2 +∂1 +−−→ A → 0 +where A ∼= KO∗(R)Λ0 +f ⊕ KO∗(C)Λ0 +g as described in Section 2.3. +In particular, the spectral sequence has three non-zero columns (for 0 ≤ p ≤ 2) and +is periodic in q (with period 8). Furthermore, a quick examination of the structure of A +(cf. Figure 1) reveals that Ai = 0 for i = 3, 5, 7 and also that Ai is free for i = 0, 4, 6. +Consequently, E2 +p,q = 0 for q = 3, 5, 7. Moreover, since E2 +2,q = ker ∂2 is a subgroup of Ai, we +must have E2 +2,q free for q = 0, 4, 6. Indeed, E∞ +2,q = ker d2 +2,q is a subgroup of E2 +2,q, so E∞ +2,q must +also be free for q = 0, 4, 6. On the other hand, KOi(C∗ +R(Λ, γ)) is finite in all degrees, so it +must be that E∞ +2,q = 0 for q = 0, 4, 6. + +THE STABLE EXOTIC CUNTZ ALGEBRAS ARE HIGHER-RANK GRAPH ALGEBRAS +7 +From what we have said so far, the E∞ +p,q page of the spectral sequence is as follows: +... +... +... +7 +0 +0 +0 +6 +∗ +∗ +0 +5 +0 +0 +0 +4 +∗ +∗ +0 +3 +0 +0 +0 +2 +∗ +∗ +∗ +1 +∗ +∗ +∗ +0 +∗ +∗ +0 +q/p +0 +1 +2 +This E∞ page identifies a filtration of KO∗(C∗ +R(Λ, γ)), in which the subquotients of KOj(C∗ +R(Λ, γ)) +appear along the diagonal p + q = j of the E∞ page. +By hypothesis we have KO0(C∗ +R(Λ, γ)) = Z2(n−1). However, the only non-zero group along +the diagonal p + q = 0 is E∞ +0,0. Thus E∞ +0,0 = Z2(n−1). Similarly, since KO7(C∗ +R(Λ, γ)) = Z2 +and KO6(C∗ +R(Λ, γ)) = Z2, we must have E∞ +1,6 = Z2 = E∞ +0,6: +... +... +... +7 +0 +0 +0 +6 +Z2 +Z2 +0 +5 +0 +0 +0 +4 +∗ +∗ +0 +3 +0 +0 +0 +2 +∗ +∗ +∗ +1 +∗ +∗ +∗ +0 +Z2(n−1) +∗ +0 +q/p +0 +1 +2 +Now, we consider the natural transformation η: A → A of degree 1. Because A is a +direct sum of copies of K +CR(R) and K +CR(C), which data already includes the map η, the +natural transformation η : KO∗(C∗ +R(Λ, γ)) → KO∗+1(C∗ +R(Λ, γ)) exists at the level of the +chain complex (6), passes to a map on the E2-page, then to a map on the E∞-page, and +finally converges to the map η: KOi(C∗ +R(Λ, γ)) → KOi+1(C∗ +R(Λ, γ)) described in Section 2.2. +This means that the map η on KO∗(C∗ +R(Λ, γ)) respects the filtration of KOi(C∗ +R(Λ, γ)) +associated with the spectral sequence; and the resulting maps on the subquotients are the +same as that on the E∞ page. +In particular, the diagonals of the E∞-page that yield +KO6(C∗ +R(Λ, γ)) and KO7(C∗ +R(Λ, γ)) give the commutative diagram below. The vertical η +maps on the left and right come from A as described above. +0 +� E∞ +0.6 +� +η +� +KO6(C∗ +R(Λ, γ)) +� +η +� +E∞ +1,5 +� +η +� +0 +0 +� E∞ +0.7 +� KO7(C∗ +R(Λ, γ)) +� E∞ +1,6 +� 0 +⇐⇒ +0 +� Z2 +� +η +� +Z2 +� +η +� +0 +� +η +� +0 +0 +� 0 +� Z2 +� Z2 +� 0 + +8 +JEFFREY L. BOERSEMA AND SARAH L. BROWNE AND ELIZABETH GILLASPY +Since the nonzero horizontal maps must be isomorphisms, the commutative diagram forces +the vertical map η6: KO6(C∗ +R(Λ, γ)) → KO7(C∗ +R(Λ, γ)) in the center of the diagram to be +zero, which contradicts the known value of η in KO∗(E +R +n). +□ +4. Existence of a Rank-3 Graph with Involution +In this section, we will construct a 3-graph Λ with involution γ, by taking products of +1-graphs with involution, such that C∗ +R(Λ, γ) is stably isomorphic to E +R +n. In what follows, we +use the following convention for suspension of graded modules. If H = {Hi} is a Z-graded +group, then ΣH is a Z-graded group with (ΣH)i = Hi+1. Similarly, Σ−1H is a Z-graded +group with (Σ−1H)i = Hi−1. This convention is consistent with K-theory and suspensions +of C∗-algebras: K∗(SnA) = ΣnK∗(A). +For the proof of the following proposition, we will need a few more preliminaries. For a +graph Λ, a subset X ⊆ Λ0 is hereditary if whenever v ∈ X and vΛw ̸= ∅, then w ∈ X. A cycle +in Λ is a path e1e2 · · ·en with r(e1) = s(en) but, for all 1 ≤ i < n, we have s(ei) ̸= r(ei+1). +We say that a cycle has an entrance if there exists 1 ≤ j ≤ n and an edge f ̸= ej with +r(f) = r(ej). If the only hereditary subsets of Λ0 are ∅ and Λ0, and every cycle in Λ has an +entrance, then [Szy01, Theorem 12] C∗(Λ) is simple. Furthermore, if every cycle in Λ has an +entrance, then Λ is aperiodic. +Proposition 4.1. There exists a 1-graph Λ and involution γ such that K +CR(C∗ +R(Λ, γ)) ∼= +ΣK +CR(R). Furthermore, C∗ +R(Λ, γ) is simple and purely infinite. +Proof. Let Λ be the 1-graph below (which extends infinitely in both directions) and let γ be +the non-trivial involution, which fixes the vertices and edges of the infinite branch on the +left and swaps the vertices and edges of the two infinite branches on the right in the obvious +way. (In fact γ is the only non-trivial involution on Λ.) +• +�❅ +❅ +❅ +❅ +❅ +❅ +❅ +� +• +�❅ +❅ +❅ +❅ +❅ +❅ +❅ +� +• +�❅ +❅ +❅ +❅ +❅ +❅ +❅ +� +• +� +�❅ +❅ +❅ +❅ +❅ +❅ +❅ +• +� +�❅ +❅ +❅ +❅ +❅ +❅ +❅ +• +� +�❅ +❅ +❅ +❅ +❅ +❅ +❅ +•� +� +�⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +• +� +�⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +• +� +�⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +• +� +� +� +� +• +� +�⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +• +� +�⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +• +� +�⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +• +� +�⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +� +�❅ +❅ +❅ +❅ +❅ +❅ +❅ +� +• +�❅ +❅ +❅ +❅ +❅ +❅ +❅ +• +� +�❅ +❅ +❅ +❅ +❅ +❅ +❅ +• +� +�❅ +❅ +❅ +❅ +❅ +❅ +❅ +• � +� +� +• +�⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +� +• +�⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +� +• +�⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +� +We begin by showing that C∗ +R(Λ, γ) is simple and purely infinite. It is straightforward to +check that Λ has no nontrivial hereditary subsets, and that every cycle has an entrance, so +simplicity of the complex algebra C∗(Λ) follows from [Szy01, Theorem 12]. Note further that +for every vertex v ∈ Λ0, there is a vertex w with vΛw ̸= ∅ for some vertex w which supports +a loop. As Λ is aperiodic, one easily checks that the conditions of [KP00, Proposition 4.9] +are satisfied, and so C∗(Λ) is purely infinite. Consequently, [BRS11, Theorem 3.9] implies +that the real C∗-algebra C∗ +R(Λ, γ) is also simple and purely infinite. +We now show that KU0(C∗ +R(Λ, γ)) = 0 and KU1(C∗ +R(Λ, γ)) = Z. (This is the same as +calculating K∗(C∗(Λ)) and does not involve the involution γ.) Let M be the adjacency matrix +for Λ (so Mv,w is the number of edges from w to v). Then KU0(C∗ +R(Λ, γ)) ∼= coker (I − Mt), + +THE STABLE EXOTIC CUNTZ ALGEBRAS ARE HIGHER-RANK GRAPH ALGEBRAS +9 +which can be interpreted as saying that KU0(C∗ +R(Λ, γ)) is generated by vertex projection +classes [pv], which are subject only to relations of the form +[pv] = +� +w∈Λ0 +Mv,w[pw] . +Let v be one of the vertices of Λ that has a loop, and let w ̸= v be the vertex for which there +is an edge from w to v (in each case there is a unique such w). Then the formula above +gives the relation [pv] = [pv] + [pw], which implies that [pw] = 0. If [pw] = 0 we will say +that w is a zero vertex. Now if w is a zero vertex and there is only one edge to w, say from +vertex u, then it follows that u is also a zero vertex. More generally, if w is a zero vertex +and all the edges to w are known to emanate from zero vertices except possibly one edge +from vertex u, then it follows that u is also a zero vertex. Using these principles, it is now +straightforward to work through the graph and to find that every vertex is a zero vertex. +Hence KU0(C∗ +R(Λ, γ)) = 0. +We know that KU1(C∗ +R(Λ, γ)) ∼= ker(I − Mt), which is to say that +(7) +KU1(C∗ +R(Λ, γ)) ∼= NΛ := +� +α: Λ0 → Z | α(v) = +� +w∈Λ0 +Mw,v α(w) +� +. +Let v be one of the vertices of Λ that has a loop, and let w ̸= v be the vertex for which +there is an edge from v to w (in each case there is a unique such w). Then we have the +relation α(v) = α(v) + α(w), which implies that α(w) = 0 for any α ∈ NΛ. If α(w) = 0 for +all α ∈ NΛ we will say that w is a null vertex. Now if w is a null vertex and there is only one +edge emanating from w, say to vertex u, then it follows that u is also a null vertex. More +generally, if w is a null vertex and all the edges from w are known to point to null vertices +except possibly one edge to vertex u, then u is also a null vertex. Using these principles, +it is now straightforward to work through the graph and to find that every vertex is a null +vertex, except for the six vertices labelled u, v, w, x, y, z shown below. +v• +�❇ +❇ +❇ +❇ +❇ +❇ +❇ +❇ +� +u• +�❇ +❇ +❇ +❇ +❇ +❇ +❇ +❇ +� +• +�❅ +❅ +❅ +❅ +❅ +❅ +❅ +� +• +� +�❅ +❅ +❅ +❅ +❅ +❅ +❅ +• +� +�❅ +❅ +❅ +❅ +❅ +❅ +❅ +• +� +�❅ +❅ +❅ +❅ +❅ +❅ +❅ +w•� +� +�② +② +② +② +② +② +② +② +• +� +�⑤ +⑤ +⑤ +⑤ +⑤ +⑤ +⑤ +⑤ +• +� +�⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +• +� +� +� +� +• +� +�⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +• +� +�⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +• +� +�⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +• +� +�④ +④ +④ +④ +④ +④ +④ +④ +� +�❈ +❈ +❈ +❈ +❈ +❈ +❈ +❈ +� +x• +�❉ +❉ +❉ +❉ +❉ +❉ +❉ +❉ +• +� +�❆ +❆ +❆ +❆ +❆ +❆ +❆ +❆ +• +� +�❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄ +• � +� +� +y• +�⑥ +⑥ +⑥ +⑥ +⑥ +⑥ +⑥ +⑥ +� +z• +�⑥ +⑥ +⑥ +⑥ +⑥ +⑥ +⑥ +⑥ +� +• +�⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +� +Using Equation (7) and the fact that the unlabeled vertices are null vertices, we see that +any α ∈ NΛ must satisfy the equations +0 = α(u) + α(w) +α(w) = α(v) + α(x) +0 = α(w) + α(x) +α(x) = α(w) + α(y) +0 = α(x) + α(z) + +10 +JEFFREY L. BOERSEMA AND SARAH L. BROWNE AND ELIZABETH GILLASPY +Solving this system over Z, we find that α(u) is a free variable and that +α(v) = −2α(u), α(w) = −α(u), α(x) = α(u), α(y) = 2α(u), and α(z) = −α(u) . +Thus NΛ ∼= Z. Hence KU∗(C∗ +R(Λ, γ)) = K∗(C∗(Λ)) = (0, Z). +Turning to the real K-theory, we now prove that KO∗(C∗ +R(Λ, γ))) = (Z2, Z2, 0, Z, 0, 0, 0, Z). +First, we show that the real and complex E2 = E∞ page of the Evans spectral sequence for +C∗ +R(Λ, γ) is as follows. +E2 +p,q +(8) +real part +... +... +... +7 +0 +0 +6 +0 +Z +5 +0 +0 +4 +0 +0 +3 +0 +0 +2 +0 +Z +1 +Z2 +0 +0 +Z2 +0 +q/p +0 +1 +complex part +... +... +... +7 +0 +0 +6 +0 +Z +5 +0 +0 +4 +0 +Z +3 +0 +0 +2 +0 +Z +1 +0 +0 +0 +0 +Z +q/p +0 +1 +(9) +We have already discussed the complex part of this spectral sequence. For the real part we +will only discuss the computations for the rows corresponding to j = −1, 0, 1. As we will see, +this is enough to determine KO∗(C∗ +R(Λ, γ)). The other rows can be computed using similar +methods and we include them in the table above for completeness, but we will neither need +nor discuss them. +First, the spectral sequence for a 1-graph with involution always vanishes in row j = −1, +since the chain complex vanishes in that degree. To compute row j = 1 of the spectral +sequence, we refer to [BG22, Theorem 3.14] and Table 1 above, which indicates that E2 +0,1 +and E2 +1,1 are the cokernel and kernel of the map +(∂1)1 = I − Mt +11 : Z +Λ0 +f +2 +→ Z +Λ0 +f +2 +where Λ0 +f is the set of fixed vertices of (Λ, γ) and M11 is the restriction of the incidence +matrix to those vertices. So it suffices to consider the graph consisting of the fixed points of +Λ, shown here. +• +� +�❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄ +• +� +�❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄ +•x +� +�❆ +❆ +❆ +❆ +❆ +❆ +❆ +❆ +•y +� +�❈ +❈ +❈ +❈ +❈ +❈ +❈ +❈ +� +• +� +�⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +• +� +�⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +• +� +�⑥ +⑥ +⑥ +⑥ +⑥ +⑥ +⑥ +⑥ +• +� +�⑥ +⑥ +⑥ +⑥ +⑥ +⑥ +⑥ +•z +� +Using this graph, and the same sort of analysis that we did in the complex case, we find that +coker (I − Mt +11) = Z2. More precisely, working modulo 2 we find that [pv] = 0 for all vertices +in Λ0 +f except those labeled x, y and z in the graph above and that [px] = [py] = [pz] ̸= 0. We +also find easily that ker(I − Mt +11) = 0. +Now, for j = 0, we need to find the cokernel and kernel of the map (∂1)0 which we will +do using Table 1. First recall the partition Λ0 = Λ0 +f ⊔ Λ0 +g ⊔ Λ0 +h where Λ0 +f is the set of fixed + +THE STABLE EXOTIC CUNTZ ALGEBRAS ARE HIGHER-RANK GRAPH ALGEBRAS +11 +vertices (the branch on the left of Λ), Λ0 +g is the set of vertices of the “upper right” branch of +Λ, and Λ0 +h is the set of vertices of the “lower right” branch. With this structure on Λ, the +(infinite) matrix B = I − Mt can be written in block form as +B = I − Mt = + + +B11 +B12 +B12 +B21 +B22 +B23 +B21 +B23 +B33 + + +where, for example, B12 keeps track of edges from vertices in Λ0 +f to vertices in Λ0 +g. Using +Table 1, we see that +(∂1)0 : ZΛ0 +f ⊕ ZΛ0 +g → ZΛ0 +f ⊕ ZΛ0 +g +is given by +(∂1)0 = +� +B11 +2B12 +B21 +B22 + B23 +� +. +We will use a new graph Λ′ to analyze this map. The graph Λ′, shown below, is obtained +from Λ by keeping the vertices from Λ0 +f and Λ0 +g. For each edge in Λ from a vertex in Λ0 +g to +a vertex in Λ0 +f, we create a corresponding edge in Λ′ and for each edge in Λ from a vertex +in Λ0 +f to a vertex in Λ0 +g we create 2 corresponding edges in Λ′. Also, for each edge from a +vertex v = γ(u) ∈ Λ0 +h to a vertex w ∈ Λ0 +g we obtain an edge in Λ′ from u to w. +•x +� +�❆ +❆ +❆ +❆ +❆ +❆ +❆ +❆ +� +⑥⑥⑥⑥⑥⑥⑥⑥ +•z +� +�❈ +❈ +❈ +❈ +❈ +❈ +❈ +❈ +� +⑥⑥⑥⑥⑥⑥⑥⑥ +• +�❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄ +� +• +�❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄ +� +� +• +� +• +� +•y +(2) +� +� +•w +� +� +�⑤ +⑤ +⑤ +⑤ +⑤ +⑤ +⑤ +⑤ +• +� +�⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +• +� +� +� +By construction the adjacency matrix M′ for the graph Λ′ satisfies +I − (M′)t = +� +B11 +2B12 +B21 +B22 + B23 +� +. +Therefore, we can use the graph Λ′ to find the cokernel and kernel of (∂1)0. Using the same +logic and terminology we used when calculating the complex K-theory, we see that w is +a zero vertex because it emits an edge to a vertex v which supports a loop, and the edge +from w to v is the only non-loop edge which points to v. Indeed, every vertex along the +bottom row of the graph Λ′ except y is a zero vertex. The fact that these zero vertices +(with the exception of w) only receive edges from one (potentially) nonzero vertex on the +top row of Λ′ implies that every vertex in the top row except x and z are also zero vertices. +Now, w is a zero vertex, but since there are two edges from y to w we obtain the relation +[pw] = [pw] + 2[py] which implies that 2[py] = 0, but [py] ̸= 0. Finally, from the relations +[px] = −[py] and [pz] = [py] we conclude that coker (∂1)0 = Z2. +To compute ker(∂1)0, we also proceed as in the computations for the complex case: All +of the vertices in the bottom row of Λ′, save w, are null vertices. Moreover, if a null vertex +v emits n edges to a single potentially non-null vertex u, we must have nα(u) = 0 for any +α ∈ NΛ′, and as α(u) ∈ Z we conclude that u must also be null. It follows that w is null, as +are all of the vertices in the top row of Λ′. That is, ker(∂1)0 = {0}. +Now, with the three rows that we’ve identified, the spectral sequence (8) implies that +KO0(C∗ +R(Λ, γ)) ∼= KO1(C∗ +R(Λ, γ)) ∼= Z2. We claim that using this we can compute KOi(C∗ +R(Λ, γ)) +for 2 ≤ i ≤ 7 using the long exact sequence (5) and other aspects of CR-structure. The +fact that KU6(C∗ +R(Λ, γ)) = KU0(C∗ +R(Λ, γ)) = 0 implies that η0 is injective, and hence an + +12 +JEFFREY L. BOERSEMA AND SARAH L. BROWNE AND ELIZABETH GILLASPY +isomorphism, and also that η−1 is surjective. Since KU2(C∗ +R(Λ, γ)) = 0 it follows that η1 is +surjective. Thus if KO2(C∗ +R(Λ, γ)) has a non-zero element, then it would have to be in the +image of η1 ◦ η0 ◦ η−1 = η3. But η3 = 0 for all real C∗-algebras. Thus KO2(C∗ +R(Λ, γ)) = 0. +Since KO2(C∗ +R(Λ, γ)) = 0, the long exact sequence implies that c3: KO3(C∗ +R(Λ, γ)) → +KU3(C∗ +R(Λ, γ)) = Z is injective. This forces KO3(C∗ +R(Λ, γ)) = Z. Moreover, as im r1 = +ker η1 = Z2 we must have r1 : Z → Z2 the unique nonzero map. Since im c3 ∼= ker r1, we +conclude that c3 is multiplication by 2. The relation rc = 2 then implies that r3 = 1. +Continuing this process using the long exact sequence, we compute that KO∗(C∗ +R(Λ, γ))) = +(Z2, Z2, 0, Z, 0, 0, 0, Z). The module maps η, r, c, ψ are then completely determined by these +groups and the long exact sequence (5); that is, K +CR(C∗ +R(Λ, γ)) and hence K +CRT(C∗ +R(Λ, γ)) +coincide with ΣK +CRT(R). +□ +Lemma 4.2. Suppose that (Λ1, γ1) and (Λ2, γ2) are higher-rank graphs with involutions. +Then (Λ1 × Λ2, γ1 × γ2) is a higher-rank graph with involution and +C∗ +R(Λ1 × Λ2, γ1 × γ2) ∼= C∗ +R(Λ1, γ1) ⊗R C∗ +R(Λ2, γ2) . +Proof. Assume that Λ1 and Λ2 have rank k1 and k2 respectively. +From [KP00, Proposi- +tion 1.8] the product Λ1 × Λ2 is a graph of rank k1 + k2, with degree functor d(λ1, λ2) = +d(λ1) + d(λ2). Furthermore, there is an involution γ on Λ1 × Λ2 defined by γ(λ1, λ2) = +(γ1(λ1), γ2(λ2)). +From [KP00, Corollary 3.5], there is an isomorphism φ: C∗(Λ1 × Λ2) → C∗(Λ1) ⊗ C∗(Λ2) +defined by φ(s(λ1,λ2)) = sλ1 ⊗sλ2. To finish the proof, we need only show that φ preserves the +real structures (4) of C∗(Λ1 × Λ2) and C∗(Λ1) ⊗ C∗(Λ2) which are induced by the graphical +involutions γi. This is straightforward: +φ(�γ(s(λ1,λ2)) = φ(s∗ +(γ1(λ1),γ2(λ2)))) += s∗ +γ1(λ1) ⊗ s∗ +γ2(λ2) += �γ1(sλ1) ⊗ �γ2(sλ2) += � +γ1 ⊗ γ2(sλ1 ⊗ sλ2) += � +γ1 ⊗ γ2(φ(s(λ1,λ2))) . +□ +Theorem 4.3. Let n be an odd integer, n ≥ 3. There exists a rank-3 graph with invo- +lution (Λn, γn) such that C∗ +R(Λn, γn) ∼= K +R ⊗R E +R +n. +Furthermore, there exists a projection +p ∈ C∗ +R(Λn, γn) such that pC∗ +R(Λn, γn)p ∼= E +R +n. +Proof. Let (Λ, γ) be the 1-graph given by Proposition 4.1 and let (En, γn) be the finite 1- +graph with involution from Example 6.2 of [Boe17]. Then both C∗ +R(Λ, γ) and C∗ +R(En, γn) are +simple and purely infinite and we have +K +CR(C∗ +R(Λ, γ)) ∼= ΣK +CR(R) +and +K +CR(C∗ +R(En, γn)) ∼= Σ6K +CR(En) . +This implies by [BRS11, Proposition 2.1] that +K +CRT(C∗ +R(Λ, γ)) ∼= ΣK +CRT(R) +and +K +CRT(C∗ +R(En, γn)) ∼= Σ6K +CRT(En) . +Let (Λn, γn) be the product rank-3 graph with involution +(Λn, γn) = (Λ, γ) × (Λ, γ) × (En, γn). + +THE STABLE EXOTIC CUNTZ ALGEBRAS ARE HIGHER-RANK GRAPH ALGEBRAS +13 +Lemma 4.2 then implies that +C∗ +R(Λn, γn) ∼= C∗ +R(Λ, γ) ⊗R C∗ +R(Λ, γ) ⊗R C∗ +R(En, γn). +Now, K +CRT(C∗ +R(Λ, γ)) is a free CRT -module, since it is isomorphic to a suspension of K +CRT(R) +(see [Boe02, Section 2.1]). Therefore the K¨unneth formula for the K-theory of real C∗- +algebras (Proposition 3.5 and Theorem 4.2 of [Boe02]) gives +K +CRT(C∗ +R(Λn, γn)) ∼= K +CRT(C∗ +R(Λ, γ)) ⊗CRT K +CRT(C∗ +R(Λ, γ)) ⊗CRT K +CRT(C∗ +R(En, γn)) +∼= Σ2K +CRT(R) ⊗CRT Σ6K +CRT(En) +∼= K +CRT(En) . +Note that C∗ +R(Λn, γn) is a stable, simple, purely infinite, real C∗-algebra, thanks to Propo- +sition 4.1 and [Boe17, Example 6.2]. We also know that KR ⊗R En is a a stable, simple, +purely infinite, real C∗-algebra, because its complexification K ⊗ On is simple and purely +infinite (see Theorem 3.9 of [BRS11]). Thus the first statement of the theorem follows by +the classification of real Kirchberg algebras, [BRS11, Theorem 10.2, Part (1)]. +To prove the second statement, by [BRS11, Proposition 3.13] there is a projection p ∈ +C∗ +R(Λn, γn) such that [p] is a generator of KO0(C∗ +R(Λn, γn)) = Z2(n−1). Then +K +CR(pC∗ +R(Λn, γn)p) ∼= K +CR(C∗ +R(Λn, γn)) ∼= K +CR(En) +(where the first isomorphism is by [Boe06, Proposition 9]). Furthermore the class of the +identity [p] ∈ KO0(pC∗ +R(Λ, γ)p) ∼= Z2(n−1) corresponds under this isomorphism to the class +of the identity [1] ∈ KO0(En) ∼= Z2(n−1). Therefore by [BRS11, Theorem 10.2, Part (2)], we +have pC∗ +R(Λ, γ)p ∼= En. +□ +5. Appendix – real Kirchberg suspension algebras +In the previous section, we introduced a graph with involution for which K +CR(C∗(Λ, γ)) ∼= +ΣK +CR(R). We consider this algebra as a sort of real Kirchberg suspension, since it is a real +purely infinite simple stable nuclear C∗-algebra satisfying the UCT, and with the same KK- +type as the suspension algebra SR ∼= C0((0, 1), R). By repeatedly taking the product of this +graph with itself, which corresponds to repeatedly tensoring this algebra with itself, we can +obtain a higher-rank graph, the real C∗-algebra of which is a real Kirchberg algebra with the +same KK-type as SiR for any i. These tensor products will be higher-rank graph algebras +of rank i. It is natural to ask which of these suspensions can be obtained from a 1-graph +with involution. In this section, we will answer this question completely, providing a full +characterization of the integers i (mod 8) for which there exists a 1-graph with involution +(Λ, γ) such that K +CR(C∗(Λ, γ)) ∼= ΣiK +CR(R) ∼= K +CR(SiR). For the positive results, we will +exhibit directly the appropriate graph or graph with involution. +Proposition 5.1. For each i = {−2, −1, 0, 1} there exists a 1-graph with involution (Λ, γ) +such that K +CR(C∗(Λ, γ)) ∼= ΣiK +CR(R). Furthermore, C∗ +R(Λ, γ) is simple and purely infinite. +Sketch of proof. For each i we show below a graph or graph with involution that satisfies +K +CR(C∗(Λ, γ)) ∼= ΣiK +CR(R). The K-theory calculations, not shown, are carried out using +the same techniques as in the proof of Proposition 4.1. + +14 +JEFFREY L. BOERSEMA AND SARAH L. BROWNE AND ELIZABETH GILLASPY +i = −2. The graph Λ is shown below; we equip it with the non-trivial involution γ which +interchanges the right-hand branches. +• � +❅ +❅ +❅ +❅ +❅ +❅ +❅ +�⑦⑦⑦⑦⑦⑦⑦ +� +• � +❅ +❅ +❅ +❅ +❅ +❅ +❅ +�⑦⑦⑦⑦⑦⑦⑦ +� +• � +❅ +❅ +❅ +❅ +❅ +❅ +❅ +�⑦⑦⑦⑦⑦⑦⑦ +� +• +� +�⑦⑦⑦⑦⑦⑦⑦ +� +❅ +❅ +❅ +❅ +❅ +❅ +❅ +• +� +�⑦⑦⑦⑦⑦⑦⑦ +� +❅ +❅ +❅ +❅ +❅ +❅ +❅ +• +� +�⑦⑦⑦⑦⑦⑦⑦ +� +❅ +❅ +❅ +❅ +❅ +❅ +❅ +•� +� +� • +� • +� • +� +� +� • +� +� +• +� • +� • +� +�⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +� +�❅ +❅ +❅ +❅ +❅ +❅ +❅ +• +� • +� • +� • +� +� +• +� +⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +�❅❅❅❅❅❅❅ +� +• +� +⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +�❅❅❅❅❅❅❅ +� +• +� +⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +�❅❅❅❅❅❅❅ +� +i = −1. The graph Λ is shown below, with trivial involution γ = id . +• +�❅ +❅ +❅ +❅ +❅ +❅ +❅ +� +• +�❅ +❅ +❅ +❅ +❅ +❅ +❅ +� +• +�❅ +❅ +❅ +❅ +❅ +❅ +❅ +� +• +�❅ +❅ +❅ +❅ +❅ +❅ +❅ +� +• +�⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +• +� +�⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +• +� +�⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +• +� +�⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +• +� +� +� +i = 0. The graph Λ is shown below, with trivial involution γ = id . +• +�⑦⑦⑦⑦⑦⑦⑦ +� +� +❅ +❅ +❅ +❅ +❅ +❅ +❅ +• +�⑦⑦⑦⑦⑦⑦⑦ +� +� +❅ +❅ +❅ +❅ +❅ +❅ +❅ +• +�⑦⑦⑦⑦⑦⑦⑦ +� +� +❅ +❅ +❅ +❅ +❅ +❅ +❅ +• +�⑦⑦⑦⑦⑦⑦⑦ +� +� +❅ +❅ +❅ +❅ +❅ +❅ +❅ +• +� • +� • +� • +� • +� +� +i = 1. The graph Λ is shown below with non-trivial involution γ, as in Proposition 4.1. +• +�❅ +❅ +❅ +❅ +❅ +❅ +❅ +� +• +�❅ +❅ +❅ +❅ +❅ +❅ +❅ +� +• +�❅ +❅ +❅ +❅ +❅ +❅ +❅ +� +• +� +�❅ +❅ +❅ +❅ +❅ +❅ +❅ +• +� +�❅ +❅ +❅ +❅ +❅ +❅ +❅ +• +� +�❅ +❅ +❅ +❅ +❅ +❅ +❅ +•� +� +�⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +• +� +�⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +• +� +�⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +• +� +� +� +� +• +� +�⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +• +� +�⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +• +� +�⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +• +� +�⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +� +�❅ +❅ +❅ +❅ +❅ +❅ +❅ +� +• +�❅ +❅ +❅ +❅ +❅ +❅ +❅ +• +� +�❅ +❅ +❅ +❅ +❅ +❅ +❅ +• +� +�❅ +❅ +❅ +❅ +❅ +❅ +❅ +• � +� +� +• +�⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +� +• +�⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +� +• +�⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +� +For the i = −2 graph, one can determine all of the groups KOi(C∗ +R(Λ, γ)) from the +associated spectral sequence, except for KO2(C∗ +R(Λ, γ)). In that case, the spectral sequence +KO2(C∗ +R(Λ, γ)) has the filtration 0 → Z → KO2(C∗ +R(Λ, γ)) → Z2 → 0. +Although this +filtration by itself does not determine KO2(C∗ +R(Λ, γ)), the long exact sequence (5) forces +KO2(C∗ +R(Λ, γ)) = Z. Moreover, the module maps r, c, η, ψ are uniquely determined by (5). +□ + +THE STABLE EXOTIC CUNTZ ALGEBRAS ARE HIGHER-RANK GRAPH ALGEBRAS +15 +Proposition 5.2. For 2 ≤ i ≤ 5, there does not exist a 1-graph (Λ, γ) with involution such +that K +CR(C∗ +R(Λ, γ)) ∼= ΣiK +CR(R). +Proof. Suppose that (Λ, γ) is a graph with involution and K +CR(C∗ +R(Λ, γ)) ∼= ΣiK +CR(R). +The real Pimsner-Voiculescu sequence (or equivalently, the real Evans spectral sequence) +for K +CR(C∗(Λ, γ)) implies that KO−1(C∗ +R(Λ, γ)) and KO−3(C∗ +R(Λ, γ)) are free abelian groups. +But recall that KO1(R) ∼= KO2(R) ∼= Z2. Thus the group (Σ2KO(R))−1 = KO1(R) has +torsion, implying that K +CR(C∗ +R(Λ, γ)) ≇ Σ2KO +CR(R), hence i ̸= 2. Similarly, the groups +(Σ3KO(R))−1, (Σ4KO(R))−3, and (Σ5KO(R))−3 have torsion, showing that i ̸= 3, 4, 5. +□ +References +[BG22] +Jeffrey L. Boersema and Elizabeth Gillaspy, K-theory for real k-graph C∗-algebras, Ann. K- +Theory 7 (2022), no. 2, 395–440. +[BHRS02] +T. Bates, J.-H. Hong, I. Raeburn, and W. Szyma´nski, The ideal structure of the C∗-algebras of +infinite graphs, Illinois J. Math. 46 (2002), no. 4, 1159–1176. +[Boe02] +Jeffrey L. Boersema, Real C∗-algebras, united K-theory, and the K¨unneth formula, K-Theory +26 (2002), no. 4, 345–402. +[Boe06] +, The range of united K-theory, J. Funct. Anal. 235 (2006), no. 2, 701–718. +[Boe14] +, The K-theory of real graph C∗-algebras, Rocky Mountain J. Math. 44 (2014), 397–417. +[Boe17] +, The real C∗-algebra of a graph with involution, M¨unster J. Math. 10 (2017), 485–521. +[BRS11] +Jeffrey L. Boersema, Efren Ruiz, and P. J. Stacey, The classification of real purely infinite simple +C∗-algebras, Doc. Math. 16 (2011), 619–655. MR 2837543 +[EFG+21] C. Eckhardt, K. Fieldhouse, D. Gent, E. Gillaspy, I. Gonzales, and D. Pask, Moves on k-graphs +preserving Morita equivalence, Canad. J. Math. (2021), to appear, arXiv:2006.13441. +[Eva08] +D.G. Evans, On the K-theory of higher rank graph C∗-algebras, New York J. Math. 14 (2008), +1–31. +[Hew96] +Beatrice Hewitt, On the homotopical classification of KO-module spectra, Ph.D. thesis, Univer- +sity of Illinois at Chicago, 1996. +[HRSW13] R. Hazlewood, I. Raeburn, A. Sims, and S.B.G. Webster, Remarks on some fundamental results +about higher-rank graphs and their C∗-algebras, Proc. Edinb. Math. Soc. (2) 56 (2013), no. 2, +575–597. +[HS04] +Jeong Hee Hong and Wojciech Szyma´nski, The primitive ideal space of the C∗-algebras of infinite +graphs, J. Math. Soc. Japan 56 (2004), 45–64. +[KP00] +A. Kumjian and D. Pask, Higher rank graph C∗-algebras, New York J. Math. 6 (2000), 1–20. +[RS04] +Iain Raeburn and Wojciech Szyma´nski, Cuntz-Krieger algebras of infinite graphs and matrices, +Trans. Amer. Math. Soc. 356 (2004), no. 1, 39–59. +[RSS15] +E. Ruiz, A. Sims, and A. P. W. Sørensen, UCT-Kirchberg algebras have nuclear dimension one, +Adv. Math. 279 (2015), 1–28. +[Szy01] +Wojciech Szyma´nski, Simplicity of Cuntz-Krieger algebras of infinite matrices, Pacific J. Math. +199 (2001), no. 1, 249–256. + diff --git a/NtE0T4oBgHgl3EQfTQBT/content/tmp_files/load_file.txt b/NtE0T4oBgHgl3EQfTQBT/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..61641265ac7fc47abaedc1b202491d1330807857 --- /dev/null +++ b/NtE0T4oBgHgl3EQfTQBT/content/tmp_files/load_file.txt @@ -0,0 +1,1060 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf,len=1059 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='02233v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='OA] 5 Jan 2023 THE STABLE EXOTIC CUNTZ ALGEBRAS ARE HIGHER-RANK GRAPH ALGEBRAS JEFFREY L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' BOERSEMA AND SARAH L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' BROWNE AND ELIZABETH GILLASPY Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' For each odd integer n ≥ 3, we construct a rank-3 graph Λn with involution γn whose real C∗-algebra C∗ R (Λn, γn) is stably isomorphic to the exotic Cuntz algebra E R n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' This construction is optimal, as we prove that a rank-2 graph with involution (Λ, γ) can never satisfy C∗ R (Λ, γ) ∼ME E R n, and the first author reached the same conclusion for rank-1 graphs (directed graphs) in [Boe17, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Our construction relies on a rank-1 graph with involution (Λ, γ) whose real C∗-algebra C∗ R (Λ, γ) is stably isomorphic to the suspension SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' In the Appendix, we show that the i-fold suspension SiR is stably isomorphic to a graph algebra iff −2 ≤ i ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Introduction For every odd integer n ≥ 3, the (complex) Cuntz algebra On has two real forms: the real Cuntz algebra O R n and the exotic Cuntz algebra En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' While the existence of En follows from the classification of simple purely infinite real C∗-algebras [Boe06, BRS11], the non-constructive nature of the existence portion of this classification theorem [Boe06, Theorem 1] means that we know very little about En beyond its K-theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' In particular, until now there has been no construction or representation of En in terms of familiar C∗-algebraic objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' In this paper, we give an explicit realization of the stabilized exotic Cuntz algebras KR ⊗R En as higher-rank graph algebras associated to rank-3 graphs with involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Given the extensive literature on the properties of higher-rank graph C∗-algebras, we anticipate that this concrete description will facilitate an improved understanding of these elusive algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Higher-rank graphs, or k-graphs, are a k-dimensional generalization of directed graphs which were introduced by Kumjian and Pask in [KP00].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Many of the properties of (complex) directed graph C∗-algebras, such as their K-theory [RS04] and their ideal structure [BHRS02, HS04], are visible from the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' While the structure of k-graph C∗-algebras is more intricate than that of graph C∗-algebras, k-graph C∗-algebras also encompass a broader range of examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Indeed [RSS15], every complex UCT Kirchberg algebra is a direct limit of 2-graph C∗-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' The real C∗-algebra C∗ R(Λ, γ) of a higher-rank graph with involution (Λ, γ) was recently introduced by the first and third authors in [BG22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' In that paper, the authors also generalized the work of [Eva08] and [Boe17] to describe a spectral sequence which converges to the CR K-theory of these real C∗-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' The main result (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='3) of the present paper, that the exotic Cuntz algebra is stably isomorphic to the C∗-algebra of a 3-graph with involution, is the best possible in terms of the rank of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' In [Boe17], the first author made an extensive analysis of the K-theory of the real C∗-algebra C∗ R(Λ, γ) of a rank-1 graph (directed graph) with involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' 1 In particular, [Boe17, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='3] establishes that the exotic Cuntz algebra cannot be isomorphic or stably isomorphic to the real C∗-algebra C∗ R(Λ) of a directed graph Λ, or to the real C∗- algebra C∗ R(Λ, γ) of a graph with involution, since KO7(En) = Z2 but KO7(C∗ R(Λ, γ)) is always torsion-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='1 below uses the K-theory spectral sequence for real higher- rank graph C∗-algebras ([BG22, Section 3]) to show that En ̸∼ME C∗ R(Λ, γ)) for any rank-2 1This class of C∗-algebras includes the real C∗-algebras C∗ R (Λ) of a directed graph, introduced in [Boe14], as C∗ R (Λ) ∼= C∗ R (Λ, γtriv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' 1 2 JEFFREY L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' BOERSEMA AND SARAH L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' BROWNE AND ELIZABETH GILLASPY graph with involution (Λ, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' However, we construct in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='3 a family of rank-3 graphs with involution (Λn, γn) such that C∗ R(Λn, γn) ∼= En ⊗R KR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Our construction combines the directed graphs with involution (En, γn) of [Boe17, Exam- ple 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='2], which satisfy C∗ R(En, γn) ∼ME S6En, with a directed graph with involution (Λ, γ) such that (Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='1) C∗ R(Λ, γ) is a real Kirchberg algebra which is KK-equivalent to the suspension algebra SR ∼= C0((0, 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' To be precise, the 3-graph Λn which gives C∗ R(Λn, γn) ∼= En ⊗R KR is a product graph, Λn = En × Λ × Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Prompted by the graph with involution of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='1, we consider in Section 5 the question of which suspensions SiR are KK-equivalent to the real C∗-algebra of a graph with involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' For −2 ≤ i ≤ 1 we exhibit an example of a graph with involution (Λ, γ) such that C∗ R(Λ, γ) is KK-equivalent to SiR, and we show in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='2 that SiR ̸∼KK C∗ R(Λ, γ) if 2 ≤ i ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' (However, we can realize these suspensions as 2-graph or 3-graph algebras, by taking products of the graphs which do realize suspensions of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=') Many key questions remain open for further investigation about the class of real C∗- algebras that can be obtained using higher-rank graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' For example, it is still unknown whether or not En itself can be realized as a rank-k graph-with-involution algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Similarly, it remains unknown which real Kirchberg algebras can be realized by higher-rank graphs with involution (as opposed to inductive limits of such objects);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' we would particularly like to find a K-theoretic characterization of such algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Acknowledgments: E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' was partially supported by NSF grant 1800749.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Higher-rank graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' [KP00, Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='1] A higher-rank graph of rank k, or a k-graph, is a countable small category Λ equipped with a degree functor d: Λ → Nk such that, if a morphism λ ∈ Λ satisfies d(λ) = m + n, then there exist unique morphisms µ, ν ∈ Λ such that λ = µν, d(µ) = m and d(ν) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Write ei for the standard ith basis vector of Nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' The morphisms of degree ei can be advantageously viewed as the “edges of color i” in Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' In this perspective, if e is an edge of color i and f is an edge of color j, their composition ef ∈ Λ satisfies d(ef) = ei + ej = ej + ei, so we must be able to rewrite ef = f ′e′ for some morphisms e′, f ′ ∈ Λ with d(f ′) = ej and d(e′) = ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Indeed, by [HRSW13, Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='4 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='5], a k-graph can be equivalently thought of as arising from a directed graph G, with k colors of edges and with a factorization rule on multicolored paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' That is, given any two colors (“red” and “blue”) and any two vertices v, w in G, the factorization rule identifies each red-blue path ef from v to w with an equivalent blue-red path f ′e′ from v to w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' We would like the quotient of the space G∗ of directed paths in G by the equivalence relation ∼ generated by the factorization rule to be a k-graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' For this to occur, the factorization rule must also satisfy certain consistency conditions which ensure that, for each path in G∗, its equivalence class under ∼ corresponds to a k-dimensional hyper-rectangle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' see [EFG+21, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='3] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' As our work in this paper does not depend on these consistency conditions, we will not reproduce them here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' That said, we remark that in THE STABLE EXOTIC CUNTZ ALGEBRAS ARE HIGHER-RANK GRAPH ALGEBRAS 3 a rank-1 graph, the factorization rule is nonexistent, and so a 1-graph is precisely the space of paths of a directed graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Let Λ be a k-graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Given n ∈ Nk and objects v, w ∈ Λ, we write (1) Λn = {λ ∈ Λ : d(λ) = n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' By the factorization rule, for every λ ∈ Λ, there are unique v, w ∈ Λ0 with vλ = λw = vλw = λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' That is, we can identify Λ0 with the objects of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' If λ = vλw, we write v = r(λ) and w = s(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Thus, expanding on Equation (1), we have vΛn = {λ ∈ Λ : r(λ) = v and d(λ) = n} Λnw = {λ ∈ Λ : s(λ) = w and d(λ) = n}, (2) as well as the obvious variations such as vΛnw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' A k-graph Λ has k adjacency matrices Mi ∈ MΛ0(N), which are given by (3) Mi(v, w) = #vΛeiw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' In the graphical picture, λ ∈ Λ(n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=',nk) means that λ represents the ∼-equivalence class of a path with ni edges of color i, for each 1 ≤ i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' That is, Λ0 consists of the length-0 paths, ie, the vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Then Mi(v, w) is the number of edges of color i from vertex w to vertex v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' If Λ1 is a k1-graph and Λ2 is a k2-graph, then [KP00, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='8] their (Cartesian) product Λ1 ×Λ2 is a (k1 +k2)-graph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' the degree functor is given by d(λ1, λ2) = (d(λ1), d(λ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' We have (Λ1 × Λ2)0 = Λ0 1 × Λ0 2 and s(λ1 × λ2) = (s(λ1), s(λ2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' In this paper we will focus on k-graphs which are row-finite and source-free (or have no sources).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' We say a k-graph Λ is row-finite if |vΛn| < ∞ for all n ∈ Nk and v ∈ Λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' The k-graph has no sources if vΛn ̸= ∅ for all v, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' It is straightforward to check that if Λ1, Λ2 are row-finite and source-free, then so is Λ1 × Λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' For a row-finite source-free k-graph Λ, its (complex) C∗-algebra is the universal C∗-algebra generated by a Cuntz–Krieger Λ-family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' [KP00, Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='5] Given a row-finite source-free k-graph Λ, a Cuntz–Krieger Λ-family is a collection {tλ}λ∈Λ of partial isometries in a C∗-algebra A which satisfy the fol- lowing conditions: (CK1) For each v ∈ Λ0, tv is a projection, and tvtw = δv,wtv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' (CK2) For each λ ∈ Λ, t∗ λtλ = ts(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' (CK3) For each λ, µ ∈ Λ, tλtµ = tλµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' (CK4) For each v ∈ Λ0 and each n ∈ Nk, tv = � λ∈vΛn tλt∗ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' We define C∗(Λ) to be the universal (complex) C∗-algebra generated by a Cuntz–Krieger family, in the sense that for any Cuntz–Krieger Λ-family {tλ}λ∈Λ, there is a surjective ∗- homomorphism C∗(Λ) → C∗({tλ}λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' We write {sλ}λ∈Λ for the generators of C∗(Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' By using the Cuntz–Krieger relations, one can compute that C∗(Λ) = span{sλs∗ µ : s(λ) = s(µ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='5(iv) of [KP00] also establishes that C∗(Λ1 × Λ2) ∼= C∗(Λ1) ⊗ C∗(Λ2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' the isomorphism takes s(λ1,λ2) to sλ1 ⊗ sλ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' We will use one more ingredient – an involution – to construct the real C∗-algebras asso- ciated to higher-rank graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' An involution γ on a k-graph Λ is a degree-preserving functor γ : Λ → Λ which satisfies γ ◦ γ = id Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' 4 JEFFREY L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' BOERSEMA AND SARAH L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' BROWNE AND ELIZABETH GILLASPY As established in [BG22, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='4], the real C∗-algebra associated to a k-graph Λ and an involution γ : Λ → Λ is (4) C∗ R(Λ, γ) = spanR{zsλs∗ µ + zsγ(λ)s∗ γ(µ) | z ∈ C, λ, µ ∈ Λ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Equivalently, we have C∗ R(Λ, γ) = {a ∈ C∗(Λ) | �γ(a) = a∗}, where �γ is the antimultiplicative C∗-involution uniquely determined by �γ(sλ) = s∗ γ(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' For any involution γ on Λ, we have C∗(Λ) ∼= C ⊗R C∗ R(Λ, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' CRT K-theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' In our work, we will use the full united K-theory K CRT(A) (introduced in [Boe02]) as well as the abbreviated variation K CR(A) which contains just the real and complex parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='2 of [BRS11] shows that the category of real purely infinite simple C∗-algebras, whose complexifications are simple and in the UCT class, is classified up to isomorphism by either of these invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' We tend to use K CR(A) since it is simpler and usually sufficient, but we will also need to use K CRT(A) on occasion since that is the context in which we have the K¨unneth formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Specifically, recall that for a real C∗-algebra A, K CR(A) = {KO∗(A), KU∗(A)} K CRT(A) = {KO∗(A), KU∗(A), KT∗(A)} where KO∗(A) is the standard 8-periodic real K-theory for a real C∗-algebra and KU∗(A) = K∗(C ⊗C A) is the 2-periodic K-theory of the complexification of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Meanwhile KT∗(A) is the 4-periodic self-conjugate K-theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' These invariants also include the additional CR and CRT -module structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' In particular for K CR(A) there are natural transformations ri : KUi(A) → KOi(A) induced by the standard inclusion C → M2(R) ci : KOi(A) → KUi(A) induced by the standard inclusion R → C ψi : KUi(A) → KUi(A) induced by conjugation C → C ηi : KOi(A) → KOi+1(A) induced by multiplication by η ∈ KO1(R) = Z2 ξi : KOi(A) → KOi+1(A) induced by multiplication by ξ ∈ KO4(R) = Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' This additional structure tends to aid in the computations of KO∗(A) because the natural transformations satisfy the relations rc = 2 cr = 1 + ψ 2η = 0 rψ = r ψ2 = id η3 = 0 ψc = c ψβU = −βUψ ξ = rβ2 Uc and they fit into a long exact sequence (5) · · rβ−1 U −−−→ KOi(A) η−→ KOi+1(A) c−→ KUi+1(A) rβ−1 U −−−→ KOi−1(A) η−→ · · · These two invariants K CR(A) and K CRT(A) contain the same information by results of [Hew96] (summarized, for example, in [BRS11, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' K-theory for higher-rank graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' For the real C∗-algebra C∗ R(Λ, γ) of a higher- rank graph with involution, [BG22, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='10] establishes the existence of a spectral sequence {Er, dr} of CR-modules that converges to K CR(C∗ R(Λ, γ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' The complex part of this spectral sequence (Er p,q) U coincides with the Evans spectral sequence [Eva08] and converges to KU∗(C∗ R(Λ, γ)) = K∗(C∗(Λ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' The real part of this spectral sequence (Er p,q) O converges to KO∗(C∗ R(Λ, γ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' THE STABLE EXOTIC CUNTZ ALGEBRAS ARE HIGHER-RANK GRAPH ALGEBRAS 5 The E2 page of the spectral sequence arises from the homology of a certain chain complex C based on the combinatorial information of Λ and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' We will use the spectral sequence only in the rank-1 and rank-2 cases, where these chain complexes have the following straightforward descriptions which we recall from [BG22, Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='14 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' First let Λ0 f be the set of vertices fixed by the involution γ and let Λ0 g ⊔ Λ0 h be any partition of Λ0\\Λ0 f satisfying γ(Λ0 g) = Λ0 h and γ(Λ0 h) = Λ0 g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Then let A = K CR(R)Λ0 f ⊕ K CR(C)Λ0 g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' For a rank-1 graph with involution the chain complex C is given by 0 → A ∂1 −→ A → 0, where ∂1 = ρ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' For a rank-2 graph with involution the chain complex C is given by 0 → A ∂2 −→ A2 ∂1 −→ A → 0, where ∂1 = � ρ1 ρ2� and ∂2 = � −ρ2 ρ1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Here the maps ρi are determined by the adjacency structure of Λ as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' For 1 ≤ i ≤ k, the complex part (ρi) U 0 : ZΛ0 → ZΛ0 is represented by the matrix Bi = I − Mt i , where Mi is the adjacency matrix of the graph Λ for the edges of degree ei, and (ρi) U 1 = 0 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' The real parts of this map (ρi) O j , for 0 ≤ j ≤ 7, can similarly be determined for each i by some variations of Bi as shown in Table 3 of [BG22] and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='4 of [Boe17], which we also reproduce here as Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='complex part ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='\uf8eb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='\uf8ed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='B11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='B12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='B12 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='\uf8f8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='Z|Λ0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='f | ⊕ Z|Λ0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='g| ⊕ Z|Λ0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='h → Z|Λ0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='f| ⊕ Z|Λ0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='g| ⊕ Z|Λ0 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Chain complex maps for real K-theory For reference, the groups of K CR(R) and K CR(C) are shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' The natural trans- formations η, c, r, and ψ that are part of the structure of united K-theory are uniquely determined from these groups and the long exact sequence (5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' they are also shown in Ta- bles 1 and 2 of [BG22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' In particular we note that for KO∗(R), the map ηi is non-trivial exactly for i = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' 6 JEFFREY L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' BOERSEMA AND SARAH L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' BROWNE AND ELIZABETH GILLASPY 0 1 2 3 4 5 6 7 KO∗(R) Z Z2 Z2 0 Z 0 0 0 KU∗(R) Z 0 Z 0 Z 0 Z 0 0 1 2 3 4 5 6 7 KO∗(C) Z 0 Z 0 Z 0 Z 0 KU∗(C) Z2 0 Z2 0 Z2 0 Z2 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Non-Existence of a Rank-2 Graph with Involution Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Let n be an odd integer, n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' There does not exist a (row-finite, source- free) rank-2 graph with involution (Λ, γ) such that K CR(C∗ R(Λ, γ)) ∼= K CR(En).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Suppose that (Λ, γ) is a row-finite, source-free rank-2 graph with involution and that K CR(C∗ R(Λ, γ)) ∼= K CR(En).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' For reference we reproduce the groups of K CR(En) here: 0 1 2 3 4 5 6 7 KO∗(En) Z2(n−1) Z2 Z2 0 Z(n−1)/2 0 Z2 Z2 KU∗(En) Zn−1 0 Zn−1 0 Zn−1 0 Zn−1 0 Note that since KU7(En) = 0 and since im η6 = ker c7 from the long exact sequence (5) relat- ing KO∗(A) and KU∗(A), we see immediately that η6: Z2 → Z2 must be an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Now, we consider only the real part of the spectral sequence from [BG22, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' This is a spectral sequence converging to KO∗(C∗ R(Λ, γ)), where the E2-page consists of the homology of the chain complex (6) 0 → A ∂2 −−→ A2 ∂1 −−→ A → 0 where A ∼= KO∗(R)Λ0 f ⊕ KO∗(C)Λ0 g as described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' In particular, the spectral sequence has three non-zero columns (for 0 ≤ p ≤ 2) and is periodic in q (with period 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Furthermore, a quick examination of the structure of A (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Figure 1) reveals that Ai = 0 for i = 3, 5, 7 and also that Ai is free for i = 0, 4, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Consequently, E2 p,q = 0 for q = 3, 5, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Moreover, since E2 2,q = ker ∂2 is a subgroup of Ai, we must have E2 2,q free for q = 0, 4, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Indeed, E∞ 2,q = ker d2 2,q is a subgroup of E2 2,q, so E∞ 2,q must also be free for q = 0, 4, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' On the other hand, KOi(C∗ R(Λ, γ)) is finite in all degrees, so it must be that E∞ 2,q = 0 for q = 0, 4, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' THE STABLE EXOTIC CUNTZ ALGEBRAS ARE HIGHER-RANK GRAPH ALGEBRAS 7 From what we have said so far, the E∞ p,q page of the spectral sequence is as follows: .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' 7 0 0 0 6 ∗ ∗ 0 5 0 0 0 4 ∗ ∗ 0 3 0 0 0 2 ∗ ∗ ∗ 1 ∗ ∗ ∗ 0 ∗ ∗ 0 q/p 0 1 2 This E∞ page identifies a filtration of KO∗(C∗ R(Λ, γ)), in which the subquotients of KOj(C∗ R(Λ, γ)) appear along the diagonal p + q = j of the E∞ page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' By hypothesis we have KO0(C∗ R(Λ, γ)) = Z2(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' However, the only non-zero group along the diagonal p + q = 0 is E∞ 0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Thus E∞ 0,0 = Z2(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Similarly, since KO7(C∗ R(Λ, γ)) = Z2 and KO6(C∗ R(Λ, γ)) = Z2, we must have E∞ 1,6 = Z2 = E∞ 0,6: .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' 7 0 0 0 6 Z2 Z2 0 5 0 0 0 4 ∗ ∗ 0 3 0 0 0 2 ∗ ∗ ∗ 1 ∗ ∗ ∗ 0 Z2(n−1) ∗ 0 q/p 0 1 2 Now, we consider the natural transformation η: A → A of degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Because A is a direct sum of copies of K CR(R) and K CR(C), which data already includes the map η, the natural transformation η : KO∗(C∗ R(Λ, γ)) → KO∗+1(C∗ R(Λ, γ)) exists at the level of the chain complex (6), passes to a map on the E2-page, then to a map on the E∞-page, and finally converges to the map η: KOi(C∗ R(Λ, γ)) → KOi+1(C∗ R(Λ, γ)) described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' This means that the map η on KO∗(C∗ R(Λ, γ)) respects the filtration of KOi(C∗ R(Λ, γ)) associated with the spectral sequence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' and the resulting maps on the subquotients are the same as that on the E∞ page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' In particular, the diagonals of the E∞-page that yield KO6(C∗ R(Λ, γ)) and KO7(C∗ R(Λ, γ)) give the commutative diagram below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' The vertical η maps on the left and right come from A as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' 0 � E∞ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='6 � η � KO6(C∗ R(Λ, γ)) � η � E∞ 1,5 � η � 0 0 � E∞ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='7 � KO7(C∗ R(Λ, γ)) � E∞ 1,6 � 0 ⇐⇒ 0 � Z2 � η � Z2 � η � 0 � η � 0 0 � 0 � Z2 � Z2 � 0 8 JEFFREY L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' BOERSEMA AND SARAH L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' BROWNE AND ELIZABETH GILLASPY Since the nonzero horizontal maps must be isomorphisms, the commutative diagram forces the vertical map η6: KO6(C∗ R(Λ, γ)) → KO7(C∗ R(Λ, γ)) in the center of the diagram to be zero, which contradicts the known value of η in KO∗(E R n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Existence of a Rank-3 Graph with Involution In this section, we will construct a 3-graph Λ with involution γ, by taking products of 1-graphs with involution, such that C∗ R(Λ, γ) is stably isomorphic to E R n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' In what follows, we use the following convention for suspension of graded modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' If H = {Hi} is a Z-graded group, then ΣH is a Z-graded group with (ΣH)i = Hi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Similarly, Σ−1H is a Z-graded group with (Σ−1H)i = Hi−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' This convention is consistent with K-theory and suspensions of C∗-algebras: K∗(SnA) = ΣnK∗(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' For the proof of the following proposition, we will need a few more preliminaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' For a graph Λ, a subset X ⊆ Λ0 is hereditary if whenever v ∈ X and vΛw ̸= ∅, then w ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' A cycle in Λ is a path e1e2 · · ·en with r(e1) = s(en) but, for all 1 ≤ i < n, we have s(ei) ̸= r(ei+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' We say that a cycle has an entrance if there exists 1 ≤ j ≤ n and an edge f ̸= ej with r(f) = r(ej).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' If the only hereditary subsets of Λ0 are ∅ and Λ0, and every cycle in Λ has an entrance, then [Szy01, Theorem 12] C∗(Λ) is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Furthermore, if every cycle in Λ has an entrance, then Λ is aperiodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' There exists a 1-graph Λ and involution γ such that K CR(C∗ R(Λ, γ)) ∼= ΣK CR(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Furthermore, C∗ R(Λ, γ) is simple and purely infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Let Λ be the 1-graph below (which extends infinitely in both directions) and let γ be the non-trivial involution, which fixes the vertices and edges of the infinite branch on the left and swaps the vertices and edges of the two infinite branches on the right in the obvious way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' (In fact γ is the only non-trivial involution on Λ.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='⑦ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='⑦ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='We begin by showing that C∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='R(Λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' γ) is simple and purely infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' It is straightforward to check that Λ has no nontrivial hereditary subsets, and that every cycle has an entrance, so simplicity of the complex algebra C∗(Λ) follows from [Szy01, Theorem 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Note further that for every vertex v ∈ Λ0, there is a vertex w with vΛw ̸= ∅ for some vertex w which supports a loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' As Λ is aperiodic, one easily checks that the conditions of [KP00, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='9] are satisfied, and so C∗(Λ) is purely infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Consequently, [BRS11, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='9] implies that the real C∗-algebra C∗ R(Λ, γ) is also simple and purely infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' We now show that KU0(C∗ R(Λ, γ)) = 0 and KU1(C∗ R(Λ, γ)) = Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' (This is the same as calculating K∗(C∗(Λ)) and does not involve the involution γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=') Let M be the adjacency matrix for Λ (so Mv,w is the number of edges from w to v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Then KU0(C∗ R(Λ, γ)) ∼= coker (I − Mt), THE STABLE EXOTIC CUNTZ ALGEBRAS ARE HIGHER-RANK GRAPH ALGEBRAS 9 which can be interpreted as saying that KU0(C∗ R(Λ, γ)) is generated by vertex projection classes [pv], which are subject only to relations of the form [pv] = � w∈Λ0 Mv,w[pw] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Let v be one of the vertices of Λ that has a loop, and let w ̸= v be the vertex for which there is an edge from w to v (in each case there is a unique such w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Then the formula above gives the relation [pv] = [pv] + [pw], which implies that [pw] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' If [pw] = 0 we will say that w is a zero vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Now if w is a zero vertex and there is only one edge to w, say from vertex u, then it follows that u is also a zero vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' More generally, if w is a zero vertex and all the edges to w are known to emanate from zero vertices except possibly one edge from vertex u, then it follows that u is also a zero vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Using these principles, it is now straightforward to work through the graph and to find that every vertex is a zero vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Hence KU0(C∗ R(Λ, γ)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' We know that KU1(C∗ R(Λ, γ)) ∼= ker(I − Mt), which is to say that (7) KU1(C∗ R(Λ, γ)) ∼= NΛ := � α: Λ0 → Z | α(v) = � w∈Λ0 Mw,v α(w) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Let v be one of the vertices of Λ that has a loop, and let w ̸= v be the vertex for which there is an edge from v to w (in each case there is a unique such w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Then we have the relation α(v) = α(v) + α(w), which implies that α(w) = 0 for any α ∈ NΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' If α(w) = 0 for all α ∈ NΛ we will say that w is a null vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Now if w is a null vertex and there is only one edge emanating from w, say to vertex u, then it follows that u is also a null vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' More generally, if w is a null vertex and all the edges from w are known to point to null vertices except possibly one edge to vertex u, then u is also a null vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Using these principles, it is now straightforward to work through the graph and to find that every vertex is a null vertex, except for the six vertices labelled u, v, w, x, y, z shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='⑥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='� �⑧ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='⑧ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='⑧ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='⑧ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='⑧ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='⑧ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='⑧ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='⑧ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='Using Equation (7) and the fact that the unlabeled vertices are null vertices,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' we see that any α ∈ NΛ must satisfy the equations 0 = α(u) + α(w) α(w) = α(v) + α(x) 0 = α(w) + α(x) α(x) = α(w) + α(y) 0 = α(x) + α(z) 10 JEFFREY L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' BOERSEMA AND SARAH L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' BROWNE AND ELIZABETH GILLASPY Solving this system over Z, we find that α(u) is a free variable and that α(v) = −2α(u), α(w) = −α(u), α(x) = α(u), α(y) = 2α(u), and α(z) = −α(u) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Thus NΛ ∼= Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Hence KU∗(C∗ R(Λ, γ)) = K∗(C∗(Λ)) = (0, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Turning to the real K-theory, we now prove that KO∗(C∗ R(Λ, γ))) = (Z2, Z2, 0, Z, 0, 0, 0, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' First, we show that the real and complex E2 = E∞ page of the Evans spectral sequence for C∗ R(Λ, γ) is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' E2 p,q (8) real part .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' 7 0 0 6 0 Z 5 0 0 4 0 0 3 0 0 2 0 Z 1 Z2 0 0 Z2 0 q/p 0 1 complex part .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' 7 0 0 6 0 Z 5 0 0 4 0 Z 3 0 0 2 0 Z 1 0 0 0 0 Z q/p 0 1 (9) We have already discussed the complex part of this spectral sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' For the real part we will only discuss the computations for the rows corresponding to j = −1, 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' As we will see, this is enough to determine KO∗(C∗ R(Λ, γ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' The other rows can be computed using similar methods and we include them in the table above for completeness, but we will neither need nor discuss them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' First, the spectral sequence for a 1-graph with involution always vanishes in row j = −1, since the chain complex vanishes in that degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' To compute row j = 1 of the spectral sequence, we refer to [BG22, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='14] and Table 1 above, which indicates that E2 0,1 and E2 1,1 are the cokernel and kernel of the map (∂1)1 = I − Mt 11 : Z Λ0 f 2 → Z Λ0 f 2 where Λ0 f is the set of fixed vertices of (Λ, γ) and M11 is the restriction of the incidence matrix to those vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' So it suffices to consider the graph consisting of the fixed points of Λ, shown here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' � �❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ � �❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ x � �❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ y � �❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ � � �⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ � �⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ � �⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ � �⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ z � Using this graph, and the same sort of analysis that we did in the complex case, we find that coker (I − Mt 11) = Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' More precisely, working modulo 2 we find that [pv] = 0 for all vertices in Λ0 f except those labeled x, y and z in the graph above and that [px] = [py] = [pz] ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' We also find easily that ker(I − Mt 11) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Now, for j = 0, we need to find the cokernel and kernel of the map (∂1)0 which we will do using Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' First recall the partition Λ0 = Λ0 f ⊔ Λ0 g ⊔ Λ0 h where Λ0 f is the set of fixed THE STABLE EXOTIC CUNTZ ALGEBRAS ARE HIGHER-RANK GRAPH ALGEBRAS 11 vertices (the branch on the left of Λ), Λ0 g is the set of vertices of the “upper right” branch of Λ, and Λ0 h is the set of vertices of the “lower right” branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' With this structure on Λ, the (infinite) matrix B = I − Mt can be written in block form as B = I − Mt = \uf8eb \uf8ed B11 B12 B12 B21 B22 B23 B21 B23 B33 \uf8f6 \uf8f8 where, for example, B12 keeps track of edges from vertices in Λ0 f to vertices in Λ0 g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Using Table 1, we see that (∂1)0 : ZΛ0 f ⊕ ZΛ0 g → ZΛ0 f ⊕ ZΛ0 g is given by (∂1)0 = � B11 2B12 B21 B22 + B23 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' We will use a new graph Λ′ to analyze this map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' The graph Λ′, shown below, is obtained from Λ by keeping the vertices from Λ0 f and Λ0 g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' For each edge in Λ from a vertex in Λ0 g to a vertex in Λ0 f, we create a corresponding edge in Λ′ and for each edge in Λ from a vertex in Λ0 f to a vertex in Λ0 g we create 2 corresponding edges in Λ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Also, for each edge from a vertex v = γ(u) ∈ Λ0 h to a vertex w ∈ Λ0 g we obtain an edge in Λ′ from u to w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' x � �❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ � ⑥⑥⑥⑥⑥⑥⑥⑥ z � �❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ � ⑥⑥⑥⑥⑥⑥⑥⑥ �❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ � �❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ � � � � y (2) � � w � � �⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ � �⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ � � � By construction the adjacency matrix M′ for the graph Λ′ satisfies I − (M′)t = � B11 2B12 B21 B22 + B23 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Therefore, we can use the graph Λ′ to find the cokernel and kernel of (∂1)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Using the same logic and terminology we used when calculating the complex K-theory, we see that w is a zero vertex because it emits an edge to a vertex v which supports a loop, and the edge from w to v is the only non-loop edge which points to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Indeed, every vertex along the bottom row of the graph Λ′ except y is a zero vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' The fact that these zero vertices (with the exception of w) only receive edges from one (potentially) nonzero vertex on the top row of Λ′ implies that every vertex in the top row except x and z are also zero vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Now, w is a zero vertex, but since there are two edges from y to w we obtain the relation [pw] = [pw] + 2[py] which implies that 2[py] = 0, but [py] ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Finally, from the relations [px] = −[py] and [pz] = [py] we conclude that coker (∂1)0 = Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' To compute ker(∂1)0, we also proceed as in the computations for the complex case: All of the vertices in the bottom row of Λ′, save w, are null vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Moreover, if a null vertex v emits n edges to a single potentially non-null vertex u, we must have nα(u) = 0 for any α ∈ NΛ′, and as α(u) ∈ Z we conclude that u must also be null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' It follows that w is null, as are all of the vertices in the top row of Λ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' That is, ker(∂1)0 = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Now, with the three rows that we’ve identified, the spectral sequence (8) implies that KO0(C∗ R(Λ, γ)) ∼= KO1(C∗ R(Λ, γ)) ∼= Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' We claim that using this we can compute KOi(C∗ R(Λ, γ)) for 2 ≤ i ≤ 7 using the long exact sequence (5) and other aspects of CR-structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' The fact that KU6(C∗ R(Λ, γ)) = KU0(C∗ R(Λ, γ)) = 0 implies that η0 is injective, and hence an 12 JEFFREY L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' BOERSEMA AND SARAH L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' BROWNE AND ELIZABETH GILLASPY isomorphism, and also that η−1 is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Since KU2(C∗ R(Λ, γ)) = 0 it follows that η1 is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Thus if KO2(C∗ R(Λ, γ)) has a non-zero element, then it would have to be in the image of η1 ◦ η0 ◦ η−1 = η3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' But η3 = 0 for all real C∗-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Thus KO2(C∗ R(Λ, γ)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Since KO2(C∗ R(Λ, γ)) = 0, the long exact sequence implies that c3: KO3(C∗ R(Λ, γ)) → KU3(C∗ R(Λ, γ)) = Z is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' This forces KO3(C∗ R(Λ, γ)) = Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Moreover, as im r1 = ker η1 = Z2 we must have r1 : Z → Z2 the unique nonzero map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Since im c3 ∼= ker r1, we conclude that c3 is multiplication by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' The relation rc = 2 then implies that r3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Continuing this process using the long exact sequence, we compute that KO∗(C∗ R(Λ, γ))) = (Z2, Z2, 0, Z, 0, 0, 0, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' The module maps η, r, c, ψ are then completely determined by these groups and the long exact sequence (5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' that is, K CR(C∗ R(Λ, γ)) and hence K CRT(C∗ R(Λ, γ)) coincide with ΣK CRT(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Suppose that (Λ1, γ1) and (Λ2, γ2) are higher-rank graphs with involutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Then (Λ1 × Λ2, γ1 × γ2) is a higher-rank graph with involution and C∗ R(Λ1 × Λ2, γ1 × γ2) ∼= C∗ R(Λ1, γ1) ⊗R C∗ R(Λ2, γ2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Assume that Λ1 and Λ2 have rank k1 and k2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' From [KP00, Proposi- tion 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='8] the product Λ1 × Λ2 is a graph of rank k1 + k2, with degree functor d(λ1, λ2) = d(λ1) + d(λ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Furthermore, there is an involution γ on Λ1 × Λ2 defined by γ(λ1, λ2) = (γ1(λ1), γ2(λ2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' From [KP00, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='5], there is an isomorphism φ: C∗(Λ1 × Λ2) → C∗(Λ1) ⊗ C∗(Λ2) defined by φ(s(λ1,λ2)) = sλ1 ⊗sλ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' To finish the proof, we need only show that φ preserves the real structures (4) of C∗(Λ1 × Λ2) and C∗(Λ1) ⊗ C∗(Λ2) which are induced by the graphical involutions γi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' This is straightforward: φ(�γ(s(λ1,λ2)) = φ(s∗ (γ1(λ1),γ2(λ2)))) = s∗ γ1(λ1) ⊗ s∗ γ2(λ2) = �γ1(sλ1) ⊗ �γ2(sλ2) = � γ1 ⊗ γ2(sλ1 ⊗ sλ2) = � γ1 ⊗ γ2(φ(s(λ1,λ2))) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' □ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Let n be an odd integer, n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' There exists a rank-3 graph with invo- lution (Λn, γn) such that C∗ R(Λn, γn) ∼= K R ⊗R E R n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Furthermore, there exists a projection p ∈ C∗ R(Λn, γn) such that pC∗ R(Λn, γn)p ∼= E R n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Let (Λ, γ) be the 1-graph given by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='1 and let (En, γn) be the finite 1- graph with involution from Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='2 of [Boe17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Then both C∗ R(Λ, γ) and C∗ R(En, γn) are simple and purely infinite and we have K CR(C∗ R(Λ, γ)) ∼= ΣK CR(R) and K CR(C∗ R(En, γn)) ∼= Σ6K CR(En) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' This implies by [BRS11, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='1] that K CRT(C∗ R(Λ, γ)) ∼= ΣK CRT(R) and K CRT(C∗ R(En, γn)) ∼= Σ6K CRT(En) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Let (Λn, γn) be the product rank-3 graph with involution (Λn, γn) = (Λ, γ) × (Λ, γ) × (En, γn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' THE STABLE EXOTIC CUNTZ ALGEBRAS ARE HIGHER-RANK GRAPH ALGEBRAS 13 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='2 then implies that C∗ R(Λn, γn) ∼= C∗ R(Λ, γ) ⊗R C∗ R(Λ, γ) ⊗R C∗ R(En, γn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Now, K CRT(C∗ R(Λ, γ)) is a free CRT -module, since it is isomorphic to a suspension of K CRT(R) (see [Boe02, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Therefore the K¨unneth formula for the K-theory of real C∗- algebras (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='5 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='2 of [Boe02]) gives K CRT(C∗ R(Λn, γn)) ∼= K CRT(C∗ R(Λ, γ)) ⊗CRT K CRT(C∗ R(Λ, γ)) ⊗CRT K CRT(C∗ R(En, γn)) ∼= Σ2K CRT(R) ⊗CRT Σ6K CRT(En) ∼= K CRT(En) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Note that C∗ R(Λn, γn) is a stable, simple, purely infinite, real C∗-algebra, thanks to Propo- sition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='1 and [Boe17, Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' We also know that KR ⊗R En is a a stable, simple, purely infinite, real C∗-algebra, because its complexification K ⊗ On is simple and purely infinite (see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='9 of [BRS11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Thus the first statement of the theorem follows by the classification of real Kirchberg algebras, [BRS11, Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='2, Part (1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' To prove the second statement, by [BRS11, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='13] there is a projection p ∈ C∗ R(Λn, γn) such that [p] is a generator of KO0(C∗ R(Λn, γn)) = Z2(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Then K CR(pC∗ R(Λn, γn)p) ∼= K CR(C∗ R(Λn, γn)) ∼= K CR(En) (where the first isomorphism is by [Boe06, Proposition 9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Furthermore the class of the identity [p] ∈ KO0(pC∗ R(Λ, γ)p) ∼= Z2(n−1) corresponds under this isomorphism to the class of the identity [1] ∈ KO0(En) ∼= Z2(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Therefore by [BRS11, Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='2, Part (2)], we have pC∗ R(Λ, γ)p ∼= En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Appendix – real Kirchberg suspension algebras In the previous section, we introduced a graph with involution for which K CR(C∗(Λ, γ)) ∼= ΣK CR(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' We consider this algebra as a sort of real Kirchberg suspension, since it is a real purely infinite simple stable nuclear C∗-algebra satisfying the UCT, and with the same KK- type as the suspension algebra SR ∼= C0((0, 1), R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' By repeatedly taking the product of this graph with itself, which corresponds to repeatedly tensoring this algebra with itself, we can obtain a higher-rank graph, the real C∗-algebra of which is a real Kirchberg algebra with the same KK-type as SiR for any i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' These tensor products will be higher-rank graph algebras of rank i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' It is natural to ask which of these suspensions can be obtained from a 1-graph with involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' In this section, we will answer this question completely, providing a full characterization of the integers i (mod 8) for which there exists a 1-graph with involution (Λ, γ) such that K CR(C∗(Λ, γ)) ∼= ΣiK CR(R) ∼= K CR(SiR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' For the positive results, we will exhibit directly the appropriate graph or graph with involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' For each i = {−2, −1, 0, 1} there exists a 1-graph with involution (Λ, γ) such that K CR(C∗(Λ, γ)) ∼= ΣiK CR(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Furthermore, C∗ R(Λ, γ) is simple and purely infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Sketch of proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' For each i we show below a graph or graph with involution that satisfies K CR(C∗(Λ, γ)) ∼= ΣiK CR(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' The K-theory calculations, not shown, are carried out using the same techniques as in the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' 14 JEFFREY L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' BOERSEMA AND SARAH L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' BROWNE AND ELIZABETH GILLASPY i = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' The graph Λ is shown below;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' we equip it with the non-trivial involution γ which interchanges the right-hand branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' � ❅ ❅ ❅ ❅ ❅ ❅ ❅ �⑦⑦⑦⑦⑦⑦⑦ � � ❅ ❅ ❅ ❅ ❅ ❅ ❅ �⑦⑦⑦⑦⑦⑦⑦ � � ❅ ❅ ❅ ❅ ❅ ❅ ❅ �⑦⑦⑦⑦⑦⑦⑦ � � �⑦⑦⑦⑦⑦⑦⑦ � ❅ ❅ ❅ ❅ ❅ ❅ ❅ � �⑦⑦⑦⑦⑦⑦⑦ � ❅ ❅ ❅ ❅ ❅ ❅ ❅ � �⑦⑦⑦⑦⑦⑦⑦ � ❅ ❅ ❅ ❅ ❅ ❅ ❅ � � � • � • � • � � � • � � � • � • � �⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ � �❅ ❅ ❅ ❅ ❅ ❅ ❅ � • � • � • � � � ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ �❅❅❅❅❅❅❅ � � ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ �❅❅❅❅❅❅❅ � � ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ �❅❅❅❅❅❅❅ � i = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' The graph Λ is shown below, with trivial involution γ = id .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' �❅ ❅ ❅ ❅ ❅ ❅ ❅ � �❅ ❅ ❅ ❅ ❅ ❅ ❅ � �❅ ❅ ❅ ❅ ❅ ❅ ❅ � �❅ ❅ ❅ ❅ ❅ ❅ ❅ � �⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ � �⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ � �⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ � �⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ � � � i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' The graph Λ is shown below, with trivial involution γ = id .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' �⑦⑦⑦⑦⑦⑦⑦ � � ❅ ❅ ❅ ❅ ❅ ❅ ❅ �⑦⑦⑦⑦⑦⑦⑦ � � ❅ ❅ ❅ ❅ ❅ ❅ ❅ �⑦⑦⑦⑦⑦⑦⑦ � � ❅ ❅ ❅ ❅ ❅ ❅ ❅ �⑦⑦⑦⑦⑦⑦⑦ � � ❅ ❅ ❅ ❅ ❅ ❅ ❅ � • � • � • � • � � i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' The graph Λ is shown below with non-trivial involution γ, as in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' �❅ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='❅ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='❅ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='❅ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='⑦ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='⑦ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='⑦ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='⑦ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='For the i = −2 graph,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' one can determine all of the groups KOi(C∗ R(Λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' γ)) from the associated spectral sequence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' except for KO2(C∗ R(Λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' γ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' In that case, the spectral sequence KO2(C∗ R(Λ, γ)) has the filtration 0 → Z → KO2(C∗ R(Λ, γ)) → Z2 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Although this filtration by itself does not determine KO2(C∗ R(Λ, γ)), the long exact sequence (5) forces KO2(C∗ R(Λ, γ)) = Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Moreover, the module maps r, c, η, ψ are uniquely determined by (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' □ THE STABLE EXOTIC CUNTZ ALGEBRAS ARE HIGHER-RANK GRAPH ALGEBRAS 15 Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' For 2 ≤ i ≤ 5, there does not exist a 1-graph (Λ, γ) with involution such that K CR(C∗ R(Λ, γ)) ∼= ΣiK CR(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Suppose that (Λ, γ) is a graph with involution and K CR(C∗ R(Λ, γ)) ∼= ΣiK CR(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' The real Pimsner-Voiculescu sequence (or equivalently, the real Evans spectral sequence) for K CR(C∗(Λ, γ)) implies that KO−1(C∗ R(Λ, γ)) and KO−3(C∗ R(Λ, γ)) are free abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' But recall that KO1(R) ∼= KO2(R) ∼= Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Thus the group (Σ2KO(R))−1 = KO1(R) has torsion, implying that K CR(C∗ R(Λ, γ)) ≇ Σ2KO CR(R), hence i ̸= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Similarly, the groups (Σ3KO(R))−1, (Σ4KO(R))−3, and (Σ5KO(R))−3 have torsion, showing that i ̸= 3, 4, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' □ References [BG22] Jeffrey L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' Boersema and Elizabeth Gillaspy, K-theory for real k-graph C∗-algebras, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' K- Theory 7 (2022), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE0T4oBgHgl3EQfTQBT/content/2301.02233v1.pdf'} +page_content=' 2, 395–440.' metadata={'source': 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+Gregorio Bernabé +gbernabe@um.es +University of Murcia +Murcia, Spain +José Manuel García +jmgarcia@um.es +University of Murcia +Murcia, Spain +Michael F.P. O’Boyle +mob@inf.ed.ac.uk +University of Edinburgh +Edinburgh, United Kingdom +Abstract +Dedicated tensor accelerators demonstrate the importance +of linear algebra in modern applications. Such accelerators +have the potential for impressive performance gains, but +require programmers to rewrite code using vendor APIs — a +barrier to wider scale adoption. Recent work overcomes this +by matching and replacing patterns within code, but such +approaches are fragile and fail to cope with the diversity of +real-world codes. +We develop ATC, a compiler that uses program synthesis +to map regions of code to specific APIs. The mapping space +that ATC explores is combinatorially large, requiring the +development of program classification, dynamic analysis, +variable constraint generation and lexical distance matching +techniques to make it tractable. +We apply ATC to real-world tensor and linear algebra +codes and evaluate them against four state-of-the-art ap- +proaches. We accelerate between 2.6x and 7x more programs, +leading to over an order of magnitude performance improve- +ment. +ACM Reference Format: +Pablo Antonio Martínez, Jackson Woodruff, Jordi Armengol-Estapé, +Gregorio Bernabé, José Manuel García, and Michael F.P. O’Boyle. +2023. Matching linear algebra and tensor code to specialized hard- +ware accelerators. In Proceedings of the 32nd ACM SIGPLAN Interna- +tional Conference on Compiler Construction (CC ’23), February 25–26, +2023, Montréal, QC, Canada. ACM, New York, NY, USA, 13 pages. +https://doi.org/10.1145/3578360.3580262 +1 +Introduction +Linear algebra is a fundamental building block of many of +today’s critical applications; from weather modeling [13] to +CC ’23, February 25–26, 2023, Montréal, QC, Canada +© 2023 Copyright held by the owner/author(s). +This is the author’s version of the work. It is posted here for your personal +use. Not for redistribution. The definitive Version of Record was published in +Proceedings of the 32nd ACM SIGPLAN International Conference on Compiler +Construction (CC ’23), February 25–26, 2023, Montréal, QC, Canada, https: +//doi.org/10.1145/3578360.3580262. +ubiquitous DNN [22] workloads. Its importance is reflected +in the large number of accelerator libraries and hardware +devices devoted to fast linear algebra. These range from +specialized devices such as Google’s TPU [34] to the tensor +cores on NVIDIA [12] among many others [5, 8, 25, 31, 33]. +While such devices promise significant performance for an +important class of applications [19], their uptake is limited by +their programmability [24]. Typically, these accelerators and +libraries are accessed via calls to specialized APIs, meaning +existing code has to be rewritten. Given the volume [35] +and variety [39] of existing legacy code, such rewriting is a +significant undertaking [19]. +The combined importance of linear algebra acceleration +and the difficulty of rewriting legacy code to accelerators has +led to recent work which attempts to automate the process. +These techniques search user code for matrix multiplica- +tions using constraints [20, 28] or polyhedral analyses [9] +and replace regions of code with appropriate API calls or +instructions. +However, as we show in Section 8.1, these approaches are +fragile. Constraints capture only a limited set of program +patterns and small variations in the user code defeat them. +While they work well on curated benchmarks, they perform +poorly on real-world code [20, 63], defeated by function calls, +optimized code and inline assembler. +Neural classification (e.g. [18]) can effectively detect code +despite these challenges. However, it does not provide a path +to acceleration, but requires further steps. These include gen- +erating variable mappings and checking for equivalence [63] +which has shown promising results for Fourier Transforms. +However, one of the key challenges in matching code to +APIs is the cost of searching for user program variables that +map to API formal parameters. As the width of the API and +complexity of the user program increase, this becomes com- +binatorially expensive. As we show in Section 8.3 existing +approaches [63] fail to scale to the challenges that linear +algebra APIs present. +1 +arXiv:2301.11659v1 [cs.PL] 27 Jan 2023 + +CC ’23, February 25–26, 2023, Montréal, QC, Canada +P.A. Martínez, J. Woodruff, J. Armengol-Estapé, G. Bernabé, J.M. García, M.F.P. O’Boyle +We present ATC, a compiler that applies program synthe- +sis to compile general user-code to linear algebra accelerators. +We identify and solve key challenges +enabling the detect/synthesize paradigm to scale to the +more complex APIs of linear algebra acceleration. In addi- +tion, ATC employs a trained platform predictor to determine +whether acceleration is profitable or not. +We applied our approach to 50 GitHub GEMM and 15 +convolution projects and discovered between 2.6 and 7x more +linear operators compared to KernelFaRer [20], IDL [28], +Polly [30] or FACC[63]. This resulted in more than an order +of magnitude performance improvement. +This paper makes the following contributions: +• We present ATC, which maps matrix multiplication +and convolution programs to hardware accelerators, +up to 7x more frequently than existing techniques. +• We introduce novel heuristics to reduce the mapping +search space by four orders of magnitude. +• We develop novel dynamic analyses to determine higher- +level information about variables, enabling synthesis +without costly whole-program analyses. +2 +Motivation +Figure 1. Example application of API replacement. The +above program is taken from the parboil benchmark [56], a +widely-used benchmark suite, which is transformed into a +call to an optimized matrix-multiplication accelerator API. +Figure 2. GEMM code optimized for AVX2 found on GitHub +consisting of 120 lines of hand-optimized intrinsics and how +ATC matches the code to the accelerator API +2.1 +Exisiting Match and replace +IDL and KernelFaRer. Both aim to detect linear algebra +operations in user programs and replace them with an appro- +priate accelerator library call. To illustrate this consider the +code in Figure 1. This shows a straight-forward matrix mul- +tiplication program fragment, from the parboil benchmark +suite [56]. They aim to detect this matrix-multiplication and +replace it with a call to the library, shown at the bottom of +the diagram. +To replace code with an API call they have to both de- +tect the code performing a matrix multiplication and also +determine which user program variables correspond to the +arguments of the API call. Both approaches are able to detect +that this is a matrix multiplication, and can determine the +mapping between user variables and API parameters. +2.2 +Examples of complex GEMM programs +Unfortunately, in practice, user code can be complex such +that code structure or pattern-based approaches inevitably +fail. +As an example, consider the code found on GitHub shown +in Figure 2 which implements a matrix-multiplication al- +gorithm (only a fragment of the 120 lines of user code are +shown here). The code structure is complex and difficult to +understand as it makes extensive use of inline assembler in- +trinsics which defeats the code structure analysis approaches +of IDL and KernelFaRer, preventing acceleration. +2 + +Matching linear algebra and tensor code to specialized hardware accelerators +CC ’23, February 25–26, 2023, Montréal, QC, Canada +OJCLONE +DATASET ++ GEMM +PROGRAMS ++ CONV +PROGRAMS +NEURAL EMBEDDINGS +FUNCTION +CANDIDATES +F1 +F2 +F3 +... FN +IO +EQUIVALENCE +IO +DETECTION +ACCELERATABLE +FUNCTION +MATCHES +GENERATION +MISMATCH +SOLVER +PROGRAM +SYNTHESIS +SET OF MATCHES +MATCHES HEURISTCS +MATCH ALGORITHM +LEVENSHTEIN +VALID MATCHES +PERFORMANCE +ANALYSIS +ACCELERATOR +SAMPLING +FUNCTION +SAMPLING +SVM CLASSIFIER +PROGRAM CLASSIFICATION +ACCELERATOR API +USER CODE +ACCELERATED CODE +Acceleratable Candidate Detection +IO Detection +Matches Generation +Matches +Reduction +Profitability Detection +Figure 3. ATC compiler architecture +2.3 +Our approach - ATC +Rather than relying on code structure to guide detection, +ATC uses behavioral equivalence to determine if a section of +code is a linear algebra operation. Firstly, ATC uses neural +program classification [18] to detect that the code in Figure +2 is probably a GEMM. It then searches variable matches to +determine the potential source and output arrays. As the +search space is combinatorially large, we introduce scal- +able, algorithm-independent heuristics (which we discuss in +Section 5) that keep the number of mappings manageable. +Next, ATC generates different input values for the arrays +and records the output. After generating many randomized +inputs, it observes that it has the equivalent behavior to the +corresponding API and is able to replace the AVX2 code with +the GEMM call at the bottom of Figure 2. +Legality. Now, IO behavioral equivalence is not proof that +a section of code is a particular linear algebra operation - +similarly IDL and KernelFaRer do not prove equivalence. For +proof, bounded model checking based on Kleene [14] can be +deployed. In practice, as demonstrated in our experimental +section, IO equivalence gives no false positives. For further +guarantees, we can ask for programmer sign-off or employ +model checking. +Profitable. Once we have detected and can replace a sec- +tion of code with an accelerator call, we need to determine if +it is profitable to do. Due to hardware evolution, we do not +use a hard-wired heuristic to determine profitability. Instead, +we learn, off-line, a simple predictive model to determine if +the target accelerator is faster than a CPU implementation. +The model is called at runtime, determining if offloading is +worthwhile. +FACC. Behavioral equivalence is also employed in FACC +[63]. Unfortunately, it is restricted to FFTs and one-dimensional +arrays, and cannot detect the replacement in Figure 1. There- +fore, we extended FACC to FACC* to consider GEMMs and +multi-dimensional arrays. This, however, exposes its weak +variable binding model which is combinatorial in the number +of user array variables and their dimensionality. Furthermore, +it relies on program synthesis to determine the length of ar- +rays, which scales poorly to problems with many potential +length parameters for arrays such as GEMM. +FACC also relies on brittle inter-procedural liveness analy- +ses to determine the liveness status of variables. This restricts +it to running only at link time, rendering it invalid for use +in shared libraries. We will see in Section 8 that the com- +bination of these issues results in excessively large search +spaces. +3 +System overview +Figure 3 gives a system flow overview of ATC. We first de- +tect regions of code that are likely to be linear algebraic +operations using a neural program classifier. The classifier is +trained ahead of time, based on programs that are equivalent +to the accelerator and prior examples of linear algebra code. +Once candidate code sections have been identified, we ap- +ply program analysis to match user program variables with +the particular API formal parameters. Given the combina- +torially large search space, we develop novel techniques to +make the problem tractable. +For each candidate matching, we generate multiple data +inputs, execute the user code section and record the output +values. If the input/output pairs correspond to the input/out- +put behavior of the accelerator API, we can say they are +behavioral equivalent and candidates for replacement. +3 + +CC ’23, February 25–26, 2023, Montréal, QC, Canada +P.A. Martínez, J. Woodruff, J. Armengol-Estapé, G. Bernabé, J.M. García, M.F.P. O’Boyle +While candidate user code may be replaceable with a call +to an accelerator API, it may not be profitable. Therefore, we +employ a simple ML classifier, trained offline, and invoked +at runtime to see if acceleration is appropriate for the user +code for the runtime known array sizes. +3.1 +Neural Program Classification +To detect potentially acceleratable parts of a program, we use +prior work in neural program classification [18]. A network is +trained with multiple instances of different program classes. +We use the OJClone dataset [43], which includes 105 classes +of different programs, and add examples of the programs that +we want to detect e.g. GEMMs and convolutions, gathered +from benchmark suite repositories other than GitHub. +At compile time a new candidate program is divided into +functions, which are presented to the neural classifier. The +classifier assigns each function in the program a probability +of belonging to a certain class. We consider the most proba- +ble class, which in the case of a GEMM or convolution is then +considered for variable matching and eventual code replace- +ment as described in the following sections. Classification +is fast (≤ 1.5 sec) and has negligible impact on compilation +time (see Section 8.3). +4 +Variable Matching +To check if a section of user code is behaviorally equivalent +to the API, we have to match up the user program variables +with API formal parameters. We first detect what variables +are livein/liveout (Section 4.1) and then the dimensions of +arrays (Section 4.2). +4.1 +Detecting livein and liveout variables +Detecting livein and liveout variables via standard static +analysis is +straightforward for well-structured programs but fails for +more diverse real-world codes, which may use assembly code +or intrinsic functions. +ATC uses dynamic analysis to determine which variables +are livein and liveouts inside a function. In C, variables are +passed by value so non-pointers variables are always livein. +In the case of pointers (or arrays), we generate random inputs +with arbitrary sizes. If the values in memory change after +executing the program, the array is considered liveout. +This allows us to detect which variables are livein or live- +out, but not both livein and liveout at the same time. We gen- +erate a new random input for liveout variables and re-execute +the function. If the output differs from the first execution, it +is both livein and liveout. We implement this algorithm as a +just-in-time compiler pass in LLVM [37]. +4.2 +Detecting the dimensions of arrays +Detecting arrays length enables offloading of appropriately- +sized regions of codes, so it is a critical step in ATC. For +Load/StoreInst +Array A? +Out of +bound? +Replace Load/Store +to/from index 0 +Perform the +instruction +Exit with +error code +YES +NO +YES +NO +Figure 4. Dimension detection algorithm overview for a +target example array called A. +Algorithm 1 Dimensions detection algorithm +1: for arr in function do +2: +fakeLoadAndStoresExcept(𝑎𝑟𝑟) +3: +replaceLoadAndStores(𝑎𝑟𝑟) +4: +repeat +5: +𝑐 = getNextCombination(𝑎𝑟𝑟) +6: +ffi_call(𝐴,𝑉 ) +7: +if not failed then +8: +𝑓 𝑜𝑢𝑛𝑑 = 𝑇𝑟𝑢𝑒 +9: +end if +10: +until not found +11: +Add 𝑐 to 𝐶 +12: end for +13: return 𝐶 +some programs, lengths can be found using static analysis +(e.g. [49]), but this fails in more complex cases. We use run- +time analysis to determine which program variables define +array size using a modified form of runtime array bound +checking. For each set of variables that could define an ar- +ray’s size (typically, from the argument list), we set such +variables to a fixed value. We then execute the user code that +is modified to check runtime array accesses. +First, the compiler selects a target array to find its size. +Then, to generate the modified program, we tweak the load +and store instructions in the user program, replacing them +with custom function calls in the IR. If a load or store does +not access the array we are interested in, we modify it to +load and store at a constant, safe location. If it does, the +instruction is replaced with a function call that will check at +runtime if the access is out of bounds. If so, the program exits +with a custom error code. If not, we have found a valid array +size. The basic idea is depicted in Figure 4. This is used by +our JIT analysis as shown in Algorithm 1 and implemented +in LLVM. +This way, the compiler can assign different input sizes to a +given array and check the exit code. Therefore, the compiler +iterates over all the possible dimensions combinations until +one of the executions does not end with the custom error exit +code. That means that the program was completed without +any illegal access to the target array, which indicates that it +is the right dimension of the array. +4 + +Matching linear algebra and tensor code to specialized hardware accelerators +CC ’23, February 25–26, 2023, Montréal, QC, Canada +Algorithm 2 Automatic matching algorithm +1: function dimsMatch(𝑓 1𝑎, 𝑓 2𝑎, 𝑝,𝑛) +2: +𝑆 = ∅ +3: +𝑖𝑑𝑥 ← 0 +4: +for 𝑎𝑟𝑔𝑠1 in f1a do +5: +𝑎𝑟𝑔𝑠2 = f2a[p[idx]] +6: +Add {𝑎𝑟𝑔𝑠1,𝑎𝑟𝑔𝑠2} to 𝑆 +7: +𝑖𝑑𝑥 ← 𝑖𝑑𝑥 + 1 +8: +end for +9: +return Size(S) = 𝑛 +10: end function +11: +12: function outMatch(𝑓 1𝑜, 𝑓 2𝑜, 𝑝) +13: +idx = IndexOf(𝑓 2𝑜, 1) +14: +return IndexOf(𝑝, idx) = IndexOf(𝑓 1𝑜, 1) +15: end function +16: +17: function findMatchings(𝑓 1𝑎, 𝑓 2𝑎, 𝑓 1𝑜, 𝑓 2𝑜,𝑛) +18: +𝐵 = ∅ +19: +for p in permutations(0...𝑛) do +20: +if dimsMatch(f1a, f2a, p) and +21: +outMatch(f1o, f2o, p) then +22: +Add 𝑝 to 𝐵 +23: +end if +24: +end for +25: +return 𝐵 +26: end function +5 +Reducing the matchings search space +To match code to APIs, the compiler generates different can- +didates for the variable to formal parameter mappings and +then tests them using IO equivalence. For small APIs, all map- +pings can be explored, but the combinatorial cost makes it +prohibitive for real-world accelerator APIs. We develop tech- +niques that reduce the mapping space by exploiting arrays +information and human coding styles. +5.1 +Exploiting array information +Using array dimensions (Section 4.2), we can reduce the num- +ber of possible matches that must be checked, as assigning +one array to another means that the dimensions of each array +must line up. +5.1.1 +Automatic matching algorithm. We first gener- +ate all 𝑛! permutations of the 𝑛 array variables to 𝑛 parame- +ters mapping. We discard all permutations where variable +livenesses do not match. Then for each candidate user array +and parameter array pair, we generate the constraints defin- +ing how their dimensions match. If we find contradictory +constraints for any permutation, we discard it. The algorithm +is shown in Algorithm 2. +5.1.2 +Automatic Matching Algorithm: Example. To il- +lustrate this, Figure 5 shows an example where we have two +X(x0*x1) +Y(x1*x2) +Z(x2*x0) +A(y0*y1) +B(y1*y2) +C(y2*y0) +[0,1,2] +A: +U: +x0 -> y0 +x1 -> y1 +x2 -> y2 +Liveout A:[0,0,1] U:[0,0,1] +[1,0,2] +x0 -> y1 +x1 -> y2 +x1 -> y0 +x2 -> y1 +x2 -> y2 +x0 -> y0 +[2,0,1] +x0 -> y2 +x1 -> y0 +x2 -> y1 +Figure 5. Example application of the matching algorithm. +The right match is found the algorithm automatically. Per- +mutations in red means they are invalid, while the green +permutation means valid. +functions with three 2D arrays each. First, the algorithm +generates all the permutations between 0 and 𝑛 − 1 (𝑛 = 3 in +this example). Then, for each permutation, it tries matching +each variable in every array in the user code with the cor- +responding variable in the array of the API (here we show +only three of the six possible permutations). +In the first case (with the permutation [0, 1, 2]), the algo- +rithm tries matching the array variables of the user program +𝑋,𝑌,𝑍 with API parameters 𝐴, 𝐵,𝐶 . We then examine each +of the variables defining each of the corresponding arrays. +Comparing 𝑋 and 𝐴 gives a match of 𝑥0 → 𝑦0 and 𝑥1 → 𝑦1. +For the second array variable 𝑌 and API parameter 𝐵, we +have 𝑥1 → 𝑦1 and 𝑥2 → 𝑦2 and for the third variable pair +𝑍,𝐶 we have 𝑥2 → 𝑦2 and 𝑥0 → 𝑦0. All of these are con- +sistent with 𝑛=3 constraint, which satisfies the condition +(dimsMatch in Algorithm 2). Liveout information is also sat- +isfied so this permutation is added as a potential mapping. +In the second permutation [1, 0, 2], where 𝑋,𝑌,𝑍 maps +to 𝐵,𝐴,𝐶, the constraints are inconsistent e.g. 𝑥1 → 𝑦2 and +𝑥1 → 𝑦0 leading to 6 ≥ 3, so it is not a valid match. In the +third and last example, constraints are equal to 𝑛, but the +liveout arrays do not match. Thus, the only valid match is +the one found in the first permutation. +5.2 +Using argument names +Programs are developed by humans, so we can assume that +the functions that humans write follow common patterns. +We exploit this by analyzing the argument names of the API +and the user program to find lexical similarities. +To compare argument names, we use the Levenshtein dis- +tance [38] to compute the distance between each of the user +programs and API arguments. Figure 6 shows the definition +of the Levenshtein distance, which calculation is based on +the minimal number of modifications needed to transform +one word into another, representing how close are those +5 + +CC ’23, February 25–26, 2023, Montréal, QC, Canada +P.A. Martínez, J. Woodruff, J. Armengol-Estapé, G. Bernabé, J.M. García, M.F.P. O’Boyle +𝑙𝑒𝑣(𝑎,𝑏) = + + +|𝑎| +if |𝑏| = 0, +|𝑏| +if |𝑎| = 0, +𝑙𝑒𝑣(𝑡𝑎𝑖𝑙(𝑎),𝑡𝑎𝑖𝑙(𝑏)) +if 𝑎[0] = 𝑏[0], +1 + 𝑚𝑖𝑛 + + +𝑙𝑒𝑣(𝑡𝑎𝑖𝑙(𝑎),𝑏) +𝑙𝑒𝑣(𝑎,𝑡𝑎𝑖𝑙(𝑏)) +𝑙𝑒𝑣(𝑡𝑎𝑖𝑙(𝑎),𝑡𝑎𝑖𝑙(𝑏)) +otherwise +(1) +Figure 6. Levenshtein recursive definition +gemm_api(float* tc_A , float* tc_B , float* tc_C , +int tc_m , int tc_n , int tc_k , +int tc_lda , int tc_ldb , int tc_ldc , +float +tc_alpha , float +tc_beta) { +gemm(int M, int N, int K, float +alpha , +float *A, int lda , float *B, int ldb , +float beta , float *C, int ldc) { +tc A +... +tc lda +... +M +1 +... +3 +... +N +1 +... +3 +... +K +1 +... +3 +... +alpha +4 +... +3 +... +A +0 +... +2 +... +lda +2 +... +0 +... +........ +....... +... +....... +... +Figure 7. Levenshtein distance calculation for the arguments +of the tensor core API (above) and an example user program. +words. After computing the distance, the compiler selects +the combination that minimizes the Levenshtein distance. +Figure 7 shows an application example of the Levenshtein +distance to a real case of GEMM matching. For calculating +the distance, we strip the API suffix (tc_) and convert all +names to lowercase. Results show that the most probable +mapping for tc_A is A in the user code, and for tc_lda is +lda, which are the right matches. +5.3 +IO generation +Once we have a candidate match we generate random inputs +of different sizes and test for input-output (IO) equivalence. +We use 30 inputs of varying sizes. Although IO behavioral +equivalence is not proof, we can increase the number of tests +for increased confidence. No existing technique such as IDL +or KernelFaReR can prove that a matched piece of code is +provably equivalent to an API and therefore rely on user +sign-off. +5.3.1 +Behavioral Equivalence and the Limits of Veri- +fication. ATC, like prior work on floating-point accelera- +tors [63], uses behavioral equivalence. The downside of this +strategy is that it requires programmer sign-off to make any +substitution. However, due to the complexities of verifying +floating-point programs [63], verification of such liftings are +some way off. +In summary, the key challenges that all competing tech- +niques face are: +• Floating-point numbers often raise challenges in theo- +rem provers as they are challenging to reason about. +• Floating-point functions may have different accuracies +in different input ranges, meaning that the obvious +checks of correctness (even within bounds) are difficult +to apply. +The backend of ATC is not tied to using behavioral equiva- +lence. As we will see, the use of such behavioral equivalence +results in no false positives. Further development of theorem +prover technologies would mean that the weak behavioral +equivalence in ATC could easily be replaced with a theorem +prover guaranteeing correctness and enabling automatic +transformations. +6 +Automatic profitability detection +We assume that user code runs faster when replaced by a +platform-specific library. The question is whether it is best +to run on a CPU or accelerator version (XPU) of the library. +This in turn depends on the input size, which is only known +at runtime. We use a predictive model based on empirical +data to enable accurate predictions as platforms and libraries +evolve by retraining the model. +SVM. We use the well-known support vector machine +(SVM) classifier with a polynomial kernel of degree 3 with +gamma=1 and𝐶=100. We sample the CPU and the accelerator +with a common dataset of input sizes, which produces a +dataset that is small enough to be processed in less than +five minutes, but large enough to be highly accurate. Data +is labeled with 0 or 1 meaning that the CPU or the XPU is +faster. The model is then trained and deployed at runtime, +when matrix sizes are known, The training phase is done +only once, at “factory time”, and the resulting model when +deployed has negligible (≤ 0.3𝑚𝑠𝑒𝑐) runtime overhead (see +Section 8.2). +7 +Setup +We evaluate GEMM and convolution acceleration on special- +ized platforms. For GEMM, we used an Intel i7-11700 (CPU) +with an NVIDIA Quadro RTX 5000 (tensor cores) (XPU). For +convolution, we used the Google Cloud Platform (GCP) ser- +vices equipped with a TPUv3 with 8 TPU cores. Compilation +benchmarks in Section 8.3 are executed in an AMD EPYC +7413. +The Intel/NVIDIA platform runs CentOS 8.3 with kernel +4.18.0. LLVM was downloaded from the official Git repository, +using commit 329fda3. User codes were compiled using gcc +11.2.0 with -O3 -march=native flags. We used cuBLAS 11.2 +and MKL 2020.2.254 for compiling codes to the XPU and +CPU, respectively. For compiling convolution programs to +6 + +Matching linear algebra and tensor code to specialized hardware accelerators +CC ’23, February 25–26, 2023, Montréal, QC, Canada +Algorithm +Code +LoC +Nº Args +Optimizations +Constraints +C struct? +Direct +1 +35 +12 +None +None +No +2 +36 +10 +OpenMP +FW = FH = 3 +No +3 +34 +8 +OpenMP +FW = FH = 3 +No +4 +43 +11 +None +FW = FH = 3 +No +5 +39 +8 +OpenMP +FW = FH = 3 +No +6 +76 +16 +None +N = 1 +No +7 +209 +18 +Vectorized +N = 1 +Yes +8 +102 +12 +None +None +No +9 +42 +16 +None +None +No +im2col+ +gemm +10 +189 +15 +None +N = 1 +Yes +11 +286 +15 +BLAS +N = 1 +Yes +12 +179 +17 +BLAS +FW = FH +Yes +Winograd +13 +687 +17 +Intrinsics + OpenMP +FW = FH = 3 +No +14 +254 +12 +None +N = 1 +Yes +15 +782 +12 +Intrinsics + OpenMP +FW = FH = 3 +No +Table 1. List of convolution codes +the CPU, we used oneDNN v1.96. The TPU system runs +Debian 10 with kernel 4.19.0-14. +7.1 +User code +We explored GitHub looking for C and C++ GEMM codes, +analyzing more than 400 programs from which we selected +50 programs. We discarded the rest of them because of wrong +implementations, compilation errors or duplicated code. The +final list of programs is shown in Table 8. We categorize +the codes as follows: Naive: naive implementations with the +traditional 3-loop structure; Naive Parallel: as Naive but with +simple outer loop parallelization; Unrolled: naive implemen- +tation with unrolled loops; Kernel Calls: implementations +that divide the loops into different function calls; Blocked: +tiled implementations; Goto: implementations of the Goto +algorithm [29]; Strassen: implementations of the Strassen +algorithm [55]; Intrinsics: implementations using Intel intrin- +sics. +In addition, we selected 50 non-GEMM projects to check +whether any of the approaches gave false positives. +Convolutions. We explored GitHub looking for C and +C++ 4D convolution implementations. We analyzed around +50 programs from which we a selected list of 15 programs +based on the same methodology used for selecting GEMMs. +The list of convolution programs is shown in Table 1. We +have included codes from the most relevant convolution +implementations: Direct: the direct convolution algorithm; +im2col+gemm: an algorithm that casts the input as matrices +(im2col) and later uses a GEMM, as in Caffe [32]; Winograd: +the Winograd algorithm. +7.2 +Methods +We evaluate our approach against 4 well known schemes: +IDL: Idioms are described using an idiom description lan- +guage [28], which is translated into a set of constraints over +LLVM IR. +KernelFaRer: Uses different pattern matching to detect spe- +cific code constructs, matching specific matrix-multiplication +structures [20]. +Polly: Detects static control parts (SCoPs) in the code using +the polyhedral model [30]. It does not replace the code with +a call to an optimized library. +FACC*: FACC uses neural embeddings and behavioral syn- +thesis to detect candidates for acceleration [63]. It is limited +to 1D arrays so we developed an extended version, FACC*, +which supports multi-dimensional arrays. +8 +Results +8.1 +Detection +Figure 9 shows the percentage of GEMM programs matched +by each technique across each of 8 categories listed in Table 8. +IDL. The constraint based scheme [28] only matches 6 +out of 50 cases. These programs are largely naive implemen- +tations of GEMM, with a simple loop structure. It is able to +manage 2 programs containing unrolled loops but fails on +anything more complex. Matching more diverse cases would +require writing a new IDL constraint description for each +sub-class. +KernelFaRer. This code matching approach [20] is more +successful, matching 11 GEMMs due to a more robust pattern +matcher. For straightforward sequential implementations, it +is able to match all but one of the cases. However, any code +variation, including loop unrolling, defeats it. +Polly. Although it does not match and replace GEMMs, +it can detect SCoPs which may be candidates for replace- +ment with appropriate API calls. It is less successful than +KernelFaRer in detecting naive implementations but is more +robust across other more complex categories including one +parallel and unrolled versions and 2 blocked cases. It slightly +outperforms KernelFaRer, matching 13 vs. 11 out of 50 cases. +FACC*. Unlike the other approaches, FACC* performed +poorly on naive implementations, but better on others. Here, +the size of the mapping search space is the limiting factor. It +was able to find 10 cases in the available time (timeout ≤ 10 +mins). We examine the reasons for this in Section 8.3. +ATC. Our approach is significantly more robust across all +categories, matching 42 out of 50 cases. It is able to detect all +naive implementations and the majority within each other +category. It detects more naive parallel implementations, +unrolled and blocked programs than Polly and is the only +technique to detect GEMMs in codes containing kernel calls +and intrinsic instructions. +8.1.1 +Accuracy. Figure 10 provides a summary of ATC’s +success and failure by type. In 8 cases ATC failed to detect +that the program contained a GEMM. In one case, program +23, this is due to there being too many candidate matches, +280 which is above our timeout threshold of 100 candidates. +The remaining cases are due to overly aggressive search +7 + +CC ’23, February 25–26, 2023, Montréal, QC, Canada +P.A. Martínez, J. Woodruff, J. Armengol-Estapé, G. Bernabé, J.M. García, M.F.P. O’Boyle +Algorithm +Code +LoC +Layout +Sizes +Optimizations +Naive +1 +22 +Column-major +Squared +None +2 +127 +Both +Any +None +3 +18 +Row-major +Any +None +4 +41 +Column-major +Squared +None +5 +11 +Row-major +Any +None +6 +11 +Row-major +Any +None +7 +30 +Row-major +Any +None +8 +18 +Column-major +Any +None +9 +40 +Column-major +Any +None +10 +39 +Column-major +Any +None +11 +43 +Row-major +Any +None +12 +11 +Row-major +Squared +None +Naive +parallel +13 +39 +Row-major +Squared +OpenMP +14 +28 +Column-major +Squared +OpenMP +15 +164 +Row-major +Any +OpenMP +16 +22 +Row-major +Multiple of nthreads +C++ threads +17 +107 +Row-major +Squared +C++ threads +Unrolled +18 +57 +Row-major +Any +None +19 +50 +Row-major +Any +None +20 +63 +Row-major +Squared +OpenMP +21 +38 +Row-major +Squared, multiple of bs +None +Kernel Calls +22 +46 +Column-major +Any +None +23 +115 +Column-major +Any +OpenMP +24 +61 +Column-major +Any +None +25 +105 +Column-major +Any +Unrolled +Algorithm +Code +LoC +Layout +Sizes +Optimizations +Kernel Calls +26 +164 +Column-major +Any +Unrolled +Blocked +27 +104 +Row-major +Any +Block +28 +30 +Row-major +Squared +OpenMP +29 +52 +Column-major +Any +None +30 +35 +Row-major +Squared +None +31 +38 +Column-major +Squared +None +32 +42 +Row-major +Multiple of bs +Unrolled +33 +49 +Row-major +Squared +None +34 +18 +Row-major +Squared +None +35 +21 +Row-major +Squared +None +Goto +36 +247 +Column-major +Squared +Intrinsics (SSE) +37 +89 +Row-major +Squared +None +Strassen +38 +210 +Row-major +Squared +None +39 +315 +Row-major +Squared, power of 2 +None +40 +162 +Row-major +Squared +None +Intrinsics +41 +102 +Row-major +Squared +Intrinsics (AVX2) +42 +91 +Row-major +Multiple of 8 +Intrinsics (AVX2) +43 +82 +Row-major +Multiple of 8 +Intrinsics (AVX2) +44 +58 +Row-major +Any +Intrinsics (SSE) +45 +112 +Row-major +Multiple of bs +Intrinsics (AVX2) +46 +136 +Row-major +Multiple of bs +Intrinsics (AVX2) +47 +120 +Row-major +Any +Intrinsics (AVX2) +48 +143 +Row-major +Multiple of bs +Intrinsics (AVX2) +49 +57 +Row-major +Multiple of bs +Intrinsics (AVX2) +50 +60 +Row-major +Any +Intrinsics (SSE) +Figure 8. List of GEMM codes +Naive +Naive p. +Unrrolled +Kernels +Blocked +Goto +Strassen +Intrinsics +All +0 +20 +40 +60 +80 +100 +4 +9 +11 +1 +12 +0 +1 +0 +3 +4 +2 +1 +0 +1 +3 +0 0 0 0 +4 +0 +2 +0 +2 +6 +0 0 0 +1 1 +0 0 0 +2 +3 +0 0 0 0 +9 +6 +131110 +42 +% of matched codes +IDL +POLLY +KFR +FACC* +ATC +Figure 9. Percentage of matched GEMM codes by different techniques. +0 +20 +40 +60 +80 +% of programs +Matched +Too many candidates +Missed matches +Figure 10. Percentage of matched GEMM codes by ATC +divided by failure reason. +pruning, missing a legal match. Improved search heuristics +are likely to improve program coverage. +False positives. None of the methods classified any of the +50 non-GEMMs as a GEMM. Across all methods, there were +no false positives. +8.2 +Performance +The performance of each approach is shown in Figure 11. +Polly is not included here as although it can detect SCoPs, it +does not explicitly identify them as GEMMs for API replace- +ment. We show two bars for KernelFaRer, which correspond +to the strategy of GEMM code with an optimized CPU im- +plementation as described in [20] and KFR (XPU) which is +our extension, replacing the CPU library with the optimized +XPU implementation. IDL and FACC* directly target the ac- +celerator, while ATC chooses the CPU or accelerator based +on its SVM platform predictor. This runtime prediction cost +is negligible ≤ 0.3𝑚𝑠𝑒𝑐 and included in Figure 11. +What is immediately clear is that detecting more GEMMs +leads to better overall speedup. In the Naive category, KFR +and ATC are both able to achieve good performance, with +a speedup of 726x and 1031x, respectively. The gap is nar- +rowed when using KFR (XPU). However, KFR is unable to +detect GEMMs in any other category leading to just a 6.2x +8 + +Matching linear algebra and tensor code to specialized hardware accelerators +CC ’23, February 25–26, 2023, Montréal, QC, Canada +Naive +Naive p. Unrrolled Kernels +Blocked +Goto +Strassen Intrinsics +All +1 +10 +100 +1000 +10000 +Speedup +IDL +KFR (CPU) +KFR (XPU) +FACC* +ATC +Figure 11. Geometric mean speedup obtained by IDL, KernelFaRer, FACC* and ATC in GEMM programs with 𝑛 = 8192. +speedup overall while ATC achieves 344.0x. Unsurprisingly, +there is more performance available on naive sequential im- +plementations than in those cases where the programmer +has spent effort in optimizing the program. +8.3 +Candidate search complexity and compile time +One of the key challenges in matching code to APIs is search- +ing for program variables that map to API formal parameters. +As the width of the API and complexity of the user program +increase, this becomes combinatorially expensive. Figure 12 +evaluates FACC* naive matching of variables and our ap- +proach based on the Levenshtein distance. Naive matching +varies considerably from just 4 candidates to over 1 million. +Our approach greatly reduces the number of candidates for +the majority of the programs. There is one special case, code +23, where we reduce the number of candidates, but it is still +too high. +Figure 13 shows the compilation time of ATC. The initial +neural classifier has a negligible constant execution time of +1.3 seconds, while the other phases’ compilation time grows +with the number of candidates. +As the number of candidates begins to increase compi- +lation time becomes prohibitively expensive. Code 23 has +280 candidates which would take 35 mins more to evaluate. +We limit the number of candidates considered to 100 which +corresponds to a timeout of ≤ 10 minutes. +8.4 +Profitability accuracy +To measure the accuracy of the SVM platform predictor, we +built a model offline and tested it on unseen data values. +Table 2 summarizes the SVM accuracy with different input +sizes and shapes. The SVM achieves a global accuracy of +99.7%, where the misprediction occurs between 𝑚 = 2000 +and 𝑚 = 8000 which is the “edge” between the CPU and the +XPU. In all other intervals, the prediction is always correct. +The best accuracy is achieved with non-squared matrices, +while square matrices give slightly lower accuracy. Overall, +this is a highly accurate predictor with a negligible runtime +overhead of ‘ ≤ 0.3𝑚𝑠𝑒𝑐. +Parameter +Value +(mnk) +m +Global +Accuracy +2000 +4000 +6000 +8000 +10000 +111 +100% +100% +100% +70.0% +100% +93.8% +123 +100% +78.9% +100% +100% +100% +95.9% +312 +100% +84.3% +100% +100% +100% +96.9% +136 +100% +89.5% +100% +100% +100% +97.9% +Table 2. SVM accuracy for different sizes. 111 means m = 1 +× m, n = 1 × m, k = 1 × m. 123 means m = 1 × m, n = 2 × m, +k = 3 × m etc +8.5 +Convolutions +Our approach is generic and can be applied to other APIs +other than GEMMs. As an example, we consider tensor con- +volutions which are a significant component of DNN work- +loads. While IDL, KernelFaRer, Polly and FACC* were unable +to detect any of the convolutions, ATC detected 10 of the +15 convolutions as shown in Figure 14; we were unable to +match 5 due to the excessive number of candidates. +Figure 15 shows the performance achieved by replacing +with library code for each of the programs we are able to +accelerate. Across all codes, the SVM predicts that the TPU +accelerator outperforms the CPU, giving an average 17.8x +performance improvement across the programs. +9 +Related work +Matching in Programs. Matching high-level program +structure has been used to discover parallelism [23], het- +erogenous offloading [6, 44] and many other core compiler +tasks [27]. Constraint languages make these tasks easier [10, +27, 28] but their constraints are very sensitive to code struc- +ture [20]. +For matrix multiplications in particular, KernelFaRer [20] +provides a more robust approach, detecting characteristics +that define matrix multiplications. Polyhedral analyses can +also be used to target matrix multiplication accelerators [9, +58], but both these techniques fail to scale to the diversity +9 + +CC ’23, February 25–26, 2023, Montréal, QC, Canada +P.A. Martínez, J. Woodruff, J. Armengol-Estapé, G. Bernabé, J.M. García, M.F.P. O’Boyle +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +1 +101 +102 +103 +104 +105 +106 +Code +Candidates generated +4s +45s +10m +1h +12h +2d +20d +Approx. compilation time +FACC* +ATC +Threshold +Figure 12. Comparison of the number of candidates generated for matching GEMM codes: FACC* vs our approach. +1 +3 +8 +48 +0 +20 +40 +60 +80 +100 +Code 2 +Code 21 +Code 7 +Code 1 +Number of candidates +Time (s) +IO Testing +Tests +Generation +Candidates +Generation +Neural +Embeddings +280 +0 +500 +1,000 +1,500 +2,000 +Code 23 +Figure 13. Compilation time for different number of candi- +dates. +Direct +i2mcol+gemm +Winograd +All +0 +20 +40 +60 +80 +100 +6 +3 +3 +0 +1 +2 +10 +5 +% of matched codes +Matched +Not matched +Figure 14. Matched convolution codes by ATC. +1 +3 +4 +5 +6 +8 +10 +11 +13 +15 +All +1 +10 +100 +1000 +Code +Speedup +Speedup (CPU) +Speedup (TPU) +ATC +Figure 15. ATC speedup in convolution programs with ℎ = +𝑤 = 224, 𝑘𝑤 = 𝑘ℎ = 11, 𝑐 = 3, 𝑘 = 96 and 𝑛 = 100. +of real code. FACC [63] uses IO equivalence, which is ro- +bust to program structure, but only addresses the challenges +of FFTs and does not scale to longer function signatures +used for GEMM. To support any accelerator type, the com- +piler should support multi-dimensional arrays, while FACC +only supports 1D arrays. Because in 1D arrays and FFTs the +search space in matching the API parameters is small, FACC +does not include anything to reduce it. With more complex +programs and domains, this limitation makes compiling pro- +grams intractable. +Mask [51] uses symbolic execution to prove equivalence, +which does not work well for floating-point problems. Fuzzy +classification techniques based on code clone detection [40, +57], domain-classification [59], pattern matching [15], code +embeddings [2, 3, 21] and identifiers [36, 47] can be used +to help compile to accelerators [63]. These classification +strategies are able to classify diverse code structures, but do +not provide a compilation strategy for using an accelerator +on their own. +A large class of techniques focus on migrating between +APIs. These techniques often use program synthesis [16], +NLP [46] and code embeddings [45, 48]. These techniques +are unable to extract existing code into APIs. +Compiling for GEMM Accelerators. Existing compila- +tion strategies largely focus on lowering code from intrinsics +to accelerators using rewrite rules [52, 53, 62] and synthesis +techniques [17]. +Existing approaches to extracting matrix multiplications [20, +28] are brittle. Synthesis-based techniques [1, 7, 41] and +rewriting-based techniques [11, 54] have been developed +to extract these DSLs that can then be lowered: but they +largely require flexible DSLs, rather than APIs presented by +hardware accelerators. +Performance Prediction. Predicting code the performance +of hardware accelerators is challenging, as the break-even +point may depend on many different arguments within a +function’s interface [4]. LogCA [4] introduces static perfor- +mance comparison models for hardware accelerators and +similar models have been applied in offloading tasks [64]. Ma- +chine learning has often been applied in profitability settings, +10 + +Matching linear algebra and tensor code to specialized hardware accelerators +CC ’23, February 25–26, 2023, Montréal, QC, Canada +such as OpenCL Kernels [60, 61] and OpenMP [42]. Similar +techniques have been applied to FPGAs, by estimating pow- +er/performance [26] and tracking actual performance [50]. +10 +Conclusions +This work presented ATC, a flexible domain-agnostic com- +piler that matches legacy linear algebra code to accelerators. +By using IO behavioral equivalence and smart search space +reduction, we are able to match over 80% of challenging +real-world programs to accelerator APIs, significantly out- +performing all alternative approaches. +Supporting new domains different from GEMM and convo- +lution is easy because ATC focuses on behavior rather than +code structure, which makes it very flexible and extensible. +Furthermore, to support other accelerators in GEMM or con- +volution, only the accelerator API is needed: ATC adapts to +the new specification automatically. +Future work will examine how to further reduce the search +space using online learning and to expand the complexity +of user code considered. Longer-term, we wish to automati- +cally target a range of accelerators with diverse functionality, +matching and transforming user code to maximize perfor- +mance. +Acknowledgments +Grant TED2021-129221B-I00 funded by MCIN/AEI/10.13039/ +501100011033 and by the “European Union NextGenera- +tionEU/PRTR”. +References +[1] Maaz Bin Safeer Ahmad, Jonathan Ragan-Kelley, Alvin Cheung, and +Shoaib Kamil. 2019. Automatically translating image processing li- +braries to halide. ACM Transactions on Graphics 38 (Nov. 2019), 1–13. +Issue 6. doi: 10.1145/3355089.3356549. +[2] Miltiadis Allamanis, Earl T. Barr, Christian Bird, and Charles Sutton. +2015. Suggesting accurate method and class names, In the 2015 10th +Joint Meeting. Proceedings of the 2015 10th Joint Meeting on Foundations +of Software Engineering - ESEC/FSE 2015. doi: 10.1145/2786805.2786849. +[3] Uri Alon, Meital Zilberstein, Omer Levy, and Eran Yahav. 2019. +code2vec: learning distributed representations of code. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='estape@ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='uk University of Edinburgh Edinburgh, United Kingdom Gregorio Bernabé gbernabe@um.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='es University of Murcia Murcia, Spain José Manuel García jmgarcia@um.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='es University of Murcia Murcia, Spain Michael F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' O’Boyle mob@inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='uk University of Edinburgh Edinburgh, United Kingdom Abstract Dedicated tensor accelerators demonstrate the importance of linear algebra in modern applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Such accelerators have the potential for impressive performance gains, but require programmers to rewrite code using vendor APIs — a barrier to wider scale adoption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Recent work overcomes this by matching and replacing patterns within code, but such approaches are fragile and fail to cope with the diversity of real-world codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' We develop ATC, a compiler that uses program synthesis to map regions of code to specific APIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' The mapping space that ATC explores is combinatorially large, requiring the development of program classification, dynamic analysis, variable constraint generation and lexical distance matching techniques to make it tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' We apply ATC to real-world tensor and linear algebra codes and evaluate them against four state-of-the-art ap- proaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' We accelerate between 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='6x and 7x more programs, leading to over an order of magnitude performance improve- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' ACM Reference Format: Pablo Antonio Martínez, Jackson Woodruff, Jordi Armengol-Estapé, Gregorio Bernabé, José Manuel García, and Michael F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' O’Boyle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Matching linear algebra and tensor code to specialized hard- ware accelerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' In Proceedings of the 32nd ACM SIGPLAN Interna- tional Conference on Compiler Construction (CC ’23), February 25–26, 2023, Montréal, QC, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' ACM, New York, NY, USA, 13 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='1145/3578360.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='3580262 1 Introduction Linear algebra is a fundamental building block of many of today’s critical applications;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' from weather modeling [13] to CC ’23, February 25–26, 2023, Montréal, QC, Canada © 2023 Copyright held by the owner/author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' This is the author’s version of the work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' It is posted here for your personal use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Not for redistribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' The definitive Version of Record was published in Proceedings of the 32nd ACM SIGPLAN International Conference on Compiler Construction (CC ’23), February 25–26, 2023, Montréal, QC, Canada, https: //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='1145/3578360.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='3580262.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' ubiquitous DNN [22] workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Its importance is reflected in the large number of accelerator libraries and hardware devices devoted to fast linear algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' These range from specialized devices such as Google’s TPU [34] to the tensor cores on NVIDIA [12] among many others [5, 8, 25, 31, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' While such devices promise significant performance for an important class of applications [19], their uptake is limited by their programmability [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Typically, these accelerators and libraries are accessed via calls to specialized APIs, meaning existing code has to be rewritten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Given the volume [35] and variety [39] of existing legacy code, such rewriting is a significant undertaking [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' The combined importance of linear algebra acceleration and the difficulty of rewriting legacy code to accelerators has led to recent work which attempts to automate the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' These techniques search user code for matrix multiplica- tions using constraints [20, 28] or polyhedral analyses [9] and replace regions of code with appropriate API calls or instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' However, as we show in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='1, these approaches are fragile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Constraints capture only a limited set of program patterns and small variations in the user code defeat them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' While they work well on curated benchmarks, they perform poorly on real-world code [20, 63], defeated by function calls, optimized code and inline assembler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Neural classification (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' [18]) can effectively detect code despite these challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' However, it does not provide a path to acceleration, but requires further steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' These include gen- erating variable mappings and checking for equivalence [63] which has shown promising results for Fourier Transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' However, one of the key challenges in matching code to APIs is the cost of searching for user program variables that map to API formal parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' As the width of the API and complexity of the user program increase, this becomes com- binatorially expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' As we show in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='3 existing approaches [63] fail to scale to the challenges that linear algebra APIs present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='11659v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='PL] 27 Jan 2023 CC ’23, February 25–26, 2023, Montréal, QC, Canada P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Martínez, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Woodruff, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Armengol-Estapé, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Bernabé, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' García, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' O’Boyle We present ATC, a compiler that applies program synthe- sis to compile general user-code to linear algebra accelerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' We identify and solve key challenges enabling the detect/synthesize paradigm to scale to the more complex APIs of linear algebra acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' In addi- tion, ATC employs a trained platform predictor to determine whether acceleration is profitable or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' We applied our approach to 50 GitHub GEMM and 15 convolution projects and discovered between 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='6 and 7x more linear operators compared to KernelFaRer [20], IDL [28], Polly [30] or FACC[63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' This resulted in more than an order of magnitude performance improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' This paper makes the following contributions: We present ATC, which maps matrix multiplication and convolution programs to hardware accelerators, up to 7x more frequently than existing techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' We introduce novel heuristics to reduce the mapping search space by four orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' We develop novel dynamic analyses to determine higher- level information about variables, enabling synthesis without costly whole-program analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 2 Motivation Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Example application of API replacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' The above program is taken from the parboil benchmark [56], a widely-used benchmark suite, which is transformed into a call to an optimized matrix-multiplication accelerator API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' GEMM code optimized for AVX2 found on GitHub consisting of 120 lines of hand-optimized intrinsics and how ATC matches the code to the accelerator API 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='1 Exisiting Match and replace IDL and KernelFaRer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Both aim to detect linear algebra operations in user programs and replace them with an appro- priate accelerator library call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' To illustrate this consider the code in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' This shows a straight-forward matrix mul- tiplication program fragment, from the parboil benchmark suite [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' They aim to detect this matrix-multiplication and replace it with a call to the library, shown at the bottom of the diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' To replace code with an API call they have to both de- tect the code performing a matrix multiplication and also determine which user program variables correspond to the arguments of the API call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Both approaches are able to detect that this is a matrix multiplication, and can determine the mapping between user variables and API parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='2 Examples of complex GEMM programs Unfortunately, in practice, user code can be complex such that code structure or pattern-based approaches inevitably fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' As an example, consider the code found on GitHub shown in Figure 2 which implements a matrix-multiplication al- gorithm (only a fragment of the 120 lines of user code are shown here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' The code structure is complex and difficult to understand as it makes extensive use of inline assembler in- trinsics which defeats the code structure analysis approaches of IDL and KernelFaRer, preventing acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 2 Matching linear algebra and tensor code to specialized hardware accelerators CC ’23, February 25–26, 2023, Montréal, QC, Canada OJCLONE DATASET + GEMM PROGRAMS + CONV PROGRAMS NEURAL EMBEDDINGS FUNCTION CANDIDATES F1 F2 F3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' FN IO EQUIVALENCE IO DETECTION ACCELERATABLE FUNCTION MATCHES GENERATION MISMATCH SOLVER PROGRAM SYNTHESIS SET OF MATCHES MATCHES HEURISTCS MATCH ALGORITHM LEVENSHTEIN VALID MATCHES PERFORMANCE ANALYSIS ACCELERATOR SAMPLING FUNCTION SAMPLING SVM CLASSIFIER PROGRAM CLASSIFICATION ACCELERATOR API USER CODE ACCELERATED CODE Acceleratable Candidate Detection IO Detection Matches Generation Matches Reduction Profitability Detection Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' ATC compiler architecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='3 Our approach - ATC Rather than relying on code structure to guide detection, ATC uses behavioral equivalence to determine if a section of code is a linear algebra operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Firstly, ATC uses neural program classification [18] to detect that the code in Figure 2 is probably a GEMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' It then searches variable matches to determine the potential source and output arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' As the search space is combinatorially large, we introduce scal- able, algorithm-independent heuristics (which we discuss in Section 5) that keep the number of mappings manageable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Next, ATC generates different input values for the arrays and records the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' After generating many randomized inputs, it observes that it has the equivalent behavior to the corresponding API and is able to replace the AVX2 code with the GEMM call at the bottom of Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Legality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Now, IO behavioral equivalence is not proof that a section of code is a particular linear algebra operation - similarly IDL and KernelFaRer do not prove equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' For proof, bounded model checking based on Kleene [14] can be deployed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' In practice, as demonstrated in our experimental section, IO equivalence gives no false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' For further guarantees, we can ask for programmer sign-off or employ model checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Profitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Once we have detected and can replace a sec- tion of code with an accelerator call, we need to determine if it is profitable to do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Due to hardware evolution, we do not use a hard-wired heuristic to determine profitability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Instead, we learn, off-line, a simple predictive model to determine if the target accelerator is faster than a CPU implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' The model is called at runtime, determining if offloading is worthwhile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' FACC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Behavioral equivalence is also employed in FACC [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Unfortunately, it is restricted to FFTs and one-dimensional arrays, and cannot detect the replacement in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' There- fore, we extended FACC to FACC* to consider GEMMs and multi-dimensional arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' This, however, exposes its weak variable binding model which is combinatorial in the number of user array variables and their dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Furthermore, it relies on program synthesis to determine the length of ar- rays, which scales poorly to problems with many potential length parameters for arrays such as GEMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' FACC also relies on brittle inter-procedural liveness analy- ses to determine the liveness status of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' This restricts it to running only at link time, rendering it invalid for use in shared libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' We will see in Section 8 that the com- bination of these issues results in excessively large search spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 3 System overview Figure 3 gives a system flow overview of ATC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' We first de- tect regions of code that are likely to be linear algebraic operations using a neural program classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' The classifier is trained ahead of time, based on programs that are equivalent to the accelerator and prior examples of linear algebra code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Once candidate code sections have been identified, we ap- ply program analysis to match user program variables with the particular API formal parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Given the combina- torially large search space, we develop novel techniques to make the problem tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' For each candidate matching, we generate multiple data inputs, execute the user code section and record the output values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' If the input/output pairs correspond to the input/out- put behavior of the accelerator API, we can say they are behavioral equivalent and candidates for replacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 3 CC ’23, February 25–26, 2023, Montréal, QC, Canada P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Martínez, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Woodruff, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Armengol-Estapé, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Bernabé, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' García, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' O’Boyle While candidate user code may be replaceable with a call to an accelerator API, it may not be profitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Therefore, we employ a simple ML classifier, trained offline, and invoked at runtime to see if acceleration is appropriate for the user code for the runtime known array sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='1 Neural Program Classification To detect potentially acceleratable parts of a program, we use prior work in neural program classification [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' A network is trained with multiple instances of different program classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' We use the OJClone dataset [43], which includes 105 classes of different programs, and add examples of the programs that we want to detect e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' GEMMs and convolutions, gathered from benchmark suite repositories other than GitHub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' At compile time a new candidate program is divided into functions, which are presented to the neural classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' The classifier assigns each function in the program a probability of belonging to a certain class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' We consider the most proba- ble class, which in the case of a GEMM or convolution is then considered for variable matching and eventual code replace- ment as described in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Classification is fast (≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='5 sec) and has negligible impact on compilation time (see Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 4 Variable Matching To check if a section of user code is behaviorally equivalent to the API, we have to match up the user program variables with API formal parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' We first detect what variables are livein/liveout (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='1) and then the dimensions of arrays (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='1 Detecting livein and liveout variables Detecting livein and liveout variables via standard static analysis is straightforward for well-structured programs but fails for more diverse real-world codes, which may use assembly code or intrinsic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' ATC uses dynamic analysis to determine which variables are livein and liveouts inside a function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' In C, variables are passed by value so non-pointers variables are always livein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' In the case of pointers (or arrays), we generate random inputs with arbitrary sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' If the values in memory change after executing the program, the array is considered liveout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' This allows us to detect which variables are livein or live- out, but not both livein and liveout at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' We gen- erate a new random input for liveout variables and re-execute the function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' If the output differs from the first execution, it is both livein and liveout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' We implement this algorithm as a just-in-time compiler pass in LLVM [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='2 Detecting the dimensions of arrays Detecting arrays length enables offloading of appropriately- sized regions of codes, so it is a critical step in ATC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' For Load/StoreInst Array A?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Out of bound?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Replace Load/Store to/from index 0 Perform the instruction Exit with error code YES NO YES NO Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Dimension detection algorithm overview for a target example array called A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Algorithm 1 Dimensions detection algorithm 1: for arr in function do 2: fakeLoadAndStoresExcept(𝑎𝑟𝑟) 3: replaceLoadAndStores(𝑎𝑟𝑟) 4: repeat 5: 𝑐 = getNextCombination(𝑎𝑟𝑟) 6: ffi_call(𝐴,𝑉 ) 7: if not failed then 8: 𝑓 𝑜𝑢𝑛𝑑 = 𝑇𝑟𝑢𝑒 9: end if 10: until not found 11: Add 𝑐 to 𝐶 12: end for 13: return 𝐶 some programs, lengths can be found using static analysis (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' [49]), but this fails in more complex cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' We use run- time analysis to determine which program variables define array size using a modified form of runtime array bound checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' For each set of variables that could define an ar- ray’s size (typically, from the argument list), we set such variables to a fixed value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' We then execute the user code that is modified to check runtime array accesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' First, the compiler selects a target array to find its size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Then, to generate the modified program, we tweak the load and store instructions in the user program, replacing them with custom function calls in the IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' If a load or store does not access the array we are interested in, we modify it to load and store at a constant, safe location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' If it does, the instruction is replaced with a function call that will check at runtime if the access is out of bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' If so, the program exits with a custom error code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' If not, we have found a valid array size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' The basic idea is depicted in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' This is used by our JIT analysis as shown in Algorithm 1 and implemented in LLVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' This way, the compiler can assign different input sizes to a given array and check the exit code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Therefore, the compiler iterates over all the possible dimensions combinations until one of the executions does not end with the custom error exit code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' That means that the program was completed without any illegal access to the target array, which indicates that it is the right dimension of the array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 4 Matching linear algebra and tensor code to specialized hardware accelerators CC ’23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' February 25–26,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 2023,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Montréal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' QC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Canada Algorithm 2 Automatic matching algorithm 1: function dimsMatch(𝑓 1𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 𝑓 2𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 𝑝,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='𝑛) 2: 𝑆 = ∅ 3: 𝑖𝑑𝑥 ← 0 4: for 𝑎𝑟𝑔𝑠1 in f1a do 5: 𝑎𝑟𝑔𝑠2 = f2a[p[idx]] 6: Add {𝑎𝑟𝑔𝑠1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='𝑎𝑟𝑔𝑠2} to 𝑆 7: 𝑖𝑑𝑥 ← 𝑖𝑑𝑥 + 1 8: end for 9: return Size(S) = 𝑛 10: end function 11: 12: function outMatch(𝑓 1𝑜,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 𝑓 2𝑜,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 𝑝) 13: idx = IndexOf(𝑓 2𝑜,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 1) 14: return IndexOf(𝑝,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' idx) = IndexOf(𝑓 1𝑜,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 1) 15: end function 16: 17: function findMatchings(𝑓 1𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 𝑓 2𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 𝑓 1𝑜,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 𝑓 2𝑜,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='𝑛) 18: 𝐵 = ∅ 19: for p in permutations(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='𝑛) do 20: if dimsMatch(f1a, f2a, p) and 21: outMatch(f1o, f2o, p) then 22: Add 𝑝 to 𝐵 23: end if 24: end for 25: return 𝐵 26: end function 5 Reducing the matchings search space To match code to APIs, the compiler generates different can- didates for the variable to formal parameter mappings and then tests them using IO equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' For small APIs, all map- pings can be explored, but the combinatorial cost makes it prohibitive for real-world accelerator APIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' We develop tech- niques that reduce the mapping space by exploiting arrays information and human coding styles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='1 Exploiting array information Using array dimensions (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='2), we can reduce the num- ber of possible matches that must be checked, as assigning one array to another means that the dimensions of each array must line up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='1 Automatic matching algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' We first gener- ate all 𝑛!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' permutations of the 𝑛 array variables to 𝑛 parame- ters mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' We discard all permutations where variable livenesses do not match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Then for each candidate user array and parameter array pair, we generate the constraints defin- ing how their dimensions match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' If we find contradictory constraints for any permutation, we discard it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' The algorithm is shown in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='2 Automatic Matching Algorithm: Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' To il- lustrate this, Figure 5 shows an example where we have two X(x0*x1) Y(x1*x2) Z(x2*x0) A(y0*y1) B(y1*y2) C(y2*y0) [0,1,2] A: U: x0 -> y0 x1 -> y1 x2 -> y2 Liveout A:[0,0,1] U:[0,0,1] [1,0,2] x0 -> y1 x1 -> y2 x1 -> y0 x2 -> y1 x2 -> y2 x0 -> y0 [2,0,1] x0 -> y2 x1 -> y0 x2 -> y1 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Example application of the matching algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' The right match is found the algorithm automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Per- mutations in red means they are invalid, while the green permutation means valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' functions with three 2D arrays each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' First, the algorithm generates all the permutations between 0 and 𝑛 − 1 (𝑛 = 3 in this example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Then, for each permutation, it tries matching each variable in every array in the user code with the cor- responding variable in the array of the API (here we show only three of the six possible permutations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' In the first case (with the permutation [0, 1, 2]), the algo- rithm tries matching the array variables of the user program 𝑋,𝑌,𝑍 with API parameters 𝐴, 𝐵,𝐶 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' We then examine each of the variables defining each of the corresponding arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Comparing 𝑋 and 𝐴 gives a match of 𝑥0 → 𝑦0 and 𝑥1 → 𝑦1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' For the second array variable 𝑌 and API parameter 𝐵, we have 𝑥1 → 𝑦1 and 𝑥2 → 𝑦2 and for the third variable pair 𝑍,𝐶 we have 𝑥2 → 𝑦2 and 𝑥0 → 𝑦0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' All of these are con- sistent with 𝑛=3 constraint, which satisfies the condition (dimsMatch in Algorithm 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Liveout information is also sat- isfied so this permutation is added as a potential mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' In the second permutation [1, 0, 2], where 𝑋,𝑌,𝑍 maps to 𝐵,𝐴,𝐶, the constraints are inconsistent e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 𝑥1 → 𝑦2 and 𝑥1 → 𝑦0 leading to 6 ≥ 3, so it is not a valid match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' In the third and last example, constraints are equal to 𝑛, but the liveout arrays do not match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Thus, the only valid match is the one found in the first permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='2 Using argument names Programs are developed by humans, so we can assume that the functions that humans write follow common patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' We exploit this by analyzing the argument names of the API and the user program to find lexical similarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' To compare argument names, we use the Levenshtein dis- tance [38] to compute the distance between each of the user programs and API arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Figure 6 shows the definition of the Levenshtein distance, which calculation is based on the minimal number of modifications needed to transform one word into another, representing how close are those 5 CC ’23, February 25–26, 2023, Montréal, QC, Canada P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Martínez, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Woodruff, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Armengol-Estapé, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Bernabé, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' García, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' O’Boyle 𝑙𝑒𝑣(𝑎,𝑏) = \uf8f1\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f2 \uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f3 |𝑎| if |𝑏| = 0, |𝑏| if |𝑎| = 0, 𝑙𝑒𝑣(𝑡𝑎𝑖𝑙(𝑎),𝑡𝑎𝑖𝑙(𝑏)) if 𝑎[0] = 𝑏[0], 1 + 𝑚𝑖𝑛 \uf8f1\uf8f4\uf8f4\uf8f2 \uf8f4\uf8f4\uf8f3 𝑙𝑒𝑣(𝑡𝑎𝑖𝑙(𝑎),𝑏) 𝑙𝑒𝑣(𝑎,𝑡𝑎𝑖𝑙(𝑏)) 𝑙𝑒𝑣(𝑡𝑎𝑖𝑙(𝑎),𝑡𝑎𝑖𝑙(𝑏)) otherwise (1) Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Levenshtein recursive definition gemm_api(float* tc_A , float* tc_B , float* tc_C , int tc_m , int tc_n , int tc_k , int tc_lda , int tc_ldb , int tc_ldc , float tc_alpha , float tc_beta) { gemm(int M, int N, int K, float alpha , float *A, int lda , float *B, int ldb , float beta , float *C, int ldc) { tc A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' tc lda .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' M 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' N 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' K 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' alpha 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' A 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' lda 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='. .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Levenshtein distance calculation for the arguments of the tensor core API (above) and an example user program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' After computing the distance, the compiler selects the combination that minimizes the Levenshtein distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Figure 7 shows an application example of the Levenshtein distance to a real case of GEMM matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' For calculating the distance, we strip the API suffix (tc_) and convert all names to lowercase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Results show that the most probable mapping for tc_A is A in the user code, and for tc_lda is lda, which are the right matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='3 IO generation Once we have a candidate match we generate random inputs of different sizes and test for input-output (IO) equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' We use 30 inputs of varying sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Although IO behavioral equivalence is not proof, we can increase the number of tests for increased confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' No existing technique such as IDL or KernelFaReR can prove that a matched piece of code is provably equivalent to an API and therefore rely on user sign-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='1 Behavioral Equivalence and the Limits of Veri- fication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' ATC, like prior work on floating-point accelera- tors [63], uses behavioral equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' The downside of this strategy is that it requires programmer sign-off to make any substitution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' However, due to the complexities of verifying floating-point programs [63], verification of such liftings are some way off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' In summary, the key challenges that all competing tech- niques face are: Floating-point numbers often raise challenges in theo- rem provers as they are challenging to reason about.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Floating-point functions may have different accuracies in different input ranges, meaning that the obvious checks of correctness (even within bounds) are difficult to apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' The backend of ATC is not tied to using behavioral equiva- lence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' As we will see, the use of such behavioral equivalence results in no false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Further development of theorem prover technologies would mean that the weak behavioral equivalence in ATC could easily be replaced with a theorem prover guaranteeing correctness and enabling automatic transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 6 Automatic profitability detection We assume that user code runs faster when replaced by a platform-specific library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' The question is whether it is best to run on a CPU or accelerator version (XPU) of the library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' This in turn depends on the input size, which is only known at runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' We use a predictive model based on empirical data to enable accurate predictions as platforms and libraries evolve by retraining the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' SVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' We use the well-known support vector machine (SVM) classifier with a polynomial kernel of degree 3 with gamma=1 and𝐶=100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' We sample the CPU and the accelerator with a common dataset of input sizes, which produces a dataset that is small enough to be processed in less than five minutes, but large enough to be highly accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Data is labeled with 0 or 1 meaning that the CPU or the XPU is faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' The model is then trained and deployed at runtime, when matrix sizes are known, The training phase is done only once, at “factory time”, and the resulting model when deployed has negligible (≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='3𝑚𝑠𝑒𝑐) runtime overhead (see Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 7 Setup We evaluate GEMM and convolution acceleration on special- ized platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' For GEMM, we used an Intel i7-11700 (CPU) with an NVIDIA Quadro RTX 5000 (tensor cores) (XPU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' For convolution, we used the Google Cloud Platform (GCP) ser- vices equipped with a TPUv3 with 8 TPU cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Compilation benchmarks in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='3 are executed in an AMD EPYC 7413.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' The Intel/NVIDIA platform runs CentOS 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='3 with kernel 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' LLVM was downloaded from the official Git repository, using commit 329fda3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' User codes were compiled using gcc 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='0 with -O3 -march=native flags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' We used cuBLAS 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='2 and MKL 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='254 for compiling codes to the XPU and CPU, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' For compiling convolution programs to 6 Matching linear algebra and tensor code to specialized hardware accelerators CC ’23, February 25–26, 2023, Montréal, QC, Canada Algorithm Code LoC Nº Args Optimizations Constraints C struct?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='Direct ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='No ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='36 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='OpenMP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='FW = FH = 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='No ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='34 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='286 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='BLAS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='N = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='Yes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='179 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='BLAS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='FW = FH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='Yes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='Winograd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='687 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='Intrinsics + OpenMP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='FW = FH = 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='No ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='254 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='N = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='Yes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='782 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='Intrinsics + OpenMP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='FW = FH = 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='No ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' List of convolution codes the CPU, we used oneDNN v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' The TPU system runs Debian 10 with kernel 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='0-14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='1 User code We explored GitHub looking for C and C++ GEMM codes, analyzing more than 400 programs from which we selected 50 programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' We discarded the rest of them because of wrong implementations, compilation errors or duplicated code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' The final list of programs is shown in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' We categorize the codes as follows: Naive: naive implementations with the traditional 3-loop structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Naive Parallel: as Naive but with simple outer loop parallelization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Unrolled: naive implemen- tation with unrolled loops;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Kernel Calls: implementations that divide the loops into different function calls;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Blocked: tiled implementations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Goto: implementations of the Goto algorithm [29];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Strassen: implementations of the Strassen algorithm [55];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Intrinsics: implementations using Intel intrin- sics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' In addition, we selected 50 non-GEMM projects to check whether any of the approaches gave false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' We explored GitHub looking for C and C++ 4D convolution implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' We analyzed around 50 programs from which we a selected list of 15 programs based on the same methodology used for selecting GEMMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' The list of convolution programs is shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' We have included codes from the most relevant convolution implementations: Direct: the direct convolution algorithm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' im2col+gemm: an algorithm that casts the input as matrices (im2col) and later uses a GEMM, as in Caffe [32];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Winograd: the Winograd algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='2 Methods We evaluate our approach against 4 well known schemes: IDL: Idioms are described using an idiom description lan- guage [28], which is translated into a set of constraints over LLVM IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' KernelFaRer: Uses different pattern matching to detect spe- cific code constructs, matching specific matrix-multiplication structures [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Polly: Detects static control parts (SCoPs) in the code using the polyhedral model [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' It does not replace the code with a call to an optimized library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' FACC*: FACC uses neural embeddings and behavioral syn- thesis to detect candidates for acceleration [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' It is limited to 1D arrays so we developed an extended version, FACC*, which supports multi-dimensional arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 8 Results 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='1 Detection Figure 9 shows the percentage of GEMM programs matched by each technique across each of 8 categories listed in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' IDL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' The constraint based scheme [28] only matches 6 out of 50 cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' These programs are largely naive implemen- tations of GEMM, with a simple loop structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' It is able to manage 2 programs containing unrolled loops but fails on anything more complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Matching more diverse cases would require writing a new IDL constraint description for each sub-class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' KernelFaRer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' This code matching approach [20] is more successful, matching 11 GEMMs due to a more robust pattern matcher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' For straightforward sequential implementations, it is able to match all but one of the cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' However, any code variation, including loop unrolling, defeats it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Polly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Although it does not match and replace GEMMs, it can detect SCoPs which may be candidates for replace- ment with appropriate API calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' It is less successful than KernelFaRer in detecting naive implementations but is more robust across other more complex categories including one parallel and unrolled versions and 2 blocked cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' It slightly outperforms KernelFaRer, matching 13 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 11 out of 50 cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' FACC*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Unlike the other approaches, FACC* performed poorly on naive implementations, but better on others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Here, the size of the mapping search space is the limiting factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' It was able to find 10 cases in the available time (timeout ≤ 10 mins).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' We examine the reasons for this in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' ATC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Our approach is significantly more robust across all categories, matching 42 out of 50 cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' It is able to detect all naive implementations and the majority within each other category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' It detects more naive parallel implementations, unrolled and blocked programs than Polly and is the only technique to detect GEMMs in codes containing kernel calls and intrinsic instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='1 Accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Figure 10 provides a summary of ATC’s success and failure by type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' In 8 cases ATC failed to detect that the program contained a GEMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' In one case, program 23, this is due to there being too many candidate matches, 280 which is above our timeout threshold of 100 candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' The remaining cases are due to overly aggressive search 7 CC ’23, February 25–26, 2023, Montréal, QC, Canada P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Martínez, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Woodruff, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Armengol-Estapé, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Bernabé, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' García, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' O’Boyle ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='Algorithm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='Code ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='Multiple of bs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='Intrinsics (AVX2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='Row-major ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='Any ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='Intrinsics (SSE) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' List of GEMM codes Naive Naive p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Unrrolled Kernels Blocked Goto Strassen Intrinsics All 0 20 40 60 80 100 4 9 11 1 12 0 1 0 3 4 2 1 0 1 3 0 0 0 0 4 0 2 0 2 6 0 0 0 1 1 0 0 0 2 3 0 0 0 0 9 6 131110 42 % of matched codes IDL POLLY KFR FACC* ATC Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Percentage of matched GEMM codes by different techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 0 20 40 60 80 % of programs Matched Too many candidates Missed matches Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Percentage of matched GEMM codes by ATC divided by failure reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' pruning, missing a legal match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Improved search heuristics are likely to improve program coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' False positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' None of the methods classified any of the 50 non-GEMMs as a GEMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Across all methods, there were no false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='2 Performance The performance of each approach is shown in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Polly is not included here as although it can detect SCoPs, it does not explicitly identify them as GEMMs for API replace- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' We show two bars for KernelFaRer, which correspond to the strategy of GEMM code with an optimized CPU im- plementation as described in [20] and KFR (XPU) which is our extension, replacing the CPU library with the optimized XPU implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' IDL and FACC* directly target the ac- celerator, while ATC chooses the CPU or accelerator based on its SVM platform predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' This runtime prediction cost is negligible ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='3𝑚𝑠𝑒𝑐 and included in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' What is immediately clear is that detecting more GEMMs leads to better overall speedup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' In the Naive category, KFR and ATC are both able to achieve good performance, with a speedup of 726x and 1031x, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' The gap is nar- rowed when using KFR (XPU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' However, KFR is unable to detect GEMMs in any other category leading to just a 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='2x 8 Matching linear algebra and tensor code to specialized hardware accelerators CC ’23, February 25–26, 2023, Montréal, QC, Canada Naive Naive p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Unrrolled Kernels Blocked Goto Strassen Intrinsics All 1 10 100 1000 10000 Speedup IDL KFR (CPU) KFR (XPU) FACC* ATC Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Geometric mean speedup obtained by IDL, KernelFaRer, FACC* and ATC in GEMM programs with 𝑛 = 8192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' speedup overall while ATC achieves 344.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='0x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Unsurprisingly, there is more performance available on naive sequential im- plementations than in those cases where the programmer has spent effort in optimizing the program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='3 Candidate search complexity and compile time One of the key challenges in matching code to APIs is search- ing for program variables that map to API formal parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' As the width of the API and complexity of the user program increase, this becomes combinatorially expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Figure 12 evaluates FACC* naive matching of variables and our ap- proach based on the Levenshtein distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Naive matching varies considerably from just 4 candidates to over 1 million.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Our approach greatly reduces the number of candidates for the majority of the programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' There is one special case, code 23, where we reduce the number of candidates, but it is still too high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Figure 13 shows the compilation time of ATC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' The initial neural classifier has a negligible constant execution time of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='3 seconds, while the other phases’ compilation time grows with the number of candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' As the number of candidates begins to increase compi- lation time becomes prohibitively expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Code 23 has 280 candidates which would take 35 mins more to evaluate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' We limit the number of candidates considered to 100 which corresponds to a timeout of ≤ 10 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='4 Profitability accuracy To measure the accuracy of the SVM platform predictor, we built a model offline and tested it on unseen data values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Table 2 summarizes the SVM accuracy with different input sizes and shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' The SVM achieves a global accuracy of 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='7%, where the misprediction occurs between 𝑚 = 2000 and 𝑚 = 8000 which is the “edge” between the CPU and the XPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' In all other intervals, the prediction is always correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' The best accuracy is achieved with non-squared matrices, while square matrices give slightly lower accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Overall, this is a highly accurate predictor with a negligible runtime overhead of ‘ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='3𝑚𝑠𝑒𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Parameter Value (mnk) m Global Accuracy 2000 4000 6000 8000 10000 111 100% 100% 100% 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='0% 100% 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='8% 123 100% 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='9% 100% 100% 100% 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='9% 312 100% 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='3% 100% 100% 100% 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='9% 136 100% 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='5% 100% 100% 100% 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='9% Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' SVM accuracy for different sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 111 means m = 1 × m, n = 1 × m, k = 1 × m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 123 means m = 1 × m, n = 2 × m, k = 3 × m etc 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='5 Convolutions Our approach is generic and can be applied to other APIs other than GEMMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' As an example, we consider tensor con- volutions which are a significant component of DNN work- loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' While IDL, KernelFaRer, Polly and FACC* were unable to detect any of the convolutions, ATC detected 10 of the 15 convolutions as shown in Figure 14;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' we were unable to match 5 due to the excessive number of candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Figure 15 shows the performance achieved by replacing with library code for each of the programs we are able to accelerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Across all codes, the SVM predicts that the TPU accelerator outperforms the CPU, giving an average 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='8x performance improvement across the programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 9 Related work Matching in Programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Matching high-level program structure has been used to discover parallelism [23], het- erogenous offloading [6, 44] and many other core compiler tasks [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Constraint languages make these tasks easier [10, 27, 28] but their constraints are very sensitive to code struc- ture [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' For matrix multiplications in particular, KernelFaRer [20] provides a more robust approach, detecting characteristics that define matrix multiplications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Polyhedral analyses can also be used to target matrix multiplication accelerators [9, 58], but both these techniques fail to scale to the diversity 9 CC ’23, February 25–26, 2023, Montréal, QC, Canada P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Martínez, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Woodruff, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Armengol-Estapé, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Bernabé, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' García, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' O’Boyle 5 10 15 20 25 30 35 40 45 50 1 101 102 103 104 105 106 Code Candidates generated 4s 45s 10m 1h 12h 2d 20d Approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' compilation time FACC* ATC Threshold Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Comparison of the number of candidates generated for matching GEMM codes: FACC* vs our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 1 3 8 48 0 20 40 60 80 100 Code 2 Code 21 Code 7 Code 1 Number of candidates Time (s) IO Testing Tests Generation Candidates Generation Neural Embeddings 280 0 500 1,000 1,500 2,000 Code 23 Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Compilation time for different number of candi- dates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Direct i2mcol+gemm Winograd All 0 20 40 60 80 100 6 3 3 0 1 2 10 5 % of matched codes Matched Not matched Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Matched convolution codes by ATC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 1 3 4 5 6 8 10 11 13 15 All 1 10 100 1000 Code Speedup Speedup (CPU) Speedup (TPU) ATC Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' ATC speedup in convolution programs with ℎ = 𝑤 = 224, 𝑘𝑤 = 𝑘ℎ = 11, 𝑐 = 3, 𝑘 = 96 and 𝑛 = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' of real code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' FACC [63] uses IO equivalence, which is ro- bust to program structure, but only addresses the challenges of FFTs and does not scale to longer function signatures used for GEMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' To support any accelerator type, the com- piler should support multi-dimensional arrays, while FACC only supports 1D arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Because in 1D arrays and FFTs the search space in matching the API parameters is small, FACC does not include anything to reduce it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' With more complex programs and domains, this limitation makes compiling pro- grams intractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Mask [51] uses symbolic execution to prove equivalence, which does not work well for floating-point problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Fuzzy classification techniques based on code clone detection [40, 57], domain-classification [59], pattern matching [15], code embeddings [2, 3, 21] and identifiers [36, 47] can be used to help compile to accelerators [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' These classification strategies are able to classify diverse code structures, but do not provide a compilation strategy for using an accelerator on their own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' A large class of techniques focus on migrating between APIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' These techniques often use program synthesis [16], NLP [46] and code embeddings [45, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' These techniques are unable to extract existing code into APIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Compiling for GEMM Accelerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Existing compila- tion strategies largely focus on lowering code from intrinsics to accelerators using rewrite rules [52, 53, 62] and synthesis techniques [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Existing approaches to extracting matrix multiplications [20, 28] are brittle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Synthesis-based techniques [1, 7, 41] and rewriting-based techniques [11, 54] have been developed to extract these DSLs that can then be lowered: but they largely require flexible DSLs, rather than APIs presented by hardware accelerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Performance Prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Predicting code the performance of hardware accelerators is challenging, as the break-even point may depend on many different arguments within a function’s interface [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' LogCA [4] introduces static perfor- mance comparison models for hardware accelerators and similar models have been applied in offloading tasks [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Ma- chine learning has often been applied in profitability settings, 10 Matching linear algebra and tensor code to specialized hardware accelerators CC ’23, February 25–26, 2023, Montréal, QC, Canada such as OpenCL Kernels [60, 61] and OpenMP [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Similar techniques have been applied to FPGAs, by estimating pow- er/performance [26] and tracking actual performance [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 10 Conclusions This work presented ATC, a flexible domain-agnostic com- piler that matches legacy linear algebra code to accelerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' By using IO behavioral equivalence and smart search space reduction, we are able to match over 80% of challenging real-world programs to accelerator APIs, significantly out- performing all alternative approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Supporting new domains different from GEMM and convo- lution is easy because ATC focuses on behavior rather than code structure, which makes it very flexible and extensible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Furthermore, to support other accelerators in GEMM or con- volution, only the accelerator API is needed: ATC adapts to the new specification automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Future work will examine how to further reduce the search space using online learning and to expand the complexity of user code considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Longer-term, we wish to automati- cally target a range of accelerators with diverse functionality, matching and transforming user code to maximize perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Acknowledgments Grant TED2021-129221B-I00 funded by MCIN/AEI/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='13039/ 501100011033 and by the “European Union NextGenera- tionEU/PRTR”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' References [1] Maaz Bin Safeer Ahmad, Jonathan Ragan-Kelley, Alvin Cheung, and Shoaib Kamil.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' [4] Muhammad Shoaib Bin Altaf and David A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Wood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' LogCA: A High-Level Performance Model for Hardware Accelerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' In Proceed- ings of the 44th Annual International Symposium on Computer Architec- ture (Toronto, ON, Canada) (ISCA ’17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Association for Computing Ma- chinery, New York, NY, USA, 375–388.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='1145/3079856.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='3080216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' [5] Michael Anderson,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Benny Chen,' metadata={'source': 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+page_content=' Amy Yang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Jiecao Yu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Hector Yuen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Ying Zhang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Aravind Anbudarai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Vandana Balan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Harsha Bojja,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Asso- ciation for Computing Machinery, New York, NY, USA, 687–702.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='1145/3519939.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content='3523439.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' [64] Gina Yuan, Shoumik Palkar, Deepak Narayanan, and Matei Zaharia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' Offload Annotations: Bringing Heterogeneous Computing to Existing Libraries and Workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' In 2020 USENIX Annual Technical Conference (USENIX ATC 20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' USENIX Association, 293–306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} +page_content=' 13' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQf3C2V/content/2301.11659v1.pdf'} diff --git a/T9E2T4oBgHgl3EQfCga9/content/tmp_files/2301.03615v1.pdf.txt b/T9E2T4oBgHgl3EQfCga9/content/tmp_files/2301.03615v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..496f1d5f21c8671e5b2756dd2f25d5a5faa98069 --- /dev/null +++ b/T9E2T4oBgHgl3EQfCga9/content/tmp_files/2301.03615v1.pdf.txt @@ -0,0 +1,3726 @@ +Study of the de Almeida-Thouless (AT) line in the one-dimensional diluted power-law +XY spin glass +Bharadwaj Vedula,1 M. A. Moore,2 and Auditya Sharma1 +1Department of Physics, Indian Institute of Science Education and Research, Bhopal, Madhya Pradesh 462066, India +2Department of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom +(Dated: January 11, 2023) +We study the AT line in the one-dimensional power-law diluted XY spin glass model, in which +the probability that two spins separated by a distance r interact with each other, decays as 1/r2σ. +Tuning the exponent σ is equivalent to changing the space dimension of a short-range model. We +develop a heat bath algorithm to equilibrate XY spins; using this in conjunction with the standard +parallel tempering and overrelaxation sweeps, we carry out large scale Monte Carlo simulations. +For σ = 0.6, which is in the mean-field regime above six dimensions – it is similar to being in 10 +dimensions – we find clear evidence for an AT line. For σ = 0.75 and σ = 0.85, which are in the +non-mean-field regime and similar to four and three dimensions respectively, our data is like that +found in previous studies of the Ising and Heisenberg spin glasses when reducing the temperature +at fixed field. For σ = 0.75, there is evidence from finite size scaling studies for an AT transition +but for σ = 0.85, the evidence for a transition is non-existent. We have also studied these systems +at fixed temperature varying the field and discovered that at both σ = 0.75 and at σ = 0.85 there +is evidence of an AT transition! Confusingly, the correlation length and spin glass susceptibility as +a function of the field are both entirely consistent with the predictions of the droplet picture and +hence the non-existence of an AT line. In the usual finite size critical point scaling studies used to +provide evidence for an AT transition, there is seemingly good evidence for an AT line at σ = 0.75 +for small values of the system size N, which is strengthening as N is increased, but for N > 2048 the +trend changes and the evidence then weakens as N is further increased. We have also studied with +fewer bond realizations the system at σ = 0.70, which is the analogue of a system with short-range +interactions just below six dimensions, and found that it is similar in its behavior to the system at +σ = 0.75 but with larger finite size corrections. The evidence from our simulations points to the +complete absence of the AT line in dimensions outside the mean-field region and to the correctness +of the droplet picture. Previous simulations which suggested there was an AT line can be attributed +to the consequences of studying systems which are just too small. The collapse of our data to the +droplet scaling form is poor for σ = 0.75 and to some extent also for σ = 0.85, when the correlation +length becomes of the order of the length of the system, due to the existence of excitations which +only cost a free energy of O(1), just as envisaged in the TNT picture of the ordered state of spin +glasses. However, for the case of σ = 0.85 we can provide evidence that for larger system sizes, +droplet scaling will prevail even when the correlation length is comparable to the system size. +I. +INTRODUCTION +While the spin glass problem at mean-field level is now +well-understood [1], questions remain as to the nature of +the ordered state in three dimensional spin glasses. A +key question is whether the ordered phase of real spin +glasses has the broken replica symmetry features found +in mean-field theory. +This question is most easily an- +swered by finding whether on application of a magnetic +field hr there is a line, the so-called de Almeida Thou- +less (AT) line [2], below which in the hr − T plane there +is replica symmetry breaking. This line exists at mean- +field level (see Fig. 1) and its possible existence in three +dimensions can be studied experimentally and with sim- +ulations. Simulational studies of the existence of replica +symmetry breaking within the zero-field spin glass state +itself are plagued by finite size effects: it is expected that +the difference between the predictions of droplet scaling +and those of replica symmetry breaking will only become +visible for very large systems (for a review see [3]). A +recent review of simulations, including studies of the ex- +istence of the AT line, can be found in Ref. [4]. +Right from the early days of spin glass studies there +have been doubts raised as to whether the AT line existed +below six dimensions. For example Bray and Roberts [6] +attempted to do an expansion in 6 − ϵ dimensions for +the critical exponents at the AT line but failed to find +a stable fixed point. They suggested that maybe that +indicated that there might be no AT line below six di- +mensions. A renormalization group calculation also gave +indications that the AT line was going away as d → 6 +from above +[7]. As it is difficult to do simulations in +dimensions around 6 to check these speculations, simula- +tors have had to turn instead to one-dimensional models +with long-range power-law interactions. +These models go back to Kotliar, Anderson and Stein +[8], who in turn were inspired by the long-range ferro- +magnet that was studied by Dyson [9, 10]. +The long- +range power-law model has the advantage that by tuning +the power-law exponent σ, one has access to both the +mean-field and the regimes with non-mean-field critical +behavior. However, the full power-law model is expen- +sive for numerics. Fortunately a clever workaround was +introduced by Leuzzi et al. [11] where instead of the in- +arXiv:2301.03615v1 [cond-mat.dis-nn] 9 Jan 2023 + +2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +T/Tc +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +hr/J +exact AT +approximate AT +SK data +σ = 0.6 (T∗ data) +σ = 0.6 (h∗ data) +Figure 1. The AT line. The solid line is the exact AT line +for the SK model, calculated as in Ref. [5]. The dashed line +is the approximation to it of Eq. (8). We have marked on +the diagram the results of our simulations on the SK model, +which were done to check our Monte Carlo procedures. The +points in red and green are the results of our simulations at +σ = 0.6, which lie in the mean-field region. The data on the +horizontal axis for σ = 0.6 are normalized to the transition +temperature Tc in zero field for that value of σ. For the XY +SK model the AT line goes to infinity as T → 0. +teractions falling off as a power law, it is the probability +of there being a bond between two spins that falls off as +a power law. The fewer bonds in the model means that a +significantly smaller computational cost is involved, thus +allowing for the simulation of larger system sizes. +While the vast literature on spin glasses is mostly fo- +cussed on Ising spins [11–16], there has been a revival of +interest in classical m-component vector spin glass mod- +els [5, 17–26] in the last decade or so. The XY model +has m = 2 and the Heisenberg model has m = 3. One +of the triggers for this revival has been the finding that +the infinite-range vector spin glass exhibits an AT line +provided a magnetic field that is random in all the com- +ponent directions is applied [5]. Furthermore analytical +studies of the AT transition in m-vector models shows +that the field theory of these AT transitions is that of the +Ising spin glass [5]. Thus it has become possible to study +the question of whether or not an Ising AT transition ex- +ists in various dimensions by studying one-dimensional +vector spin glasses with long-range interactions [18]! +In this paper, we study the one-dimensional diluted +XY spin glass subjected to a random vector magnetic +field, with the aid of large scale Monte Carlo simula- +tions. While Monte Carlo simulations are a time-tested +tool for the study of phase transitions in spin glasses, the +exorbitant cost of equilibration makes them rather chal- +lenging in practice. It has been argued that vector spins +tend to equilibrate faster compared to Ising spins [27], be- +cause of the soft nature of the spins involved, even though +the presence of more components adds to the cost. The +Heisenberg spin glass [5, 17–19, 27–32] has been the pop- +ular vector spin to have been considered, because of the +availability of the heatbath algorithm [28], which works +very efficiently to equilibrate it. The XY spin glass is +less effectively handled by the heatbath algorithm [33] +because of the technicalities involved in inverting a prob- +ability distribution for which a simple closed form ex- +pression is unavailable in the XY case. In this paper, we +develop a method, which is outlined in Appendix A to +perform this inversion numerically with the hope of ben- +efiting from the vector nature of XY spins, while simul- +taneously reducing the components to as small a number +as possible. +The improved algorithm yields mixed fruits. The gains +from the reduced number of components seems to be +largely counterbalanced by the additional resources con- +sumed by the numerical inversion. However, with the aid +of extensive computational power, we are able to access +system sizes comparable to those in the corresponding +study with Heisenberg spins. Our findings for the XY +diluted spin glass closely mimic those obtained for the +Heisenberg version of the same model [5, 17, 18] and the +Ising spin glass in three dimensions [34] when we investi- +gate crossing the possible AT line by varying T at fixed +values of hr. In the mean-field regime, (we studied here +the case of σ = 0.6, which corresponds to 10 dimen- +sions, which is above the upper critical dimension of 6), +there is clear evidence of an Almeida-Thouless line (see +Sec. V A). There is rather weak evidence for an Almeida- +Thouless line for σ = 0.85 using the commonly employed +finite size critical point scaling methods of analysis (see +Sec. V C). At this value of σ, our system should be simi- +lar to the Edwards-Anderson model in three dimensions +with short-range interactions. For the in-between case +at σ = 0.75 which lies in the non-mean-field regime, but +closer to the mean-field boundary at σ = 2/3, our data +do provide stronger evidence for a phase transition in +the presence of small magnetic fields than at σ = 0.85. +However, by varying the magnetic field hr at fixed tem- +perature T we find in Sec. V that at both σ = 0.75 and +at σ = 0.85 there is quite decent evidence for an AT +line. Confusingly, the field dependence of the correlation +length is very well-described by the Imry-Ma prediction +of the droplet picture, which implies the complete ab- +sence of the AT transition! In the droplet picture the +correlation length in a field remains finite and only di- +verges as hr → 0. However, when this correlation length +becomes comparable to the system size L the Imry-Ma +formula needs to be modified and we give in Sec. +VI +a scaling form for this modification. +It is based upon +the usual finite size scaling approach used in studying +critical phenomena, and just as for critical phenomena +we find that there are finite size corrections to this scal- +ing form. In addition to these scaling corrections there + +3 +are corrections which arise when ξSG ∼ L < L∗ which +are of different origin and are connected to TNT effects +[35, 36]. TNT effects arise from droplets of free energy +cost of O(1), which certainly exist in systems whose sizes +L < L∗ [3]. The length scale L∗ is always large and is +expected to diverge as d → 6 or as σ → 2/3. It is only +for the case of σ = 0.85 that we can reach sizes where +TNT effects seem to be getting small. These matters are +discussed in Sec. VII. +Furthermore, we can use the droplet scaling picture to +explain some of the features of the apparent AT transi- +tion which arise on performing the usual finite size crit- +ical scaling analyses, and show that these are the conse- +quence of not studying large enough systems. Unfortu- +nately these arguments will only become compelling for +system sizes which we cannot reach. Our chief evidence +for the droplet picture is its very successful prediction +of the correlation length as a function of the field in the +region when finite size and TNT effects are unimportant. +Our claim that the evidence favors the absence of the +AT line for values of σ outside the mean-field region is +consistent with the attempt [21] to calculate the AT field +at T = 0 using an expansion in 1/m. This indicated that +as d → 6 from above in the Edwards-Anderson model, the +AT field would go to zero, implying the absence of the AT +line below 6 dimensions (which in the one-dimensional +long-range model corresponds to 2/3 < σ < 1). +For +σ > 1 there is no finite temperature spin glass phase. +The plan of this paper is as follows. In Sec. II we de- +scribe the model in detail. In Sec. III we describe the +quantities which were studied in our Monte Carlo simu- +lations, the details of which are given in Appendix A. Our +data is analysed in Sec. V on the assumption that there +is an AT transition, while in Sec. VI the data is analysed +according to droplet scaling assumptions. In Sec. VII we +discuss the effect of TNT behavior on our results. Finally +in Sec. VIII we summarize our conclusions. +II. +MODEL HAMILTONIAN +The general Hamiltonian for vector spin glasses is: +H = − +� +⟨i,j⟩ +JijSi · Sj − +� +i +hi · Si , +(1) +where Si is the spin on the ith lattice site (i += +1, 2, . . . , N), which is chosen to be a unit vector. m rep- +resents the number of components of the vector Si. In +this work we concentrate on XY spins, and set m = 2. +The Cartesian components hµ +i (µ = 1, 2) of the on-site +external magnetic field are i.i.d random variables drawn +from a Gaussian distribution of zero mean and variance +h2 +r and satisfy the relation: +� +hµ +i hν +j +� +av = h2 +rδijδµν. +(2) +We use the notation ⟨· · · ⟩ for thermal average and [· · · ]av +for an average over quenched disorder throughout this +paper. +The spins are arranged on a circle so the geometric +distance between a pair of spins (i, j) is given by [16] +rij = N +π sin +� π +N |i − j| +� +, +(3) +which is the length of the chord connecting the ith and +jth spins. The interactions Jij are independent random +variables such that the probability of having a non-zero +interaction between a pair of spins (i, j) falls with the +distance rij between the spins as a power law: +P(Jij) ∝ +1 +r2σ +ij +. +(4) +If the spins i and j are linked the magnitude of the inter- +action between them is drawn from a Gaussian distribu- +tion whose mean is zero and whose standard deviation is +unity, i.e: +[Jij]av = 0 +and +� +J2 +ij +� +av = 1. +(5) +The mean number of non-zero bonds from a site is fixed +to be ˜z (co-ordination number). So, the total number +of bonds among all the spins on the lattice is fixed to +be Nb = N ˜z/2. +When ˜z = 6 this model mimics the +3D simple cubic lattice model and we use this value for +˜z for all the σ values studied. +(For σ = 0 and ˜z = +N −1, the model becomes the infinite-range Sherrington- +Kirkpatrick (SK) model [37]). +To generate the set of interaction pairs [11, 17] (i, j) +with the desired probability we pick a site i randomly and +uniformly and then choose a second site j with probabil- +ity given by: +pij = +r−2σ +ij +� +j̸=i +r−2σ +ij +. +(6) +If the spins at i and j are already connected we repeat +this process until we find a pair of sites (i, j) which have +not been connected. Once we find such a pair of spins, +we connect them with a bond whose strength Jij is a +Gaussian random variable with attributes given by Eq. +(5). We repeat this process exactly Nb times to generate +Nb pairs of interacting spins. +The advantage of the diluted model over the fully con- +nected model is that, in a fully connected model, there +are N(N − 1)/2 interactions. The ratio of the number of +interactions of the diluted model to the fully connected +model is ˜z/N which is a very small value as N becomes +large. Hence it is possible to go to much larger system +sizes with a diluted model as compared to a fully con- +nected model. +At zero-field, the mean-field spin glass transition tem- +perature for the m-component vector spin glass is given +by [17, 18, 38] +T MF +c += 1 +m +� +�� +j +� +J2 +ij +� +av +� +� +1/2 += +√ +˜z +m J. +(7) + +4 +The approximate location of the AT line for an m- +component infinite-range spin glass near the zero-field +transition temperature Tc is [5] +�hr +J +�2 += +4 +m(m + 2) +� +1 − T +Tc +�3 +. +(8) +The accuracy of this approximation for the SK model can +be judged from Fig. 1. +A one-dimensional chain with power law diluted in- +teractions for a particular value of σ is equivalent to a +short-range model [14] of effective dimension deff, where +deff = +2 +2σ − 1, +(9) +i.e., there is a one-to-one mapping between a long-range +diluted network with exponent σ and a short-range model +with space dimension deff, at least when 1/2 < σ < 2/3. +Thus when σ = 0.60, deff = 10. +For the interval +2/3 < σ < 1 other relations are required [15, 39]. For +example, for Ising spin glasses, it was suggested in Ref. +[39] that d = 4 corresponded to σ ≈ 0.790, while d = 3 +corresponded to σ ≈ 0.896. Unfortunately, the mapping +for the XY model has been less studied. +III. +CORRELATION LENGTHS AND +SUSCEPTIBILITIES +In this section we discuss the quantities which were +obtained from our Monte Carlo simulations and used to +extract a correlation length ξSG and the spin glass suscep- +tibility χSG. The simulations themselves are described in +detail in Appendix A. +The thermal average of a quantity is calculated using +multiple replicas in the following standard way: +⟨A⟩⟨B⟩⟨C⟩⟨D⟩ = ⟨A(1)B(2)C(3)D(4)⟩ +(10) +where (1),(2),(3), and (4) are four copies of the system at +the same temperature. The wave-vector-dependent spin +glass susceptibility is given by [5] +χSG(k) = 1 +N +� +i,j +1 +m +� +µ,ν +�� +χµν +ij +�2� +av eik(i−j), +(11) +where +χµν +ij = +� +Sµ +i Sν +j +� +− ⟨Sµ +i ⟩ +� +Sν +j +� +. +(12) +The spin glass correlation length is then determined from +ξSG = +1 +2 sin(kmin/2) +� +χSG(0) +χSG (kmin) − 1 +�1/(2σ−1) +(13) +where kmin = (2π/N). +IV. +FINITE-SIZE ANALYSES ASSUMING A +TRANSITION EXISTS +In this section we detail the method of finite-size anal- +ysis when a transition is assumed to exist. When study- +ing the AT line, which is a line of phase transitions in +the hr − T plane, it can be crossed on an infinite number +of trajectories. The most commonly used trajectory is +the one where hr is kept constant and the temperature +T is varied. In this work we also consider the trajectory +in which T is kept constant and hr is varied. We refer +to the zero-field transition temperature as Tc while we +denote a generic transition temperature on the AT line +by TAT(hr). Similarly we denote the field on the AT line +by hAT(T). +The spin glass susceptibility χSG ≡ χSG(0) of a finite +system of N spins has the finite size scaling form (near +the transition temperature TAT(hr)) [5]: +χSG +N 2−η = C +� +N 1/ν (T − TAT(hr)) +� +, +(2/3 ≤ σ < 1), +(14a) +χSG +N 1/3 = C +� +N 1/3 (T − TAT(hr)) +� +, +(1/2 < σ ≤ 2/3), +(14b) +where η is given by 2 − η = 2σ − 1. These forms are +examples of finite size scaling expressions which would +be expected to hold in the critical region when N → ∞, +(T − TAT(hr)) → 0, with (say) N 1/ν(T − TAT(hr)) finite. +The scaling function C will depend on the value of σ. +There are always finite size corrections to these forms. +For example, the corrections to Eq. (14b) will be of the +form +χSG +N 1/3 = C +� +N 1/3 (T − TAT(hr)) +� ++ N −ωG +� +N 1/3(T − TAT(hr)) +� +. +(15) +It has been suggested [17, 40] that the correction to scal- +ing exponent is given at least in the mean-field region +by +ω = 1/3 − (2σ − 1). +(16) +Curves of χSG/N 2−η +(χSG/N 1/3 +in the mean-field +regime) plotted for different system sizes should inter- +sect at the transition temperature TAT(hr). In reality, +finite-size corrections to Eq. (14) are always present and +cause the intersection point between the curves for size N +and 2N to depend on N. The intersection temperatures +vary as [40–43] +T ∗(N, 2N) = TAT(hr) + A +N λ , +(17) +where A is the amplitude of the leading correction, and +the exponent λ is +λ = 1/3 + ω, +(1/2 < σ ≤ 2/3), +(18a) +λ = 1/ν + ω, +(2/3 < σ < 1), +(18b) + +5 +where ω is the leading correction to the scaling exponent. +When σ = 0.6, ω = −2σ + 4/3, so λ = 5/3 − 2σ = 0.467 +[40]. In the regime when σ > 2/3 the values of both ν +and λ are not well-determined, so there we shall treat λ +as a fitting parameter. +The spin glass correlation length has a similar finite +size scaling form in the critical region +ξSG +N += X +� +N 1/ν (T − TAT(hr)) +� +, +(2/3 ≤ σ < 1), +(19a) +ξSG +N deff/6 = X +� +N 1/3 (T − TAT(hr)) +� +, +(1/2 < σ ≤ 2/3). +(19b) +ν, the correlation length critical exponent, has to be de- +termined numerically in the interval 2/3 < σ < 1. +We have also studied crossing the AT line at fixed T +and varying hr. Then Eq. (19) takes the form, +ξSG +N += X +� +N 1/ν (hr − hAT(T)) +� +, +(2/3 ≤ σ < 1), +(20a) +ξSG +N deff/6 = X +� +N 1/3 (hr − hAT(T)) +� +, +(1/2 < σ ≤ 2/3), +(20b) +where hAT(T) denotes the field at the AT line at tem- +perature T. Similarly, the spin glass susceptibility χSG +of the finite system near the AT transition line takes the +form +χSG +N 2−η = C +� +N 1/ν (hr − hAT(T)) +� +, +(2/3 ≤ σ < 1), +(21a) +χSG +N 1/3 = C +� +N 1/3 (hr − hAT(T)) +� +, +(1/2 < σ ≤ 2/3). +(21b) +In the thermodynamic limit, Eq. (19) is similar to +Eq. (20); the effect of finite size corrections to the two +can differ. For example, while the correction to scaling +exponent λ does not depend on the choice of the trajec- +tory, the magnitude of the scaling corrections can differ. +Thus in the intersection formulae when applied to fields +h∗(N, 2N) = hAT(T) + +˜A +N λ , +(22) +the coefficient ˜A will be different from A in Eq. (17). Cor- +rections to scaling of, say, Eq. (21a), are more generally +of the form +χSG +N 2−η = C +� +N 1/ν (hr − hAT(T)) +� ++ N −ωG +� +N 1/ν (hr − hAT(T)) +� +, +(23) +where ω is the correction to scaling exponent, and G is +another scaling function. This type of scaling form holds +in the limit where N 1/ν(hr −hAT(T)) is fixed as N → ∞, +which of course can only be realized approximately in +numerical studies. +A key feature of the finite size critical point scaling +analysis is that right on the AT line itself, that is when +hr = hAT(T), R = χSG/N 2−η (χSG/N 1/3 for σ ≤ 2/3) +should be finite as N → ∞. We find (see Sec. VI) that R +is at least not increasing with N, and perhaps finite (see +Fig. 37), for σ = 0.60 but for σ = 0.70, 0.75 and 0.85 it is +in fact increasing with N, at the crossing field h∗(N, 2N). +We deduce from this observation that at these values of +σ the crossings at h∗(N, 2N) are not associated with a +true critical point at all but are consequences of droplet +scaling. At a true critical point R would tend to a finite +constant as N increases, but we find it increases with N, +provided N > 1024 (or system sizes 2N > 2048) for the +case of σ = 0.75 (see Fig. 38). +In Sec. V we shall present our attempts at analysing +the data for σ = 0.6, σ = 0.75, σ = 0.85 at fixed values of +hr but varying T, and also at a fixed value of T and vary- +ing hr, on the assumption that there is an AT line and +using the finite-size scaling methods of this subsection. +We have also obtained data at fixed T and varying hr +for σ = 0.60, 0.70, 0.75, 0.85 and analysed them using +finite size generalizations of well-known droplet scaling +relations. In this case the droplet picture provides a sim- +ple set of formulae for analysing the data in the assumed +absence of an AT line. +V. +ANALYSES OF THE SIMULATION DATA +ASSUMING THERE IS AN AT LINE +We shall study the phase transitions at hr += 0, +and determine the zero-field transition temperature Tc +(= TAT(hr = 0)), and seek evidence of an AT transition +at non-zero hr using the standard critical point finite size +scaling method of determining the “crossings” or inter- +sections of the curves of, say, χSG/N z (with z = 1/3 +when σ ≤ 2/3, and with z = 2 − η = 2σ − 1 for σ ≥ 2/3) +at values of N and 2N as we reduce T through the AT +transition temperature at fixed hr, or the field hr at fixed +T in the vicinity of the AT field hAT(T) as outlined in +Sec. IV. There seems no reason to doubt the existence +of an AT line for any value of σ in the mean-field region +σ < 2/3, and our results are entirely consistent with the +existence of an AT transition at σ = 0.60. They serve +as a useful comparison for the studies in the non-mean +regime σ > 2/3, where the evidence will be found to favor +the droplet picture. We have studied N values 128, 256, +512, 1024, 2048, 4096, 8192 and 16384 for both σ = 0.60 +and σ = 0.75, but went up to N = 32768 for the case +of σ = 0.75 when the field hr was varied at fixed T. +When σ = 0.85 the largest N values used was 4096. In +this case the zero-field transition temperature Tc is quite +low and as a consequence all the investigations have to +be done also at low temperatures, where equilibration +times are long, preventing the study of larger systems. +We are mainly interested in the question as to whether + +6 +outside the mean-field region, that is for σ > 2/3, an AT +transition actually exists and whether (say) the depen- +dence of ξSG on the field hr can be understood as will +be suggested in Sec. VI on the droplet picture without +invoking an AT transition at all. If it can, this would +provide support to the argument that the droplet scaling +picture rather than replica symmetry breaking describes +spin glasses below 6 dimensions. We have analysed the +data for σ = 0.70, 0.75 and for σ = 0.85 using the usual +“crossing” method, (the finite size scaling approach out- +lined in Sec. IV), which indeed works well for σ = 0.6. +The evidence for the existence of an AT transition at +σ = 0.75 and 0.85 will be contrasted with the evidence +against an AT transition at these values of σ. +Our main focus was the case σ = 0.75. +We looked +briefly at the case σ = 0.70 to find whether or not it +might be practical to study whether σ = 2/3 is the value +of σ above which the AT line might disappear. We found +that it was similar to σ = 0.75, but that the corrections +to scaling were larger. This means that for a given level +of accuracy, larger N values are required. We studied +σ = 0.85 because it should behave similarly to physical +systems in three dimensions but we could not equilibrate +systems at the larger N values in this case because the +temperatures T of interest have to be less than Tc, which +is rather small. +A. +σ = 0.6 +We shall focus on σ = 0.60 in this subsection. It corre- +sponds according to Eq. (9) to an effective dimension of +10 dimensions, which is in the mean-field region; it lies +above the upper critical dimension of spin glasses, which +is 6 (or in the mean-field region σ < 2/3 in the one- +dimensional long-range model). It is natural to expect +that for this value of σ there will be an AT line and this +is amply confirmed by our simulations. For this value of +σ, simulations of the corresponding Ising model [12, 15] +and the Heisenberg model [17, 18] also found an AT line. +Our results for hr = 0 are given in Figs. 2 and 3. Ac- +cording to Eq. (14b), the data for χSG/N 1/3 when plot- +ted for different system sizes should intersect at the tran- +sition temperature Tc. Similarly, according to Eq. (19b), +the data of ξSG/N deff/6 with deff = 2/(2σ − 1) should in- +tersect at the same transition temperature. Fig. 2 shows +the data for different system sizes. We find the tempera- +ture T ∗(N, 2N) at which the curves corresponding to the +system sizes N and 2N intersect. We then fit this data +with Eq. (17) to find the transition temperature. The +exponent λ ≡ 5/3 − 2σ is known to equal 0.467 in this +case [17, 40]. The result is displayed in Fig. 3, where the +T ∗(N, 2N) data obtained from intersections of χSG are +fitted against N −λ with a straight line for the largest 6 +pairs of system sizes to give Tc = 0.8873 ± 0.0017. The +corresponding intersections of the ξSG data (omitting the +two smallest system sizes) gives Tc = 0.8893 ± 0.0046. +The values of Tc obtained from χSG data and ξSG data +0.75 +0.80 +0.85 +0.90 +0.95 +T +0.1 +1 +χSG/N 1/3 +σ = 0.6 +hr = 0 +N = 128 +N = 256 +N = 512 +N = 1024 +N = 2048 +N = 4096 +N = 8192 +N = 16384 +0.7 +0.8 +0.9 +T +10−3 +10−2 +10−1 +100 +101 +ξSG/N deff /6 +Figure 2. +The main figure shows the plot of χSG/N 1/3 as +a function of the temperature T for different system sizes, +for σ = 0.6 with hr = 0. The inset figure shows the corre- +sponding data for ξSG/N deff/6, with deff = 2/(2σ − 1) in the +mean-field regime (Eq. (9)). +The exponents of N are cho- +sen according to Eqs. (14b) and (19b). Both the plots show +that the curves for different system sizes intersect. The data +for the intersection temperatures T ∗(N, 2N) between pairs of +adjacent system sizes are presented in Fig. 3. +are in agreement with each other. The mean-field pre- +diction of Eq. (7) is much higher, T MF +c += +√ +6/2 = 1.2247. +Fluctuation effects not present in the SK limit must be +responsible for this large difference. +For hr = 0.1, the data is as shown in Figs. 4 and 5. +When the T ∗(N, 2N) data obtained from χSG are fitted +against N −λ with a straight line for the largest 4 pairs +of system sizes we get TAT(hr = 0.1) = 0.6735 ± 0.0120. +The corresponding ξSG data (omitting the two smallest +system sizes) gives TAT(hr = 0.1) = 0.6745 ± 0.0148. +Thus we have found that the AT line passes through +the point (T, hr) = (0.674, 0.1). To compare that with +the predictions from the SK model, we use the zero-field +transition temperature Tc = 0.887 obtained above. Then +for hr = 0.1, the predicted value of the AT transition +temperature ratio of the SK model would be TAT(hr = +0.1)/Tc = 0.74, while the Monte Carlo determined value +at σ = 0.6 is 0.7590 ± 0.0113. (For the SK model, the +Monte Carlo value of the ratio is 0.7641 ± 0.0341). Thus +while the zero-field transition temperature at σ = 0.6 is +not close to the mean-field value of Eq. (7), the SK form +of the AT line is a good approximation provided it is ex- +pressed in terms of the renormalized zero-field transition +temperature Tc (see also Fig. 1). +The AT line can be approached not only by reducing +the temperature T but also by reducing the field at fixed +T. In Figs. 6 and 7 we have constructed the crossing plots + +7 +0.000 +0.025 +0.050 +0.075 +0.100 +N −λ +0.70 +0.75 +0.80 +0.85 +T ∗(N, 2N) +σ = 0.6 +hr = 0 +χSG +−0.72N −λ + 0.89 +ξSG +−1.56N −λ + 0.89 +Figure 3. A plot of the intersection temperatures T ∗(N, 2N) +for χSG/N 1/3 and ξSG/N deff/6 obtained from the data in +Fig. 2, as a function of N −λ, for σ = 0.6 with hr = 0. The +value of the exponent λ is fixed to be 0.467 which is known +exactly [17, 40]. The fits give Tc = 0.8873 ± 0.0017 from χSG +and Tc = 0.8893 ± 0.0046 from ξSG. +0.60 +0.65 +0.70 +0.75 +T +0.2 +0.3 +0.4 +0.5 +0.6 +χSG/N 1/3 +σ = 0.6 +hr = 0.1 +N = 128 +N = 256 +N = 512 +N = 1024 +N = 2048 +N = 4096 +N = 8192 +N = 16384 +0.55 +0.60 +0.65 +0.70 +0.75 +T +0.1 +1 +10 +ξSG/N deff /6 +Figure 4. A finite size scaling plot of χSG (main figure) and +ξSG (inset figure), for σ = 0.6 in a magnetic field of hr = +0.1. Both the datasets clearly indicate that a phase transition +occurs. +The transition temperature in the thermodynamic +limit is estimated in Fig. 5. +as a function of hr for ξSG and χSG respectively. Analysis +of the crossing plots of h∗(N, 2N) in Fig. 8 shows that +the behavior is again consistent with the existence of an +AT line at least at σ = 0.60. The same value of λ was +used as when plotting T ∗(N, 2N). The h∗(N, 2N) data +for all the pairs of system sizes are fitted against N −λ +to give hAT(T = 0.6) = 0.1569 ± 0.0061 from χSG and +0.000 +0.025 +0.050 +0.075 +0.100 +N −λ +0.58 +0.60 +0.62 +0.64 +0.66 +0.68 +T ∗(N, 2N) +σ = 0.6 +hr = 0.1 +χSG +−0.23N −λ + 0.67 +ξSG +−0.58N −λ + 0.67 +Figure 5. +The intersection temperatures T ∗(N, 2N), for +σ = 0.6 with hr = 0.1 (look at Fig. 4). Both the datasets +are consistent with a spin glass transition temperature of +TAT(hr = 0.1) = 0.67. +10−2 +10−1 +100 +hr +10−7 +10−5 +10−3 +10−1 +101 +103 +ξSG/N deff/6 +σ = 0.6 +T = 0.6 +N = 128 +N = 256 +N = 512 +N = 1024 +N = 2048 +N = 4096 +N = 8192 +N = 16384 +Figure 6. A finite size scaling plot of ξSG as a function of +magnetic field hr, for σ = 0.6 at a temperature of T = 0.6. +hAT(T = 0.6) = 0.1571 ± 0.0067 from ξSG. We found +two points on the AT line, (T, hr) = (0.674, 0.1) from +T ∗(N, 2N), and (T, hr) = (0.6, 0.157) from h∗(N, 2N) +data. These points are plotted in Fig. 1 for comparison +with the exact AT line for the SK model. +B. +σ = 0.75 +The case σ = 0.75 corresponds to the non-mean-field +regime: the long-range diluted model for this value of σ +is equivalent to a short-range model with d ≈ 4 dimen- + +8 +10−2 +10−1 +100 +hr +10−2 +10−1 +100 +χSG/N 1/3 +σ = 0.6 +T = 0.6 +N = 128 +N = 256 +N = 512 +N = 1024 +N = 2048 +N = 4096 +N = 8192 +N = 16384 +Figure 7. A finite size scaling plot of χSG as a function of +magnetic field hr, for σ = 0.6 at a temperature of T = 0.6. +0.000 +0.025 +0.050 +0.075 +0.100 +N −λ +0.08 +0.10 +0.12 +0.14 +0.16 +h∗(N, 2N) +σ = 0.6 +T = 0.6 +λ = 5 +3 − 2σ = 0.467 +χSG +−0.36N −λ + 0.16 +ξSG +−0.61N −λ + 0.16 +Figure 8. The intersection fields h∗(N, 2N), for σ = 0.6 with +T = 0.6. Both the datasets are consistent with a spin glass +transition at hAT(T = 0.6) ≈ 0.16. +sions. In this regime, simulations of the corresponding +Heisenberg model [17, 18] were thought consistent with +an AT transition. +According to Eq. (14a), the data for χSG/N 2−η,where +2 − η = 2σ − 1, plotted for different system sizes should +intersect at the transition temperature Tc. Similarly, ac- +cording to Eq. (19a), the curves of ξSG/N should also +intersect at the transition temperature. The main plots +of Figs. 9 and 11 show the finite-size-scaled data of χSG, +and the corresponding inset plots show the finite-size- +scaled data of ξSG. The curves for different system sizes +show a clear tendency to intersect close to the same tem- +perature. The data for T ∗(N, 2N) are then fitted with +0.50 +0.55 +0.60 +0.65 +0.70 +T +0.1 +χSG/N 2−η +σ = 0.75 +hr = 0 +N = 128 +N = 256 +N = 512 +N = 1024 +N = 2048 +N = 4096 +N = 8192 +0.5 +0.6 +0.7 +T +0.1 +1 +ξSG/N +Figure 9. The main figure shows data for χSG/N 2−η (with +2 − η = 2σ − 1) for different system sizes, for σ = 0.75 with +hr = 0. The inset shows the data for ξSG/N. According to +Eqs. (14a) and (19a) the data should intersect at Tc, which is +shown in Fig. 10. +0.0 +0.1 +0.2 +0.3 +0.4 +N −λ +0.54 +0.56 +0.58 +0.60 +0.62 +0.64 +T ∗(N, 2N) +σ = 0.75 +hr = 0 +λ = 0.1583 +χSG +−0.15N −λ + 0.64 +ξSG +−0.15N −λ + 0.64 +Figure 10. A plot of the intersection temperatures T ∗(N, 2N) +obtained from the data in Fig. 9, for σ = 0.75 with hr = 0. +Using the value of the scaling exponent λ = 0.1583 (ob- +tained from the h∗(N, 2N) data), the T ∗(N, 2N) data are +fitted against N −λ using a straight line. The resulting values +for the transition temperature are Tc = 0.6397 ± 0.0051 from +χSG and Tc = 0.6440 ± 0.0154 from ξSG. +Eq. (17) where the value of the exponent λ is not known +in the non-mean-field regime and hence should be con- +sidered as a fitting parameter. +If there were an AT transition there would be a unique +value of λ, the same for both the ξSG and χSG intersec- +tions corresponding to both h∗(N, 2N) and T ∗(N, 2N). +The h∗(N, 2N) data obtained from χSG intersections in + +9 +0.30 +0.35 +0.40 +0.45 +0.50 +T +0.3 +0.4 +χSG/N 2−η +σ = 0.75 +hr = 0.05 +N = 128 +N = 256 +N = 512 +N = 1024 +N = 2048 +N = 4096 +N = 8192 +0.35 +0.40 +0.45 +0.50 +0.55 +T +1 +ξSG/N +Figure 11. +A finite size scaling plot of χSG (main figure) +and ξSG (inset figure) for σ = 0.75. +The magnetic field is +hr = 0.05. +0.0 +0.1 +0.2 +0.3 +0.4 +N −λ +0.2 +0.3 +0.4 +0.5 +T ∗(N, 2N) +σ = 0.75 +hr = 0.05 +λ = 0.1583 +χSG +−0.35N −λ + 0.48 +ξSG +0.73N −λ + 0.19 +Figure 12. The intersection temperatures T ∗(N, 2N) for σ = +0.75 with hr = 0.05. The data from Fig. 11 did not fit well +with Eq. (17). So we used the value of λ obtained from the +h∗(N, 2N) data and did a linear fitting which gives TAT(hr = +0.05) = 0.4832 ± 0.0394 from χSG and TAT(hr = 0.05) = +0.1894 ± 0.0383 from ξSG, and the values do not agree with +each other. +Fig. 14 (which is described later) are fitted with Eq. (22) +by considering λ, Tc and ˜A as fitting parameters. This +is a non-linear fitting procedure for which we use effi- +cient methods like the Trusted Region Reflective (TRF) +algorithm and the Levenberg-Marquardt(LM) algorithm +(for which packages are available in python) to determine +the fitting parameters, and we obtain λ = 0.1583. Since +the exponent giving the leading correction to scaling λ is +universal, we use the same value of λ with both intersec- +10−3 +10−2 +10−1 +100 +hr +10−4 +10−3 +10−2 +10−1 +100 +ξSG/N +σ = 0.75 +T = 0.55 +N = 128 +N = 256 +N = 512 +N = 1024 +N = 2048 +N = 4096 +N = 8192 +N = 16384 +N = 32768 +10−3 +10−2 +10−1 +100 +hr +10−3 +10−1 +ξSG/N +Figure 13. A finite size scaling plot of ξSG as a function of +magnetic field hr, for σ = 0.75 at a temperature of T = 0.55. +The inset shows our two largest system sizes. +10−3 +10−2 +10−1 +100 +hr +10−3 +10−2 +10−1 +χSG/N 2−η +σ = 0.75 +T = 0.55 +N = 128 +N = 256 +N = 512 +N = 1024 +N = 2048 +N = 4096 +N = 8192 +N = 16384 +N = 32768 +Figure 14. A finite size scaling plot of χSG as a function of +magnetic field hr, for σ = 0.75 at a temperature of T = 0.55. +tions h∗(N, 2N) and T ∗(N, 2N) obtained from χSG and +ξSG data. We substitute the value of λ obtained above +in Eq. (17) and fit the T ∗(N, 2N) data against N −λ with +a straight line. +As shown in Fig. 10, for hr = 0, the +χSG fit (considering all the pairs of system sizes) gives +Tc = 0.6397 ± 0.0051. The corresponding ξSG fit (omit- +ting the smallest system size) gives Tc = 0.6440±0.0154. +For hr = 0.05, the intersection temperatures data +are shown in Fig. 12. +Omitting the smallest system + +10 +0.0 +0.1 +0.2 +0.3 +0.4 +N −λ +0.005 +0.010 +0.015 +0.020 +h∗(N, 2N) +σ = 0.75 +T = 0.55 +λ = 0.1583 +χSG +0.03N −λ + 0.00 +ξSG +−0.01N −λ + 0.02 +Figure 15. The intersection fields h∗(N, 2N) for σ = 0.75 with +T = 0.55. +size, the T ∗(N, 2N) data are fitted with Eq. +(17) to +give TAT(hr = 0.05) = 0.4832 ± 0.0394 from χSG and +TAT(hr = 0.05) = 0.1894 ± 0.0383 from ξSG. Compared +to Fig. +5 which gives the equivalent plot for the case +with σ = 0.60, the data in Fig. 12 does not look like +data which is converging to the same asymptotic limit +when N is large. If the crossings were actually due to a +genuine AT transition, then the asymptotic limit should +be the same for both. +We have also studied ξSG and χSG at fixed T, but vary- +ing hr and the finite size scaling plots for these are given +in Figs. +13 and 14. +There appears to be good inter- +sections in the curves, supporting therefore the possible +existence of an AT transition at the temperature studied +T = 0.55. A plot of h∗(N, 2N) versus 1/N λ is in Fig. 15, +using the same value of λ = 0.1583. In the intersections +of ξSG there is a clear rising trend of h∗(N, 2N) with in- +creasing N until N = 1024, followed by decreasing values +of h∗(N, 2N) for N > 2048. For the case of σ = 0.60, +where there is almost certainly a genuine AT transition, +(Fig. 8) only the rising trend is seen. It is as if for the +smaller systems N < 2048 the system at σ = 0.75 is +behaving similarly to its mean-field cousin at σ = 0.60. +Note that this change of trend cannot be attributed to the +correction to scaling terms of Eq. (23). These only apply +in the limit N → ∞ with N 1/ν(hr −hAT(T)) fixed. For a +genuine AT transition the intersections h∗(N, 2N) from +both ξSG and χSG should both extrapolate as N → ∞ to +the same field hAT(T). It is hard to argue that Fig. 15 +provides good evidence for this. On the other hand, on +the droplet picture, it would be expected that h∗(N, 2N) +should extrapolate to zero. The evidence that is happen- +ing is also weak. +0.25 +0.30 +0.35 +0.40 +T +0.1 +0.2 +0.3 +χSG/N 2−η +σ = 0.85 +hr = 0 +N = 128 +N = 256 +N = 512 +N = 1024 +N = 2048 +N = 4096 +0.3 +0.325 +0.35 +0.375 +0.4 +T +1 +ξSG/N +Figure 16. A finite size scaling plot of χSG (main figure) and +ξSG (inset figure), for σ = 0.85 in a magnetic field of hr = 0 +(with 2 − η = 2σ − 1). Both the datasets clearly indicate that +a phase transition occurs. The transition temperature in the +thermodynamic limit is estimated in Fig. 17. +0.000 +0.002 +0.004 +0.006 +N −λ +0.32 +0.34 +0.36 +0.38 +T ∗(N, 2N) +σ = 0.85 +hr = 0 +λ = 1.0315 +χSG +−2.93N −λ + 0.33 +ξSG +9.78N −λ + 0.33 +Figure 17. The intersection temperatures T ∗(N, 2N), for σ = +0.85 with hr = 0. A non-linear fit of the χSG data from Fig. 16 +with the Eq. (17) using the Levenberg-Marquadt algorithm +gives λ = 1.0315. A linear fit of the data using this value of λ +gives Tc = 0.3336±0.0013 from χSG and Tc = 0.3297±0.0036 +from ξSG. +C. +σ = 0.85 +For σ = 0.85 we are further into the non-mean-field +region. +According to Eq. (9), σ = 0.85 corresponds +to a short-range model close to three dimensions. +In +this regime, simulations of the corresponding Heisenberg +model [17, 18] did not find an AT line. + +11 +0.1 +0.2 +0.3 +0.4 +T +0.1 +χSG/N 2−η +σ = 0.85 +hr = 0.05 +N = 128 +N = 256 +N = 512 +N = 1024 +N = 2048 +Figure 18. A finite size scaling plot of χSG for σ = 0.85. The +magnetic field is hr = 0.05. +0.1 +0.2 +0.3 +0.4 +T +0.1 +0.2 +χSG/N 2−η +σ = 0.85 +hr = 0.02 +N = 128 +N = 256 +N = 512 +N = 1024 +N = 2048 +Figure 19. +A finite size scaling plot of χSG, for σ = 0.85 +with hr = 0.02. +The data do not intersect even at very +low temperatures (much lower than the mean field value of +TAT(hr = 0.02) = 0.2997 obtained using Eq. (8)) indicating +that there is no phase transition in this regime. +For hr = 0, Fig. 16 clearly shows that the curves for +different system sizes are intersecting. The data for in- +tersection temperatures are shown in Fig. 17. Similar to +the case of σ = 0.75, the T ∗(N, 2N) data obtained from +χSG are fitted with Eq. (17) by considering λ, Tc, and A +as fitting parameters. We obtain λ = 1.0315 from both +TRF and LM methods. The fit using the χSG data for +0.1 +0.2 +0.3 +0.4 +T +1 +ξSG/N +σ = 0.85 +hr = 0.05 +N = 128 +N = 256 +N = 512 +N = 1024 +N = 2048 +Figure 20. +A finite size scaling plot of ξSG, for σ = 0.85 +with hr = 0.05. +The data show merging behavior at low +temperatures. +0.1 +0.2 +0.3 +0.4 +T +100 +101 +ξSG/N +σ = 0.85 +hr = 0.02 +N = 128 +N = 256 +N = 512 +N = 1024 +N = 2048 +Figure 21. +A finite size scaling plot of ξSG, for σ = 0.85 +with hr = 0.02. +The data show merging behavior at low +temperatures. +all the pairs of system sizes gives Tc = 0.3336 ± 0.0013. +The corresponding ξSG fit (omitting the smallest system +size) gives Tc = 0.3297 ± 0.0036. The two values of Tc +are quite close. +For hr = 0.05 the χSG/N 2−η data do not intersect as +shown in Fig. 18. Such a field could conceivably be above +the largest AT field even at T = 0 so we also studied a +smaller field: hr = 0.02 shown in Fig. 19. There is no + +12 +10−3 +10−2 +10−1 +100 +hr +10−2 +10−1 +100 +ξSG/N +σ = 0.85 +T = 0.3 +N = 128 +N = 256 +N = 512 +N = 1024 +N = 2048 +N = 4096 +10−3 +10−2 +10−1 +100 +hr +10−2 +10−1 +100 +ξSG/N +Figure 22. A finite size scaling plot of ξSG as a function of +magnetic field hr, for σ = 0.85 at a temperature of T = 0.3. +The inset shows our two largest system sizes. +10−3 +10−2 +10−1 +100 +hr +10−3 +10−2 +10−1 +χSG/N 2−η +σ = 0.85 +T = 0.3 +N = 128 +N = 256 +N = 512 +N = 1024 +N = 2048 +N = 4096 +Figure 23. A finite size scaling plot of χSG as a function of +magnetic field hr, for σ = 0.85 at a temperature of T = 0.3. +sign of any crossing at this field either!. The ξSG data is +less clearcut. Fig. 20 shows there are no intersections at +a field of hr = 0.05 while a merging behavior is seen for +the larger systems at hr = 0.02, as shown in Fig. 21. In +our simulations we went to very low temperatures such as +T = 0.1, which is small in comparison with the mean-field +values of TAT for hr = 0.02 and hr = 0.05 using Eq. (8), +but we still could not find any clear intersections in the +χSG or ξSG data. This suggests that there is no phase +0.000 +0.002 +0.004 +0.006 +N −λ +0.01 +0.02 +0.03 +h∗(N, 2N) +σ = 0.85 +T = 0.3 +λ = 1.0315 +χSG +0.36N −λ + 0.00 +ξSG +3.96N −λ + 0.00 +Figure 24. The intersection fields h∗(N, 2N) for σ = 0.85 with +T = 0.3. We substitue the value of the exponent λ = 1.0315 +obtained from the T ∗(N, 2N) data at hr = 0 in Eq. (22) +and fit h∗(N, 2N) data agianst N −λ to get hAT(T = 0.3) = +0.0046±0.0006 from χSG and hAT(T = 0.3) = 0.0047±0.0016 +from ξSG. +transition in this regime in the presence of a magnetic +field. Our data are consistent with the scenario where +the external magnetic field destroys the phase transition, +just as happens for a ferromagnet when a uniform field +is turned on. +Very similar features were seen for the +Heisenberg version of this model [17, 18] and in the three +dimensional Ising model [34]. +Confusingly, intersections are seen at fixed T = 0.3 +as hr is varied in the plots of ξSG/N in Fig. 22 and of +χSG/N 2−η in Fig. 23. The usual analysis of h∗(N, 2N) +is given in Fig. 24. Thus in crossing the AT line along a +trajectory of fixed T we have seen intersections, suggest- +ing there might be an AT transition. However, the large +N limit of h∗(N, 2N) in Fig. 24 in the case of σ = 0.85, +suggests that hAT(T) might actually be zero, consistent +with the droplet scaling picture. In the next section the +dependence of ξSG and χSG on hr will be explained using +the droplet scaling approach. +VI. +DATA ANALYSES ON THE DROPLET +PICTURE +In this section we give the field dependence of ξSG and +χSG according to the droplet picture [44–46], including +also their finite size modifications, and compare these +with our simulation data. +In the droplet picture one uses an Imry-Ma argument +[47] for the correlation length ξ and identifies it with +the size of the region or domain within which the spins +become re-oriented in the presence of the random field. +The free energy gained from such a reorientation by the +the random field is of order +� +q(T)hrξd/2. The size of +such domains ξ is determined by equating this free energy + +13 +10−3 +10−2 +10−1 +100 +101 +hr +10−1 +101 +103 +105 +107 +ξSG +σ = 0.75 +T = 0.55 +N = 128 +N = 256 +N = 512 +N = 1024 +N = 2048 +N = 4096 +N = 8192 +N = 16384 +N = 32768 +fit for N = 32768 +(∼ h−2.71 +r +) +Figure 25. A plot of ξSG as a function of magnetic field hr, +for σ = 0.75 at a temperature of T = 0.55. +10−3 +10−2 +10−1 +100 +101 +hr +100 +102 +104 +106 +ξSG +σ = 0.85 +T = 0.3 +N = 128 +N = 256 +N = 512 +N = 1024 +N = 2048 +N = 4096 +fit for N = 4096 +(∼ h−2.20 +r +) +Figure 26. A plot of ξSG as a function of magnetic field hr, +for σ = 0.85 at a temperature of T = 0.3. +to the free energy cost of the interface of this domain of +re-ordered spins with the rest of the system, which is of +the form Υ(T)ξθ [48]. Equating these two free energies +gives +ξ ∼ +� +Υ(T) +� +q(T)hr +�1/(d/2−θ) +. +(24) +While there is a considerable literature on the depen- +dence of the interface exponent θ on σ for the case of +10−2 +10−1 +100 +101 +hr +10−6 +10−3 +100 +103 +106 +109 +1012 +ξSG +σ = 0.6 +T = 0.6 +N = 128 +N = 256 +N = 512 +N = 1024 +N = 2048 +N = 4096 +N = 8192 +N = 16384 +fit for N = 16384 +(∼ h−6.88 +r +) +Figure 27. A plot of ξSG as a function of magnetic field hr, +for σ = 0.6 at a temperature of T = 0.6. +10−1 +101 +N 1/xhr +10−6 +10−4 +10−2 +100 +ξSG/N +σ = 0.75 +T = 0.55 +x = 2.7077 +N = 128 +N = 256 +N = 512 +N = 1024 +N = 2048 +N = 4096 +N = 8192 +N = 16384 +N = 32768 +Figure 28. +A complete finite size scaling plot of ξSG as a +function of magnetic field hr, plotted on a log-log scale, for +σ = 0.75 at a temperature of T = 0.55. +Ising spin glasses [49], we know of no equivalent studies +for the case of the XY spin glass. (Our data suggests +that its θ might be close to that of the Ising spin glass). +Eq. (24) shows that as hr → 0, the length scale be- +comes infinite; ξ diverges as ξ ∼ 1/hx +r, where +x = +1 +d/2 − θ. +(25) +The exponent x is the analogue of ν at the AT transition; + +14 +10−1 +101 +N 1/xhr +10−5 +10−4 +10−3 +10−2 +10−1 +100 +ξSG/N +σ = 0.85 +T = 0.3 +x = 2.2019 +N = 128 +N = 256 +N = 512 +N = 1024 +N = 2048 +N = 4096 +Figure 29. +A complete finite size scaling plot of ξSG as a +function of magnetic field hr, plotted on a log-log scale, for +σ = 0.85 at a temperature of T = 0.3. +10−1 +100 +101 +102 +N 1/xhr +10−6 +10−4 +10−2 +100 +ξSG/N +σ = 0.75 +T = 0.55 +x = 2.7077 +N = 16384 +N = 32768 +Figure 30. +A complete finite size scaling plot of ξSG as a +function of magnetic field hr, for σ = 0.75 at a temperature +of T = 0.55, showing our two largest system sizes. +it is as if the AT transition hAT(T) = 0. +We would +expect this formula to apply until finite size effects limit +its growth, which will occur when ξ is of O(L) (or O(N) +in our one-dimensional system). Identifying ξSG with ξ, +Figs. 25 and 26 show that the Imry-Ma fit indeed works +well at the larger fields; the data for the larger hr collapse +nicely onto a power law form as predicted by Eq. (24) +for all sizes N. It only departs from this formula when +10−1 +100 +101 +102 +N 1/xhr +10−5 +10−4 +10−3 +10−2 +10−1 +100 +ξSG/N +σ = 0.85 +T = 0.3 +x = 2.2019 +N = 2048 +N = 4096 +Figure 31. +A complete finite size scaling plot of ξSG as a +function of magnetic field hr, for σ = 0.85 at a temperature +of T = 0.3, showing our two largest system sizes. +10−1 +101 +N 1/xhr +10−5 +10−4 +10−3 +10−2 +10−1 +χSG/N z +σ = 0.75 +T = 0.55 +x = 2.7077 +xχ = 1.6114 +z = 0.5951 +N = 128 +N = 256 +N = 512 +N = 1024 +N = 2048 +N = 4096 +N = 8192 +N = 16384 +N = 32768 +Figure 32. +A complete finite size scaling plot of χSG as a +function of magnetic field hr, for σ = 0.75 at a temperature +of T = 0.55. +ξSG becomes of order N, when finite size corrections to +the Imry-Ma formula are needed. Also TNT effects (see +Sec. VII) produce corrections to the Imry-Ma formula +when ξSG is of O(N) unless N = L > L∗. The crossover +scale L∗ is thought to be large, especially as σ approaches +2/3 (or d → 6) [3]. +To allow for finite size effects on the Imry-Ma formula +we use the analogue of Eq. (21a) with hAT = 0 and ν = x + +15 +10−1 +100 +101 +102 +N 1/xhr +10−5 +10−4 +10−3 +10−2 +10−1 +χSG/N z +σ = 0.75 +T = 0.55 +x = 2.7077 +xχ = 1.6114 +z = 0.5951 +N = 16384 +N = 32768 +Figure 33. +A complete finite size scaling plot of χSG as a +function of magnetic field hr, for σ = 0.75 at a temperature +of T = 0.55, for our two largest system sizes. +10−1 +101 +N 1/xhr +10−6 +10−5 +10−4 +10−3 +10−2 +10−1 +χSG/N z +σ = 0.85 +T = 0.3 +x = 2.2019 +xχ = 1.8919 +z = 0.8592 +N = 128 +N = 256 +N = 512 +N = 1024 +N = 2048 +N = 4096 +Figure 34. +A complete finite size scaling plot of χSG as a +function of magnetic field hr, for σ = 0.85 at a temperature +of T = 0.3. +to write: +ξSG/N = X(N 1/xhr). +(26) +Our results for σ = 0.75 are shown in Fig. 28 and for σ = +0.85 are shown in Fig. +29. There are clearly finite size +corrections to this formula. It is a formula which formally +would be expected to hold in the scaling limit of N → ∞ +with N 1/xhr fixed. The crossover function X(y) ∼ 1/yx +10−1 +100 +101 +102 +N 1/xhr +10−6 +10−5 +10−4 +10−3 +10−2 +10−1 +χSG/N z +σ = 0.85 +T = 0.3 +x = 2.2019 +xχ = 1.8919 +z = 0.8592 +N = 2048 +N = 4096 +Figure 35. +A complete finite size scaling plot of χSG as a +function of magnetic field hr, for σ = 0.85 at a temperature +of T = 0.3, for our two largest system sizes. +0.3 +0.4 +0.5 +0.6 +0.7 +1/N z−(2σ−1) +0.2 +0.4 +0.6 +0.8 +h∗(N, 2N) N 1/x +χSG(σ = 0.7) +ξSG(σ = 0.7) +χSG(σ = 0.75) +ξSG(σ = 0.75) +χSG(σ = 0.85) +ξSG(σ = 0.85) +Figure 36. A plot of h∗(N, 2N)N 1/x versus 1/N ω, for σ = +0.70, 0.75 and 0.85. The values of x and ω, which is obtained +from Eq. (36) were taken from Table I. +when y is large, in order to recover Eq. (24). It goes to +a constant when y → 0. However, a closer look at our +two largest system sizes N = 16384 and N = 32768 at +σ = 0.75 (Fig. 30) and our two largest system sizes at +σ = 0.85, N = 2048 and N = 4096 (Fig. 31) shows that +the finite size corrections are becoming small, and are +smaller the further the system is away from the mean- +field region. If one moves in the other direction, towards + +16 +103 +104 +N +0.24 +0.26 +0.28 +0.30 +0.32 +0.34 +0.36 +R +σ = 0.6, T = 0.6 +Figure 37. A plot of R = χSG(h∗(N, 2N), N)/N 1/3 versus N, +for σ = 0.6 at a temperature of T = 0.6. +103 +104 +N +0.28 +0.29 +0.30 +0.31 +0.32 +0.33 +0.34 +R +σ = 0.75, T = 0.55 +Figure 38. A plot of R = χSG(h∗(N, 2N), N)/N 2σ−1 versus +N, for σ = 0.75 at a temperature of T = 0.55. +the start of the mean-field region σ = 2/3, the finite size +corrections are larger, as seen in Fig. 40 for σ = 0.70. +The finite size scaling form for these corrections to the +scaling of Eq. (26) will be of the form +ξSG +N += X(N 1/xhr) + N −ωH(N 1/xhr), +(27) +where ω is the correction to scaling exponent. However, +TNT effects (see Sec. VII) produce large further correc- +tions to these asymptotic forms when L < L∗. Since in +103 +N +0.2200 +0.2225 +0.2250 +0.2275 +0.2300 +0.2325 +0.2350 +0.2375 +R +σ = 0.85, T = 0.3 +Figure 39. A plot of R = χSG(h∗(N, 2N), N)/N 2σ−1 versus +N, for σ = 0.85 at a temperature of T = 0.3. +our studies L∗ is probably larger than the length N of +our system, at least for σ = 0.75, the scaling form of +Eq. (27) does not work in the region where ξSG is of or- +der N (see Fig. 42). For σ = 0.85 where L∗ is expected +to be smaller, Fig. 43 hints that Eq. (27) might apply as +the plots at adjacent sizes for the larger N values seem +to be getting closer together as N is increased, which is +a feature predicted by Eq. (27). +In Fig. +27 we show a similar plot to those in Figs. +25 and 26 but for the case of σ = 0.60. Notice however +that because of the AT transition at this value of σ, at +which ξSG would diverge to infinity as N → ∞ at some +finite field hr = hAT(T), a shoulder above the dashed +line has started to appear which is the beginning of this +divergence. Such a feature is absent in the figures for +both σ = 0.75 and at σ = 0.85. +The spin glass susceptibility according to the droplet +picture is a similar generalization of the finite size scaling +form of Eq. (14a): +χSG +N z = C(N 1/xhr), +(2/3 ≤ σ < 1). +(28) +The crossover function C(y) ∼ 1/yxχ when y is large, so +that then χSG ∼ ξz and becomes independent of N. Its +form is then +χSG ∼ 1/hxχ +r , +(29) +which implies that xχ = xz. +In the opposite limit as +y → 0, C(y) goes to a finite constant. The exponent z +depends upon whether we are dealing with short-range +interactions, (such as nearest-neighbor interactions) or +with the long-range interactions employed in this paper. +For short-range interactions, the average value of χ2 +ij falls + +17 +Table I. Values of the exponents xχ, x, and z obtained from +our simulations for different values of σ and T. +σ +T +xχ +x +z +0.7 +0.6 +1.5747 ± 0.0009 +3.3220 ± 0.0413 +0.4740 ± 0.0062 +0.75 +0.55 +1.6114 ± 0.0005 +2.7077 ± 0.0531 +0.5951 ± 0.0119 +0.85 +0.3 +1.8919 ± 0.0014 +2.2019 ± 0.0152 +0.8592 ± 0.0066 +off with spin separation rij as +χ2 +ij ∼ q(T)2T +Υ(T)rθ +ij +, +(30) +[45, 46]. This result applies in the zero-field spin glass +state. Then as, +χSG = 1 +N +N +� +i,j=1 +χ2 +ij, +(31) +so in d dimensions for the zero-field spin glass χSG ∼ +Ld−θ. Hence +z = d − θ +d +, +(32) +in order to recover the result χSG → N z as N 1/xhr goes +to zero. We caution that this formula for z will only hold +for short-range interactions. +With long-range interactions a “droplet” is not a single +connected region but a set of isolated islands of flipped +spins [49] and this will make the decay of χ2 +ij with rij +faster than in Eq. (30). This is an effect which has not +been studied before, and so in our problem the exponent +z has to be determined by fitting the data. The results +of our determinations of the droplet exponents x, xχ and +z for the different values of σ which we have studied are +summarised in Table 1. +The resulting excellent data collapse (at least when +ξSG < N), is shown in Figs. 32, 33, +34, and +35. The +value of z was determined from the observation that when +N 1/xhr is large, χSG should be independent of N. It is +remarkable that z determined at large values of N 1/xhr +results in a decent collapse of the data in the opposite +limit where N 1/xhr → 0. Nevertheless corrections to the +Imry-Ma scaling form are visible in the figures (and are +sizeable in the region where N 1/xhr is small when viewed +in a linear plot rather than a log scale plot, (just as in the +ξSG plots Figs. 28 and 29). In the limit when N 1/xhr is +held fixed with N → ∞ the leading correction to scaling +will be +χSG +N z = C(N 1/xhr) + N −ωG(N 1/xhr), +(33) +where G(y) is an unknown scaling function and the cor- +rection to scaling exponent ω is not known with any cer- +tainty (but see Eq. (36)). +Let us suppose that the droplet picture is correct and +that (say) the spin glass susceptibility χSG is described +by Eq. (28). This equation predicts that there will be a +crossing in the plots of χSG/N 2σ−1 used in AT line criti- +cal scaling studies. (Note we are setting 2 − η = 2σ − 1). +The correction to scaling term of Eq. (33) is not needed +for this, but this correction does strongly influence where +the crossings take place for the N values which are +reached in our simulations. The crossing arises as fol- +lows. At small values of hrN 1/x, the function C goes to a +constant. It turns out that z > (2σ − 1), so χSG/N 2σ−1 +diverges as N is increased as N z−(2σ−1) as hr → 0. On +the other hand when hrN 1/x is large, χSG → 1/hz +r, so +χSG/N 2σ−1 → 1/N 2σ−1hz +r → 0 as N goes to infinity. +Because at small fields, χSG/N 2σ−1 is larger for large N, +but at bigger hr fields it is smaller at the larger N val- +ues, so there must be a crossing point. We shall denote +the crossing value between the lines at N and 2N by +h∗(N, 2N) = H. Then H is determined by the solution +of the following +χSG(H, N) +N 2σ−1 += C(N 1/xH)N z−(2σ−1) = +χSG(H, 2N) +(2N)2σ−1 += C((2N)1/xH)(2N)z−(2σ−1). +(34) +Assuming C(y) → a − by, when y → 0, it is easy to +show then that the N dependence of h∗(N, 2N) at very +large N will be as 1/N 1/x. In reality we have no data +in this region of very large N where the corrections to +scaling term in Eq. (33) can be ignored. The corrections +to scaling are numerically small but are very important +in determining the values of h∗(N, 2N). +There is a similar crossing predicted in the plots of +ξSG/N as a function of hr when Eq. (27) holds, using the +analogue of Eq. (34). In this case it is the scaling cor- +rection which causes the curves to cross, (which requires +H(0) to be negative), and for these curves the crossings +h∗(N, 2N) at very large N will decrease as 1/N 1/x+ω, +(compare with Eq. (18b)) on taking χ(y) = c − dy and +H(y) → constant as y → 0. Once again we have no data +in this very large N regime. In Fig. 36 we have plotted +h∗(N, 2N)N 1/x versus 1/N ω, assuming that ω is given by +Eq. (36). Note that the size of the corrections to scaling +∼ 1/N ω is simply not small for the values of N which we +can study, contrary to what was assumed in the above. +h∗(N, 2N)N 1/x should go to a constant as N goes to in- +finity and it is only for the case of σ = 0.85, where the +corrections to scaling are the smallest of the three cases +studied, does that look remotely possible. For the case +of σ = 0.70 the corrections look to be very large. We +conclude that for the values of σ = 0.70 and σ = 0.75, +the crossing data on h∗(N, 2N) is not close to the large +N asymptotic form predicted by the droplet picture. But +the droplet picture does predict that the existence of such +intersections. +If we only had information on the values of the cross- +ing fields h∗(N, 2N) it would be difficult to really be sure +whether the droplet picture or the RSB picture best de- +scribed the data. The results on h∗(N, 2N) alone are in- +conclusive as regards both the AT transition line picture + +18 +100 +101 +102 +N 1/xhr +10−7 +10−5 +10−3 +10−1 +101 +ξSG/N +σ = 0.7 +T = 0.6 +x = 3.3220 +N = 8192 +N = 16384 +Figure 40. +A complete finite size scaling plot of ξSG as a +function of magnetic field hr, for σ = 0.70 at a temperature +of T = 0.6 for our two largest system sizes. +and the droplet picture. While on the droplet picture +h∗(N, 2N) are predicted to go to zero as N → ∞, the +values of h∗(N, 2N) are not convincingly going to zero as +N is increased (see Fig. 36). Fortunately, there is another +way of distinguishing the two approaches, which does +not require us to reach the N values at which h∗(N, 2N) +starts to approach zero. We define +R ≡ χSG(h∗(N, 2N), N)/N 2σ−1. +(35) +(Because we only determine χSG(hr, N) at a finite num- +ber of values of hr, we use linear interpolation to calcu- +late χSG(h∗(N, 2N), N) using the χSG(hr, N) values at +the two determined values of hr which lie on either side +of h∗(N, 2N)). On the phase transition picture, R should +approach a finite constant as N → ∞. On the droplet +picture R should increase as N z−(2σ−1) as N → ∞. For +σ = 0.60 where an AT line is expected R should go to +a constant but at the N values studied it actually still +appears to be decreasing (see Fig. 37) and has yet to be- +come constant, presumably due to finite size effects. This +indicates that trying to determine whether σ = 2/3 is the +exact value at which the crossover to droplet scaling be- +havior will also be challenging from the side below 2/3. +However, for σ = 0.75, Fig. 38 shows that R is clearly +increasing with N for large N values. But if we had had +only data for system sizes < 2048 we might have indeed +concluded that there was good evidence for an AT transi- +tion in that R seemed to be an N independent constant. +While at the sizes we can reach R is clearly increasing +with N it has yet to reach its asymptotic form of in- +crease as N z−(2σ−1). The quantity R also increases with +N for σ = 0.70 and σ = 0.85, (see for example Fig. 39). +In order for χSG to match as σ → 2/3 from either the +100 +101 +102 +N 1/xhr +10−4 +10−3 +10−2 +10−1 +χSG/N z +σ = 0.7 +T = 0.6 +x = 3.3220 +xχ = 1.5747 +z = 0.4740 +N = 8192 +N = 16384 +Figure 41. A complete finite size scaling plot of χSG a function +of magnetic field hr, for σ = 0.70 at a temperature of T = 0.6 +for our two largest system sizes. +mean-field side, (where z = 1/3) with its value in the +non-mean field region, we would expect that z should +approach 1/3 as σ → 2/3 from above. +At σ = 0.85, +z ≈ 0.8409, at σ = 0.75, z ≈ 0.6065, while at σ = 0.70, +we find z ≈ 0.4737. Thus it seems quite plausible that +z could approach 1/3 as σ → 2/3 from above. Then the +combination z − (2σ − 1) would approach zero in this +limit, which means that the divergence of R with N will +become harder and harder to see as σ approaches 2/3. +We conclude that it will be challenging to do numerical +work which shows that the AT line disappears at precisely +σ = 2/3. On the mean-field side of 2/3 the correction to +scaling exponent ω = 1/3 − (2σ − 1). It therefore seems +natural to expect that on the non-mean field regime +ω = z − (2σ − 1). +(36) +If valid, this would imply that corrections to scaling +should be larger at σ = 0.70 than at σ = 0.75, and this +is what we observed in Figs. 40 and 41, in comparison +with Figs. 30 and 33. +In the presence of a genuine AT transition, as hr is +reduced one would pass through three regions: first the +paramagnetic state at larger values of hr, then the criti- +cal region, then the low-temperature phase with RSB at +smaller values of hr. The good data collapse for all val- +ues of hr using Eq. (26), and Eq. (28) shows that at any +finite value of hr there is just one region, the paramag- +netic region. Studying “intersections” as in Sec. V is an +attempt to find the critical region. But the intersections +at finite values of hr for σ = 0.75 and σ = 0.85 are not +signs of a genuine phase transition, but at least in the +case of χSG these crossings are also just a consequence of +droplet scaling. The behavior of h∗(N, 2N) as a function + +19 +10−2 +10−1 +100 +N 1/xhr +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +ξSG/N +σ = 0.75 +T = 0.55 +x = 2.7077 +N = 128 +N = 256 +N = 512 +N = 1024 +N = 2048 +N = 4096 +N = 8192 +N = 16384 +N = 32768 +Figure 42. A finite size scaling plot of ξSG as a function of +magnetic field hr, for σ = 0.75 at a temperature of T = 0.55. +of N is greatly complicated by finite size effects and will +only become clear at much larger N values than those +which we have been able to study. +Because on the droplet picture there is no AT line and +so one is always in the paramagnetic phase at any non- +zero field (just as in a ferromagnet). +However, length +scales like ξSG become very large as hr → 0 for temper- +atures T < Tc(hr = 0). Once they become comparable +to the system dimensions L and one is in the regime +hr < h∗(N, 2N), the system will have many of the fea- +tures which might be associated with being in the bro- +ken replica symmetric phase which is envisaged to exist +below the AT line. For physical systems in three dimen- +sions the relevant length scale is not the linear dimension +of the system L, but the linear dimension of a fully equi- +librated region. This may explain why both simulations +and experiments have failed for many years to resolve the +debate. +Might it be possible to find by simulations whether the +borderline between RSB ordering and droplet ordering is +at σ = 2/3, which is the equivalent of d = 6 with short- +range interactions? To this end we looked at the case +of σ = 0.70. We found from studying the crossings of +ξSG and χSG for the zero field case that the zero field +transition temperature is ≈ 0.724. Figs. 40 and 41 show +our attempt to collapse the data with the droplet scaling +forms. Clearly the effects of corrections to scaling are +larger than was the case at σ = 0.75 in Figs. 30 and +33. This is in accord with Eq. (36) which predicts that +the correction to scaling exponent ω will go to zero as +σ → 2/3 if also z → 1/3 as expected. We conclude that +it will be difficult to provide good numerical evidence +that σ = 2/3 is the lower critical dimension of the AT +transition. +10−2 +10−1 +100 +101 +N 1/xhr +0.5 +1.0 +1.5 +2.0 +ξSG/N +σ = 0.85 +T = 0.3 +x = 2.2019 +N = 128 +N = 256 +N = 512 +N = 1024 +N = 2048 +N = 4096 +Figure 43. A finite size scaling plot of ξSG as a function of +magnetic field hr, for σ = 0.85 at a temperature of T = 0.3. +VII. +TNT VERSUS THE DROPLET SCALING +PICTURE +Newman and Stein [50] (see also the recent review +[51]), have suggested that the ordered phase of spin +glasses in finite dimensions will fall into one of 4 cate- +gories, (and which one might depend on the dimension- +ality d of the system): The RSB state is one of these, +and is somewhat similar to that envisaged by Parisi for +the SK model, but there is also the chaotic pairs state +picture of Newman and Stein. In both of these pictures +there is an AT transition. The other two pictures are +the so-called TNT picture of Krzakala and Martin [35] +and Palassini and Young [36] and the droplet scaling pic- +ture [44–46]. In neither the TNT picture nor the droplet +scaling picture is there an AT transition. In the droplet +picture the Parisi overlap function P(q) is trivial, consist- +ing of two delta functions at ±qEA in zero field, whereas +in the TNT picture the form of P(q) is quite similar to +the non-trivial (NT) form which Parisi found for the SK +model. +The TNT picture accounts for the non-trivial +form of the Parisi overlap function by postulating that +there exist droplets of the linear size L of the system, +which contain O(Ld) spins, and which do not have a free +energy of order Lθ (as they would in the droplet scaling +picture), but which have instead a free energy of O(1). It +is the presence of such droplets which makes P(q) non- +trivial, which is a feature observed in all simulations of +it to date. +In a recent paper [3] one of us argued that once the +linear dimension of the system became larger than a +crossover length L∗ the non-trivial behavior observed in +P(q) will change to the trivial form predicted by droplet +scaling. Estimates of L∗ in d = 3 suggest it might be + +20 +large, of the order of several hundred lattice spacings +and it is probably the case that to date the regime where +L > L∗ has not been reached. Furthermore it was sug- +gested that as d → 6, L∗ would grow towards infinity, +as the droplets of O(1) evolve to the O(1) excitations in +the Parisi RSB solution, where the pure states have free +energies which differ from each other by O(1). In our +one dimensional proxy system we would therefore expect +to find that L∗ is much larger when σ = 0.75 than it is +when σ = 0.85. +In this paper there are TNT-like effects visible in the +behavior of ξSG/N in the region where ξSG is of O(N) +(see Figs. +28 and 29). +When ξSG is of order N the +droplets which are important are those of size N and if +L < L∗ some of these will have free energy of O(1) rather +than Lθ. As a consequence the good scaling collapse of +the data visible when ξSG/N ≪ 1 will be lost. In Figs. +42 and 43 we have plotted ξSG/N on a linear scale versus +N 1/xhr focussing only on the region where ξSG/N is of +O(1). If the droplet scaling collapse had been good and +of the form of Eq. (27) then as N is increased the collapse +should get better and better. In fact due to TNT effects +the data in Fig. 42 for σ = 0.75 show the opposite trend, +and the lines get further apart with increasing N in the +region where ξSG is of O(N). +However, for σ = 0.85 +Fig. 43 shows the lines seem to be getting closer with +increasing N. +It suggests that for this value of σ we +are getting into the region where L > L∗ when droplet +scaling applies even when ξSG is of O(N). Data at larger +values of N than 4096 would be nice to confirm this trend +but because these simulations have to be done at quite +low temperatures compared to those for σ = 0.75 it will +be challenging to do this. Despite this limitation on the +size of N which can be reached for σ = 0.85, there is +evidence that for it, TNT and finite size scaling effects +are less troublesome than for σ = 0.75, despite the fact +that much larger values of N can be studied at this σ +value. +VIII. +SUMMARY AND CONCLUSIONS +In this paper, we have studied the phase transitions +in the one-dimensional power-law diluted XY spin glass, +both in the zero-field limit, and in the presence of a mag- +netic field random in the component directions. Whether +or not an AT line exists for various values of the param- +eter σ is a question of fundamental interest. To address +this, we have performed large scale Monte-Carlo sim- +ulations using a new heatbath algorithm, described in +Appendix A. This algorithm hopefully speeds up equi- +libration, so cutting computational costs. We certainly +do gain some advantage in terms of computational time +due to the smaller number of components of XY spins +compared to those of the Heisenberg model. Alas, the +heatbath algorithm for XY spins suffers from an intrin- +sic disadvantage. Because our algorithm has to generate +two random numbers during each Monte Carlo step, the +benefits of the smaller number of components are largely +counterbalanced by the additional labor involved in the +heatbath step. We were unable to go to larger system +sizes than in the corresponding work with Heisenberg +spins [18]. +The largest system sizes that we are able +to simulate are: N = 16384 for σ = 0.6, N = 32768 for +σ = 0.75, while the largest N for σ = 0.85 was 4096. The +total CPU time spent in generating all the data that we +presented at fixed hr and varying T was 1183636.2 hrs, +which is 135.12 years. The total CPU time consumed in +generating the data at fixed T and varying hr was 96101.6 +days which is 263.29 years. Thus despite the algorithm +not producing significant dividends, we are able to study +fairly large system sizes owing to the expenditure of a +large amount of computer time. +The results from our work are broadly in accord with +those for the corresponding Heisenberg spin glass model. +For σ = 0.6, which is in the mean-field regime, we find +a phase transition in the absence of an external mag- +netic field, and in the presence of a magnetic field, which +indicates the existence of an AT line. The location of +the AT line is close to the mean-field predictions. For +σ = 0.75, which is in the non-mean-field regime, the con- +ventional data collapse suggests the existence of an AT +line, but the behavior of the intersections as a function +of N indicate that the data is not close to its large N +asymptotic form. The estimated location of the AT field +based upon intersections that we get from our data at +σ = 0.75 is strikingly smaller than estimates based on +the mean-field theory formulas. For σ = 0.85, which is +deep in the non-mean-field regime and corresponds to a +space dimension of about 3, our data are consistent with +the absence of an AT line. In this case there is no crossing +of the curves of χSG/N 2−η versus T at various N values. +But confusingly intersections h∗(N, 2N), as a function of +hr, seem to exist, whereas intersections T ∗(N, 2N) are +absent at least for σ = 0.85. +However, for σ = 0.75 and for σ = 0.85 we found that +the droplet picture provided a much better description +of our data from that obtained assuming the existence +of an AT transition line. The Imry-Ma formula for the +field dependence of ξSG works well until ξSG becomes +comparable to the system size. A similar behavior was +reported for the Ising spin glass at σ = 0.75 in Ref. [48]. +A finite-size scaling formulation was developed to treat +the data at small fields when ξSG is comparable to the +system size N, and with it an excellent collapse of all +our data on ξSG and χSG was obtained. We showed that +droplet scaling predicts the existence of the intersections +h∗(N, 2N). Our data unfortunately does not extend to +values of N large enough to be in the asymptotic region +where the N-dependence of h∗(N, 2N) is simple. Fortu- +nately there exists a way of testing whether the intersec- +tions are due to an AT transition or are just those pre- +dicted by droplet scaling, which is to study the N depen- +dence of R = χSG/N 2σ−1, calculated at h∗(N, 2N), and +this test supports the droplet picture provided N > 1024 +at σ = 0.75. Thus it is only for large systems that one + +21 +can obtain good evidence for the droplet picture. +We now summarize our main results. The strongest +evidence for droplet scaling is the success of the Imry-Ma +formula for the field dependence of ξSG for σ = 0.75 and +σ = 0.85 (see Figs. 30 and 31). If droplet scaling works, +then no AT line is to be expected. When ξSG ∼ N there +are visible sizeable corrections to the Imry-Ma formula +which are related to TNT effects. However for σ = 0.85 +there is tentative evidence in Fig. 43 that if even larger +systems could be studied then the TNT effects might be +absent, and so there could exist a length scale L∗ above +which TNT effects become unimportant (see Ref. [3]). +If instead of droplet scaling one assumes that there is +an AT phase transition then the usual finite size scaling +plots used to determine hAT as in Fig. 15 for σ = 0.75 +are unsatisfactory: for example the values of hAT which +would be derived from the crossings of ξSG and χSG as +N becomes large look to be significantly different. In the +equivalent data plot for σ = 0.60 (see Fig. +8) they are +in good agreement. Furthermore the quantity R of Eq. +(35) should approach a constant as N → ∞ if there is a +genuine AT transition, but instead for the cases σ = 0.75 +(Fig. 38) and 0.85 (Fig. 39), it is increasing with N once +N becomes large enough. +The simulations of this paper provide numerical evi- +dence that the AT line and hence RSB is absent in spin +glasses below six dimensions. What is now needed is an +explanation of why this might be the case. Better still +would be a rigorous proof that the lower critical dimen- +sion for replica symmetry breaking is six. Our work indi- +cates that showing that σ = 2/3 is the precise value of the +critical value of σ will be challenging using simulations +as finite size effects are large in its vicinity. +ACKNOWLEDGMENTS +We are grateful to the High Performance Comput- +ing (HPC) facility at IISER Bhopal, +where large- +scale +calculations +in +this +project +were +run. +We +thank Peter Young and Dan Stein for helpful discus- +sions. +B.V is grateful to the Council of Scientific +and Industrial Research (CSIR), India, for his PhD +fellowship. +A.S acknowledges financial support from +SERB via the grant (File Number: CRG/2019/003447), +and from DST via the DST-INSPIRE Faculty Award +[DST/INSPIRE/04/2014/002461]. +Appendix A: The simulation method +We now give some technical aspects of how the simula- +tions are run. In the simulations we start with a random +initial configuration and allow it to evolve according to +the prescription given in this section. To incorporate par- +allel tempering, we simultaneously simulate NT copies of +the system over NT different temperatures ranging from +Tmin ≡ T1 to Tmax ≡ TNT . +In order to facilitate the +computation of the observables outlined in this section, +it is convenient to simulate 4 sets of NT copies (2 for +hr = 0), which we label (1),(2),(3), and (4). We perform +overrelaxation, heatbath and parallel tempering sweeps +over all these copies keeping track of the labels appropri- +ately. +For every 10 overrelaxation sweeps we perform +1 heatbath and 1 parallel tempering sweep, since the +overrelaxation sweep involves a significantly lower com- +putational cost, and is known to speed up equilibration. +The parameters of the simulations are shown in Tables II +and III. Once the system reaches equilibrium, we perform +the same number of sweeps in the measurement phase, +so Nsweep is the total number of sweeps over which the +simulation is run, inclusive of both the equilibration and +measurement phases. The last column in the table shows +the amount of computer time expended to generate the +data corresponding to the parameters in that row. +In +the measurement phase, we perform one measurement +on the system for every 4 sweeps. The following sections +contain the details of our Monte Carlo simulation pro- +cedures. +In order to equilibrate the system as quickly +as possible, we perform three kinds of sweeps: overre- +laxation or microcanonical sweeps, heatbath sweeps, and +parallel tempering sweeps. +1. +Overrelaxation sweep +We sweep sequentially through all the lattice sites and +compute the local field Hi = � +j JijSj+hi at a particular +lattice site. The new spin direction S′ +i at the ith lattice +site is taken to be the mirror image of the vector Si about +Hi, i.e., +S′ +i = −Si + 2Si · Hi +H2 +i +Hi. +(A1) +Since S′ +i · Hi = Si · Hi, the energy of the system does +not change due to these sweeps. Hence these sweeps are +also called microcanonical sweeps. These sweeps help us +in sampling out the microstates with the same energy. +The process of equilibration speeds up when we include +overrelaxation sweeps along with the other sweeps [33, +52]. +2. +Heatbath sweep +The overrelaxation sweeps generate states with the +same energy and hence they cannot directly equilibrate +the system. Therefore, we also perform a heatbath sweep +for every 10 microcanonical sweeps. Similar to the micro- +canonical case, we sweep sequentially through the lattice. +To equilibrate the system, the angle θ between Hi and +S +′ +i should be sampled out from the Boltzmann distribu- +tion given by +fΘ(θ) = e−βEi +Z += eβHiSi cos θ +Z += ew cos θ +Z +, +(A2) + +22 +where w = βHiSi and +Z = +π +� +−π +eβHiSi cos θ dθ +(A3) +is the normalizing constant. The simplest way to do this +is to equate the cumulative density function (CDF) of θ, +FΘ(θ), to that of a uniform distribution: +FΘ(θ) = +θ +� +−π +fΘ(θ′) dθ′ = Π(r1) = r1, +(A4) +where r1 is a random variable sampled from a uniform +distribution in the interval (0, 1). The value of θ can be +obtained by simply inverting this function to get +θ = F −1 +Θ (r1). +(A5) +This method works well with the Heisenberg spins +as fΘ(θ) is integrable, which gives an invertible CDF +FΘ(θ) [18, 28]. Since the probabililty density function +(PDF) fΘ(θ) for the XY spin glasses given by Eq. (A2) +is not exactly integrable, this method cannot be used. +To overcome this problem and to sample out θ from the +Boltzmann distribution (Eq. (A2)) in as few a number of +sweeps as possible, we develop a heatbath sweep based +on the rejection method [53]. We generate two random +numbers r1 ∈ uniform(−π, π) and r2 ∈ uniform(0, fmax). +If r2 < fθ(r1), we accept the move, i.e., take θ = r1. Else, +we reject the move and generate another pair of random +numbers (r1, r2). This process is repeated until we find +an acceptable value of r1. A graphical representation for +this method is shown in Fig. 44. The new spin direction +S′ +i in Cartesian co-ordinates is given by: +S′ +x = cos(θ + θH), +(A6a) +S′ +y = sin(θ + θH), +(A6b) +where θH is the angle made by the Hi vector with the X- +axis. Since the generation of random numbers is involved, +this sweep is computationally costlier than others. Hence +we perform more microcanonical sweeps than heatbath +sweeps. +3. +Parallel tempering sweep +Spin glasses have a complex free energy landscape due +to which, at low temperatures, they tend to get stuck in- +side metastable valleys, and true equilibration consumes +a lot of time. At high temperatures, the system can eas- +ily escape the valley due to thermal fluctuations, and +so equilibration is quick. To equilibrate the system in +as small a number of moves as possible, we perform +one parallel tempering sweep for every 10 overrelaxation +sweeps [28, 33]. To benefit from the parallel tempering +algorithm [54, 55], we simultaneously run the simulation +−2 +0 +2 +θ +0.0 +0.1 +0.2 +0.3 +0.4 +fΘ(θ) +fmax +(r1, r2) +(r1, fΘ(r1)) +(r1, r2) +(r1, fΘ(r1)) +w = βHS = 1.5 +Rejected +Accepted +Figure 44. Graphical representation of the rejection method. +We randomly pick a point (r1, r2) within the rectangle from a +uniform distribution. If the point lies under the fΘ(θ) curve +given by Eq. (A2), then the point is accepted, and θ is taken +to be r1. Otherwise, the point is rejected. +for NT copies of the system at NT different temperatures +T1 < T2 < T3 < · · · < TNT . The minimum temperature +T1 is the low temperature at which we are interested in +studying the behavior of the system, and the maximum +temperature TNT is high enough that the system equi- +librates very fast. We perform overrelaxation and heat- +bath sweeps separately on each of the NT copies of the +system. In the parallel tempering sweep, we compare the +energies of two spin configurations at adjacent tempera- +tures, Ti and Ti+1, starting from the smallest tempera- +ture T1. We swap these two spin configurations such that +the detailed balance condition is satisfied. The Metropo- +lis probability for such a swap is +P(T swap) = min{1, exp(∆β∆E)} +(A7) += +� +exp(∆β∆E) +(if ∆β∆E < 0), +1 +(otherwise), +(A8) +where ∆β += +1/Ti − 1/Ti+1 and ∆E += +Ei(Ti) − +Ei+1(Ti+1). In this way, a given set of spins performs +a random walk in temperature space. +4. +Checks for equilibration +In order to check whether the system has reached equi- +librium, we have used a convenient test [56] which is pos- +sible because of the Gaussian nature of the interactions +and the onsite external magnetic field. The relation +U = zJ2 +2T (ql − qs) + h2 +r +T +� +q − |S|2� +, +(A9) + +23 +Table II. Parameters of the simulations. Nsamp is the number of disorder samples, Nsweep is the number of over-relaxation +Monte Carlo sweeps for a single disorder sample. The system is equilibrated over the first half of the sweeps, and measurements +are done over the last half of the sweeps with a measurement performed every four over-relaxation sweeps. Tmin and Tmax are +the lowest and highest temperatures simulated, and NT is the number of temperatures used for parallel tempering. +σ +hr +N +Nsamp +Nsweep +Tmin +Tmax +NT +ttot(hrs) +0.6 +0 +128 +10000 +512 +0.6 +1 +18 +0.49 +0.6 +0 +256 +8000 +1024 +0.6 +1 +22 +2.23 +0.6 +0 +512 +6400 +2048 +0.6 +1 +22 +6.46 +0.6 +0 +1024 +8000 +4096 +0.6 +1 +26 +40.74 +0.6 +0 +2048 +3840 +8192 +0.6 +1 +24 +105.41 +0.6 +0 +4096 +3200 +16384 +0.6 +1 +27 +571.49 +0.6 +0 +8192 +3200 +32768 +0.6 +1 +30 +3776.85 +0.6 +0 +16384 +2600 +65536 +0.64 +0.98 +32 +18225.5 +0.6 +0.1 +128 +9600 +2048 +0.5 +0.8 +21 +7.75 +0.6 +0.1 +256 +9600 +2048 +0.5 +0.8 +21 +15.89 +0.6 +0.1 +512 +9600 +8192 +0.5 +0.8 +22 +85.67 +0.6 +0.1 +1024 +8000 +16384 +0.5 +0.8 +22 +414.47 +0.6 +0.1 +2048 +7200 +32768 +0.5 +0.8 +26 +2029.23 +0.6 +0.1 +4096 +7200 +65536 +0.5 +0.8 +24 +10014.8 +0.6 +0.1 +8192 +4380 +131072 +0.55 +0.8 +25 +34810.6 +0.6 +0.1 +16384 +7128 +262144 +0.55 +0.8 +28 +224425 +0.75 +0 +128 +12800 +1024 +0.35 +0.85 +21 +1.6 +0.75 +0 +256 +12800 +2048 +0.35 +0.85 +24 +7.21 +0.75 +0 +512 +8000 +8192 +0.35 +0.85 +24 +35.22 +0.75 +0 +1024 +8000 +16384 +0.35 +0.85 +24 +196.9 +0.75 +0 +2048 +6400 +32768 +0.35 +0.85 +25 +774.55 +0.75 +0 +4096 +4880 +65536 +0.35 +0.85 +27 +3405.2 +0.75 +0 +8192 +3000 +131072 +0.38 +0.82 +30 +14290.9 +0.75 +0.05 +128 +19200 +8192 +0.28 +0.6 +21 +45.27 +0.75 +0.05 +256 +16000 +16384 +0.28 +0.6 +20 +133.35 +0.75 +0.05 +512 +13600 +32768 +0.28 +0.6 +20 +464.77 +0.75 +0.05 +1024 +11000 +65536 +0.28 +0.6 +21 +2075.57 +0.75 +0.05 +2048 +10920 +262144 +0.28 +0.6 +24 +21314.3 +0.75 +0.05 +4096 +10800 +524288 +0.3 +0.58 +26 +123093 +0.75 +0.05 +8192 +5320 +1048576 +0.32 +0.54 +32 +364358 +0.85 +0 +128 +12800 +8192 +0.2 +0.5 +30 +17.97 +0.85 +0 +256 +12800 +16384 +0.2 +0.5 +32 +72.15 +0.85 +0 +512 +12800 +65536 +0.2 +0.5 +30 +752.36 +0.85 +0 +1024 +12800 +131072 +0.2 +0.5 +30 +3219.93 +0.85 +0 +2048 +8000 +262144 +0.2 +0.5 +30 +9504.05 +0.85 +0 +4096 +6480 +524288 +0.24 +0.48 +30 +40322.4 +0.85 +0.02 +128 +8000 +65536 +0.1 +0.4 +30 +194.33 +0.85 +0.02 +256 +4000 +131072 +0.1 +0.4 +32 +470.39 +0.85 +0.02 +512 +4400 +524288 +0.1 +0.4 +34 +4780.6 +0.85 +0.02 +1024 +3000 +2097152 +0.1 +0.4 +35 +30356.9 +0.85 +0.02 +2048 +1800 +4194304 +0.16 +0.4 +36 +84056 +0.85 +0.05 +128 +2000 +65536 +0.1 +0.4 +30 +67.07 +0.85 +0.05 +256 +4000 +131072 +0.1 +0.4 +32 +604.8 +0.85 +0.05 +512 +3500 +524288 +0.1 +0.4 +36 +4958.17 +0.85 +0.05 +1024 +3120 +2097152 +0.1 +0.4 +36 +28028.1 +0.85 +0.05 +2048 +3240 +4194304 +0.16 +0.4 +36 +151503 +is valid in equilibrium. Here +U = 1 +N [⟨H⟩]av += − 1 +N +� +�� +⟨i,j⟩ +ϵijJij ⟨Si · Sj⟩ + +� +i,µ +hµ +i ⟨Sµ +i ⟩ +� +� +av +(A10) +is the average energy per spin, q = +1 +N +� +i +[⟨Si⟩ · ⟨Si⟩]av +is +the +Edwards-Anderson +order +parameter, +ql += +1 +Nb +� +⟨i,j⟩ +� +ϵij ⟨Si · Sj⟩2� +av is the “link overlap”, and qs = +1 +Nb +� +⟨i,j⟩ +� +ϵij +� +(Si · Sj)2�� +av is the “spin overlap”, where + +24 +Table III. Parameters of the simulations done at fixed temperature T and varying field hr. N(hr) is the number of values of +field taken in the range hr(min,max). The equilibration times are different for different values of the field hr, which lie in the +range Nsweep(min,max). The number of disorder samples for different fields lie in the range Nsamp(min,max). ttot is the total +CPU time consumed in hours to generate data for a particular system size. +σ +T +N +hr(min,max) +N(hr) +Nsweep(min,max) +Nsamp(min,max) +ttot(hrs) +0.6 +0.6 +128 +(0.010, 9.000) +32 +(2048, 2048) +(2000, 64000) +11.44 +0.6 +0.6 +256 +(0.010, 9.000) +32 +(4096, 4096) +(2000, 48000) +55.5 +0.6 +0.6 +512 +(0.010, 9.000) +32 +(8192, 16384) +(2000, 80000) +163.91 +0.6 +0.6 +1024 +(0.010, 9.000) +32 +(4096, 65536) +(2000, 80000) +2375 +0.6 +0.6 +2048 +(0.010, 9.000) +32 +(16384, 131072) +(1200, 60000) +10259.8 +0.6 +0.6 +4096 +(0.010, 9.000) +32 +(65536, 2097152) +(960, 28600) +92052.1 +0.6 +0.6 +8192 +(0.010, 9.000) +32 +(65536, 4194304) +(400, 19428) +310948 +0.6 +0.6 +16384 +(0.010, 9.000) +32 +(131072, 4194304) +(488, 10689) +1021481 +0.7 +0.6 +128 +(0.010, 9.000) +31 +(1024, 1024) +(4000, 4000) +1.86 +0.7 +0.6 +256 +(0.010, 9.000) +31 +(2048, 2048) +(1000, 8000) +8.53 +0.7 +0.6 +512 +(0.010, 9.000) +31 +(4096, 4096) +(1000, 8000) +39.46 +0.7 +0.6 +1024 +(0.010, 9.000) +31 +(8192, 8192) +(1500, 8000) +288.41 +0.7 +0.6 +2048 +(0.010, 9.000) +31 +(16384, 16384) +(500, 8000) +559.99 +0.7 +0.6 +4096 +(0.010, 9.000) +31 +(32768, 32768) +(400, 8000) +3020.62 +0.7 +0.6 +8192 +(0.010, 9.000) +31 +(16384, 131072) +(2000, 6800) +17500.2 +0.7 +0.6 +16384 +(0.010, 9.000) +31 +(32768, 262144) +(640, 6400) +77734.3 +0.75 +0.55 +128 +(0.001, 9.000) +42 +(512, 1024) +(2000, 40000) +6.59 +0.75 +0.55 +256 +(0.001, 9.000) +42 +(1024, 2048) +(2000, 40000) +56.67 +0.75 +0.55 +512 +(0.001, 9.000) +42 +(4096, 4096) +(1000, 24000) +165.51 +0.75 +0.55 +1024 +(0.001, 9.000) +43 +(8192, 8192) +(1000, 12000) +255.75 +0.75 +0.55 +2048 +(0.001, 9.000) +43 +(16384, 16384) +(1000, 12000) +1000.52 +0.75 +0.55 +4096 +(0.001, 9.000) +43 +(32768, 32768) +(800, 12000) +5269.11 +0.75 +0.55 +8192 +(0.001, 9.000) +42 +(65536, 65536) +(800, 8000) +21410.3 +0.75 +0.55 +16384 +(0.001, 9.000) +42 +(131072, 131072) +(760, 6280) +62062 +0.75 +0.55 +32768 +(0.001, 9.000) +42 +(131072, 262144) +(512, 4995) +146627 +0.85 +0.3 +128 +(0.001, 9.000) +36 +(32768, 131072) +(2000, 30000) +264.97 +0.85 +0.3 +256 +(0.001, 9.000) +36 +(65536, 262144) +(1000, 25000) +1052.21 +0.85 +0.3 +512 +(0.001, 9.000) +36 +(131072, 524288) +(1000, 34800) +6161.52 +0.85 +0.3 +1024 +(0.001, 9.000) +36 +(262144, 1048576) +(1000, 20000) +17668 +0.85 +0.3 +2048 +(0.001, 9.000) +36 +(524288, 8388608) +(320, 3372) +68933.6 +0.85 +0.3 +4096 +(0.001, 9.000) +36 +(262144, 16777216) +(312, 5760) +439004 +Nb = Nz/2, and ϵij = 1 if the ith and jth spins are in- +teracting and is zero otherwise. The [· · · ]av in Eq. (A10) +is analytically evaluated by performing integration over +Jij and hµ +i [57] since they have Gaussian distributions. +On evaluating this integral using integration by parts, +we get Eq. (A9). As the system reaches equilibrium, the +two sides of Eq. (A9) approach their common equilibrium +value from opposite directions. +In simulations, we evaluate both sides of Eq. (A9) for +different number of Monte-Carlo sweeps (MCSs), which +increase in an exponential manner, each value being twice +the previous one. The averaging is done over the last half +of the sweeps. We initially start with a random spin con- +figuration, so the LHS of Eq. (A9) is small and the RHS +is very large. As the system gets closer to equilibrium, +these two values come closer to each other from opposite +directions. +When we notice that the averaged quanti- +ties satisfy Eq. (A9) within error bars, consistently for +at least the last two points, we declare that our system +has reached equilibrium. Once the system reaches equi- +librium, we perform the same number of sweeps in the +measurement phase, where we evaluate different quanti- +ties (given below) used to study the possible phase tran- +sitions of the system. +[1] Marc M´ezard, Giorgio Parisi, +and Miguel Angel Vira- +soro, Spin glass theory and beyond: An Introduction to +the Replica Method and Its Applications, Vol. 9 (World +Scientific Publishing Company, 1986). +[2] J R L de Almeida and D J Thouless, “Stability of the +sherrington-kirkpatrick solution of a spin glass model,” +Journal of Physics A: Mathematical and General 11, +983–990 (1978). + +25 +[3] M. A. Moore, “Droplet-scaling versus replica symmetry +breaking debate in spin glasses revisited,” Phys. Rev. E +103, 062111 (2021). +[4] V. Martin-Mayor, J. J. Ruiz-Lorenzo, B. Seoane, +and +A. P. 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B 53, 2537–2545 (1996). +[53] Richard C Larson and Amedeo R Odoni, Urban opera- +tions research, Monograph (1981). +[54] Koji Hukushima and Koji Nemoto, “Exchange monte +carlo method and application to spin glass simulations,” +Journal of the Physical Society of Japan 65, 1604–1608 +(1996). +[55] J. Machta, “Strengths and weaknesses of parallel tem- +pering,” Phys. Rev. E 80, 056706 (2009). +[56] Helmut G. Katzgraber, Matteo Palassini, +and A. P. +Young, “Monte Carlo simulations of spin glasses at low +temperatures,” Phys. Rev. B 63, 184422 (2001). +[57] A J Bray and M A Moore, “Some observations on the +mean-field theory of spin glasses,” Journal of Physics C: +Solid State Physics 13, 419–434 (1980). + diff --git a/T9E2T4oBgHgl3EQfCga9/content/tmp_files/load_file.txt b/T9E2T4oBgHgl3EQfCga9/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..592485b2aa1f1765f1168fa9a7ab2e9471f45946 --- /dev/null +++ b/T9E2T4oBgHgl3EQfCga9/content/tmp_files/load_file.txt @@ -0,0 +1,2168 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf,len=2167 +page_content='Study of the de Almeida-Thouless (AT) line in the one-dimensional diluted power-law XY spin glass Bharadwaj Vedula,1 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Moore,2 and Auditya Sharma1 1Department of Physics, Indian Institute of Science Education and Research, Bhopal, Madhya Pradesh 462066, India 2Department of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom (Dated: January 11, 2023) We study the AT line in the one-dimensional power-law diluted XY spin glass model, in which the probability that two spins separated by a distance r interact with each other, decays as 1/r2σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Tuning the exponent σ is equivalent to changing the space dimension of a short-range model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' We develop a heat bath algorithm to equilibrate XY spins;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' using this in conjunction with the standard parallel tempering and overrelaxation sweeps, we carry out large scale Monte Carlo simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' For σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6, which is in the mean-field regime above six dimensions – it is similar to being in 10 dimensions – we find clear evidence for an AT line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' For σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 and σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85, which are in the non-mean-field regime and similar to four and three dimensions respectively, our data is like that found in previous studies of the Ising and Heisenberg spin glasses when reducing the temperature at fixed field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' For σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75, there is evidence from finite size scaling studies for an AT transition but for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85, the evidence for a transition is non-existent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' We have also studied these systems at fixed temperature varying the field and discovered that at both σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 and at σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 there is evidence of an AT transition!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Confusingly, the correlation length and spin glass susceptibility as a function of the field are both entirely consistent with the predictions of the droplet picture and hence the non-existence of an AT line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' In the usual finite size critical point scaling studies used to provide evidence for an AT transition, there is seemingly good evidence for an AT line at σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 for small values of the system size N, which is strengthening as N is increased, but for N > 2048 the trend changes and the evidence then weakens as N is further increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' We have also studied with fewer bond realizations the system at σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='70, which is the analogue of a system with short-range interactions just below six dimensions, and found that it is similar in its behavior to the system at σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 but with larger finite size corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The evidence from our simulations points to the complete absence of the AT line in dimensions outside the mean-field region and to the correctness of the droplet picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Previous simulations which suggested there was an AT line can be attributed to the consequences of studying systems which are just too small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The collapse of our data to the droplet scaling form is poor for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 and to some extent also for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85, when the correlation length becomes of the order of the length of the system, due to the existence of excitations which only cost a free energy of O(1), just as envisaged in the TNT picture of the ordered state of spin glasses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' However, for the case of σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 we can provide evidence that for larger system sizes, droplet scaling will prevail even when the correlation length is comparable to the system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' INTRODUCTION While the spin glass problem at mean-field level is now well-understood [1], questions remain as to the nature of the ordered state in three dimensional spin glasses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' A key question is whether the ordered phase of real spin glasses has the broken replica symmetry features found in mean-field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' This question is most easily an- swered by finding whether on application of a magnetic field hr there is a line, the so-called de Almeida Thou- less (AT) line [2], below which in the hr − T plane there is replica symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' This line exists at mean- field level (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 1) and its possible existence in three dimensions can be studied experimentally and with sim- ulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Simulational studies of the existence of replica symmetry breaking within the zero-field spin glass state itself are plagued by finite size effects: it is expected that the difference between the predictions of droplet scaling and those of replica symmetry breaking will only become visible for very large systems (for a review see [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' A recent review of simulations, including studies of the ex- istence of the AT line, can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Right from the early days of spin glass studies there have been doubts raised as to whether the AT line existed below six dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' For example Bray and Roberts [6] attempted to do an expansion in 6 − ϵ dimensions for the critical exponents at the AT line but failed to find a stable fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' They suggested that maybe that indicated that there might be no AT line below six di- mensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' A renormalization group calculation also gave indications that the AT line was going away as d → 6 from above [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' As it is difficult to do simulations in dimensions around 6 to check these speculations, simula- tors have had to turn instead to one-dimensional models with long-range power-law interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' These models go back to Kotliar, Anderson and Stein [8], who in turn were inspired by the long-range ferro- magnet that was studied by Dyson [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The long- range power-law model has the advantage that by tuning the power-law exponent σ, one has access to both the mean-field and the regimes with non-mean-field critical behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' However, the full power-law model is expen- sive for numerics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Fortunately a clever workaround was introduced by Leuzzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' [11] where instead of the in- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='03615v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='dis-nn] 9 Jan 2023 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0 T/Tc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='4 hr/J exact AT approximate AT SK data σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6 (T∗ data) σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6 (h∗ data) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The AT line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The solid line is the exact AT line for the SK model, calculated as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The dashed line is the approximation to it of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' We have marked on the diagram the results of our simulations on the SK model, which were done to check our Monte Carlo procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The points in red and green are the results of our simulations at σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6, which lie in the mean-field region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The data on the horizontal axis for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6 are normalized to the transition temperature Tc in zero field for that value of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' For the XY SK model the AT line goes to infinity as T → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' teractions falling off as a power law, it is the probability of there being a bond between two spins that falls off as a power law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The fewer bonds in the model means that a significantly smaller computational cost is involved, thus allowing for the simulation of larger system sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' While the vast literature on spin glasses is mostly fo- cussed on Ising spins [11–16], there has been a revival of interest in classical m-component vector spin glass mod- els [5, 17–26] in the last decade or so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The XY model has m = 2 and the Heisenberg model has m = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' One of the triggers for this revival has been the finding that the infinite-range vector spin glass exhibits an AT line provided a magnetic field that is random in all the com- ponent directions is applied [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Furthermore analytical studies of the AT transition in m-vector models shows that the field theory of these AT transitions is that of the Ising spin glass [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Thus it has become possible to study the question of whether or not an Ising AT transition ex- ists in various dimensions by studying one-dimensional vector spin glasses with long-range interactions [18]!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' In this paper, we study the one-dimensional diluted XY spin glass subjected to a random vector magnetic field, with the aid of large scale Monte Carlo simula- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' While Monte Carlo simulations are a time-tested tool for the study of phase transitions in spin glasses, the exorbitant cost of equilibration makes them rather chal- lenging in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' It has been argued that vector spins tend to equilibrate faster compared to Ising spins [27], be- cause of the soft nature of the spins involved, even though the presence of more components adds to the cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The Heisenberg spin glass [5, 17–19, 27–32] has been the pop- ular vector spin to have been considered, because of the availability of the heatbath algorithm [28], which works very efficiently to equilibrate it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The XY spin glass is less effectively handled by the heatbath algorithm [33] because of the technicalities involved in inverting a prob- ability distribution for which a simple closed form ex- pression is unavailable in the XY case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' In this paper, we develop a method, which is outlined in Appendix A to perform this inversion numerically with the hope of ben- efiting from the vector nature of XY spins, while simul- taneously reducing the components to as small a number as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The improved algorithm yields mixed fruits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The gains from the reduced number of components seems to be largely counterbalanced by the additional resources con- sumed by the numerical inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' However, with the aid of extensive computational power, we are able to access system sizes comparable to those in the corresponding study with Heisenberg spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Our findings for the XY diluted spin glass closely mimic those obtained for the Heisenberg version of the same model [5, 17, 18] and the Ising spin glass in three dimensions [34] when we investi- gate crossing the possible AT line by varying T at fixed values of hr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' In the mean-field regime, (we studied here the case of σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6, which corresponds to 10 dimen- sions, which is above the upper critical dimension of 6), there is clear evidence of an Almeida-Thouless line (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' V A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' There is rather weak evidence for an Almeida- Thouless line for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 using the commonly employed finite size critical point scaling methods of analysis (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' V C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' At this value of σ, our system should be simi- lar to the Edwards-Anderson model in three dimensions with short-range interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' For the in-between case at σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 which lies in the non-mean-field regime, but closer to the mean-field boundary at σ = 2/3, our data do provide stronger evidence for a phase transition in the presence of small magnetic fields than at σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' However, by varying the magnetic field hr at fixed tem- perature T we find in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' V that at both σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 and at σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 there is quite decent evidence for an AT line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Confusingly, the field dependence of the correlation length is very well-described by the Imry-Ma prediction of the droplet picture, which implies the complete ab- sence of the AT transition!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' In the droplet picture the correlation length in a field remains finite and only di- verges as hr → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' However, when this correlation length becomes comparable to the system size L the Imry-Ma formula needs to be modified and we give in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' VI a scaling form for this modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' It is based upon the usual finite size scaling approach used in studying critical phenomena, and just as for critical phenomena we find that there are finite size corrections to this scal- ing form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' In addition to these scaling corrections there 3 are corrections which arise when ξSG ∼ L < L∗ which are of different origin and are connected to TNT effects [35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' TNT effects arise from droplets of free energy cost of O(1), which certainly exist in systems whose sizes L < L∗ [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The length scale L∗ is always large and is expected to diverge as d → 6 or as σ → 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' It is only for the case of σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 that we can reach sizes where TNT effects seem to be getting small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' These matters are discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Furthermore, we can use the droplet scaling picture to explain some of the features of the apparent AT transi- tion which arise on performing the usual finite size crit- ical scaling analyses, and show that these are the conse- quence of not studying large enough systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Unfortu- nately these arguments will only become compelling for system sizes which we cannot reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Our chief evidence for the droplet picture is its very successful prediction of the correlation length as a function of the field in the region when finite size and TNT effects are unimportant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Our claim that the evidence favors the absence of the AT line for values of σ outside the mean-field region is consistent with the attempt [21] to calculate the AT field at T = 0 using an expansion in 1/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' This indicated that as d → 6 from above in the Edwards-Anderson model, the AT field would go to zero, implying the absence of the AT line below 6 dimensions (which in the one-dimensional long-range model corresponds to 2/3 < σ < 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' For σ > 1 there is no finite temperature spin glass phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The plan of this paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' II we de- scribe the model in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' III we describe the quantities which were studied in our Monte Carlo simu- lations, the details of which are given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Our data is analysed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' V on the assumption that there is an AT transition, while in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' VI the data is analysed according to droplet scaling assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' VII we discuss the effect of TNT behavior on our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Finally in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' VIII we summarize our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' MODEL HAMILTONIAN The general Hamiltonian for vector spin glasses is: H = − � ⟨i,j⟩ JijSi · Sj − � i hi · Si , (1) where Si is the spin on the ith lattice site (i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' , N), which is chosen to be a unit vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' m rep- resents the number of components of the vector Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' In this work we concentrate on XY spins, and set m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The Cartesian components hµ i (µ = 1, 2) of the on-site external magnetic field are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='d random variables drawn from a Gaussian distribution of zero mean and variance h2 r and satisfy the relation: � hµ i hν j � av = h2 rδijδµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (2) We use the notation ⟨· · · ⟩ for thermal average and [· · · ]av for an average over quenched disorder throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The spins are arranged on a circle so the geometric distance between a pair of spins (i, j) is given by [16] rij = N π sin � π N |i − j| � , (3) which is the length of the chord connecting the ith and jth spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The interactions Jij are independent random variables such that the probability of having a non-zero interaction between a pair of spins (i, j) falls with the distance rij between the spins as a power law: P(Jij) ∝ 1 r2σ ij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (4) If the spins i and j are linked the magnitude of the inter- action between them is drawn from a Gaussian distribu- tion whose mean is zero and whose standard deviation is unity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='e: [Jij]av = 0 and � J2 ij � av = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (5) The mean number of non-zero bonds from a site is fixed to be ˜z (co-ordination number).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' So, the total number of bonds among all the spins on the lattice is fixed to be Nb = N ˜z/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' When ˜z = 6 this model mimics the 3D simple cubic lattice model and we use this value for ˜z for all the σ values studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (For σ = 0 and ˜z = N −1, the model becomes the infinite-range Sherrington- Kirkpatrick (SK) model [37]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' To generate the set of interaction pairs [11, 17] (i, j) with the desired probability we pick a site i randomly and uniformly and then choose a second site j with probabil- ity given by: pij = r−2σ ij � j̸=i r−2σ ij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (6) If the spins at i and j are already connected we repeat this process until we find a pair of sites (i, j) which have not been connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Once we find such a pair of spins, we connect them with a bond whose strength Jij is a Gaussian random variable with attributes given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' We repeat this process exactly Nb times to generate Nb pairs of interacting spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The advantage of the diluted model over the fully con- nected model is that, in a fully connected model, there are N(N − 1)/2 interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The ratio of the number of interactions of the diluted model to the fully connected model is ˜z/N which is a very small value as N becomes large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Hence it is possible to go to much larger system sizes with a diluted model as compared to a fully con- nected model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' At zero-field, the mean-field spin glass transition tem- perature for the m-component vector spin glass is given by [17, 18, 38] T MF c = 1 m � �� j � J2 ij � av � � 1/2 = √ ˜z m J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (7) 4 The approximate location of the AT line for an m- component infinite-range spin glass near the zero-field transition temperature Tc is [5] �hr J �2 = 4 m(m + 2) � 1 − T Tc �3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (8) The accuracy of this approximation for the SK model can be judged from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' A one-dimensional chain with power law diluted in- teractions for a particular value of σ is equivalent to a short-range model [14] of effective dimension deff, where deff = 2 2σ − 1, (9) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=', there is a one-to-one mapping between a long-range diluted network with exponent σ and a short-range model with space dimension deff, at least when 1/2 < σ < 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Thus when σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='60, deff = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' For the interval 2/3 < σ < 1 other relations are required [15, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' For example, for Ising spin glasses, it was suggested in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' [39] that d = 4 corresponded to σ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='790, while d = 3 corresponded to σ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='896.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Unfortunately, the mapping for the XY model has been less studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' CORRELATION LENGTHS AND SUSCEPTIBILITIES In this section we discuss the quantities which were obtained from our Monte Carlo simulations and used to extract a correlation length ξSG and the spin glass suscep- tibility χSG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The simulations themselves are described in detail in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The thermal average of a quantity is calculated using multiple replicas in the following standard way: ⟨A⟩⟨B⟩⟨C⟩⟨D⟩ = ⟨A(1)B(2)C(3)D(4)⟩ (10) where (1),(2),(3), and (4) are four copies of the system at the same temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The wave-vector-dependent spin glass susceptibility is given by [5] χSG(k) = 1 N � i,j 1 m � µ,ν �� χµν ij �2� av eik(i−j), (11) where χµν ij = � Sµ i Sν j � − ⟨Sµ i ⟩ � Sν j � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (12) The spin glass correlation length is then determined from ξSG = 1 2 sin(kmin/2) � χSG(0) χSG (kmin) − 1 �1/(2σ−1) (13) where kmin = (2π/N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' FINITE-SIZE ANALYSES ASSUMING A TRANSITION EXISTS In this section we detail the method of finite-size anal- ysis when a transition is assumed to exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' When study- ing the AT line, which is a line of phase transitions in the hr − T plane, it can be crossed on an infinite number of trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The most commonly used trajectory is the one where hr is kept constant and the temperature T is varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' In this work we also consider the trajectory in which T is kept constant and hr is varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' We refer to the zero-field transition temperature as Tc while we denote a generic transition temperature on the AT line by TAT(hr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Similarly we denote the field on the AT line by hAT(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The spin glass susceptibility χSG ≡ χSG(0) of a finite system of N spins has the finite size scaling form (near the transition temperature TAT(hr)) [5]: χSG N 2−η = C � N 1/ν (T − TAT(hr)) � , (2/3 ≤ σ < 1), (14a) χSG N 1/3 = C � N 1/3 (T − TAT(hr)) � , (1/2 < σ ≤ 2/3), (14b) where η is given by 2 − η = 2σ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' These forms are examples of finite size scaling expressions which would be expected to hold in the critical region when N → ∞, (T − TAT(hr)) → 0, with (say) N 1/ν(T − TAT(hr)) finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The scaling function C will depend on the value of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' There are always finite size corrections to these forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' For example, the corrections to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (14b) will be of the form χSG N 1/3 = C � N 1/3 (T − TAT(hr)) � + N −ωG � N 1/3(T − TAT(hr)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (15) It has been suggested [17, 40] that the correction to scal- ing exponent is given at least in the mean-field region by ω = 1/3 − (2σ − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (16) Curves of χSG/N 2−η (χSG/N 1/3 in the mean-field regime) plotted for different system sizes should inter- sect at the transition temperature TAT(hr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' In reality, finite-size corrections to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (14) are always present and cause the intersection point between the curves for size N and 2N to depend on N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The intersection temperatures vary as [40–43] T ∗(N, 2N) = TAT(hr) + A N λ , (17) where A is the amplitude of the leading correction, and the exponent λ is λ = 1/3 + ω, (1/2 < σ ≤ 2/3), (18a) λ = 1/ν + ω, (2/3 < σ < 1), (18b) 5 where ω is the leading correction to the scaling exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' When σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6, ω = −2σ + 4/3, so λ = 5/3 − 2σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='467 [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' In the regime when σ > 2/3 the values of both ν and λ are not well-determined, so there we shall treat λ as a fitting parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The spin glass correlation length has a similar finite size scaling form in the critical region ξSG N = X � N 1/ν (T − TAT(hr)) � , (2/3 ≤ σ < 1), (19a) ξSG N deff/6 = X � N 1/3 (T − TAT(hr)) � , (1/2 < σ ≤ 2/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (19b) ν, the correlation length critical exponent, has to be de- termined numerically in the interval 2/3 < σ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' We have also studied crossing the AT line at fixed T and varying hr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (19) takes the form, ξSG N = X � N 1/ν (hr − hAT(T)) � , (2/3 ≤ σ < 1), (20a) ξSG N deff/6 = X � N 1/3 (hr − hAT(T)) � , (1/2 < σ ≤ 2/3), (20b) where hAT(T) denotes the field at the AT line at tem- perature T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Similarly, the spin glass susceptibility χSG of the finite system near the AT transition line takes the form χSG N 2−η = C � N 1/ν (hr − hAT(T)) � , (2/3 ≤ σ < 1), (21a) χSG N 1/3 = C � N 1/3 (hr − hAT(T)) � , (1/2 < σ ≤ 2/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (21b) In the thermodynamic limit, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (19) is similar to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (20);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' the effect of finite size corrections to the two can differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' For example, while the correction to scaling exponent λ does not depend on the choice of the trajec- tory, the magnitude of the scaling corrections can differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Thus in the intersection formulae when applied to fields h∗(N, 2N) = hAT(T) + ˜A N λ , (22) the coefficient ˜A will be different from A in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Cor- rections to scaling of, say, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (21a), are more generally of the form χSG N 2−η = C � N 1/ν (hr − hAT(T)) � + N −ωG � N 1/ν (hr − hAT(T)) � , (23) where ω is the correction to scaling exponent, and G is another scaling function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' This type of scaling form holds in the limit where N 1/ν(hr −hAT(T)) is fixed as N → ∞, which of course can only be realized approximately in numerical studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' A key feature of the finite size critical point scaling analysis is that right on the AT line itself, that is when hr = hAT(T), R = χSG/N 2−η (χSG/N 1/3 for σ ≤ 2/3) should be finite as N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' We find (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' VI) that R is at least not increasing with N, and perhaps finite (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 37), for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='60 but for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='70, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 it is in fact increasing with N, at the crossing field h∗(N, 2N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' We deduce from this observation that at these values of σ the crossings at h∗(N, 2N) are not associated with a true critical point at all but are consequences of droplet scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' At a true critical point R would tend to a finite constant as N increases, but we find it increases with N, provided N > 1024 (or system sizes 2N > 2048) for the case of σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' V we shall present our attempts at analysing the data for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6, σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75, σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 at fixed values of hr but varying T, and also at a fixed value of T and vary- ing hr, on the assumption that there is an AT line and using the finite-size scaling methods of this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' We have also obtained data at fixed T and varying hr for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='60, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='70, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 and analysed them using finite size generalizations of well-known droplet scaling relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' In this case the droplet picture provides a sim- ple set of formulae for analysing the data in the assumed absence of an AT line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' ANALYSES OF THE SIMULATION DATA ASSUMING THERE IS AN AT LINE We shall study the phase transitions at hr = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' and determine the zero-field transition temperature Tc (= TAT(hr = 0)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' and seek evidence of an AT transition at non-zero hr using the standard critical point finite size scaling method of determining the “crossings” or inter- sections of the curves of,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' say,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' χSG/N z (with z = 1/3 when σ ≤ 2/3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' and with z = 2 − η = 2σ − 1 for σ ≥ 2/3) at values of N and 2N as we reduce T through the AT transition temperature at fixed hr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' or the field hr at fixed T in the vicinity of the AT field hAT(T) as outlined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' There seems no reason to doubt the existence of an AT line for any value of σ in the mean-field region σ < 2/3, and our results are entirely consistent with the existence of an AT transition at σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' They serve as a useful comparison for the studies in the non-mean regime σ > 2/3, where the evidence will be found to favor the droplet picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' We have studied N values 128, 256, 512, 1024, 2048, 4096, 8192 and 16384 for both σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='60 and σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75, but went up to N = 32768 for the case of σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 when the field hr was varied at fixed T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' When σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 the largest N values used was 4096.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' In this case the zero-field transition temperature Tc is quite low and as a consequence all the investigations have to be done also at low temperatures, where equilibration times are long, preventing the study of larger systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' We are mainly interested in the question as to whether 6 outside the mean-field region, that is for σ > 2/3, an AT transition actually exists and whether (say) the depen- dence of ξSG on the field hr can be understood as will be suggested in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' VI on the droplet picture without invoking an AT transition at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' If it can, this would provide support to the argument that the droplet scaling picture rather than replica symmetry breaking describes spin glasses below 6 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' We have analysed the data for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='70, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 and for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 using the usual “crossing” method, (the finite size scaling approach out- lined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' IV), which indeed works well for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The evidence for the existence of an AT transition at σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 will be contrasted with the evidence against an AT transition at these values of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Our main focus was the case σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' We looked briefly at the case σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='70 to find whether or not it might be practical to study whether σ = 2/3 is the value of σ above which the AT line might disappear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' We found that it was similar to σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75, but that the corrections to scaling were larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' This means that for a given level of accuracy, larger N values are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' We studied σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 because it should behave similarly to physical systems in three dimensions but we could not equilibrate systems at the larger N values in this case because the temperatures T of interest have to be less than Tc, which is rather small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6 We shall focus on σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='60 in this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' It corre- sponds according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (9) to an effective dimension of 10 dimensions, which is in the mean-field region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' it lies above the upper critical dimension of spin glasses, which is 6 (or in the mean-field region σ < 2/3 in the one- dimensional long-range model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' It is natural to expect that for this value of σ there will be an AT line and this is amply confirmed by our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' For this value of σ, simulations of the corresponding Ising model [12, 15] and the Heisenberg model [17, 18] also found an AT line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Our results for hr = 0 are given in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Ac- cording to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (14b), the data for χSG/N 1/3 when plot- ted for different system sizes should intersect at the tran- sition temperature Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Similarly, according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (19b), the data of ξSG/N deff/6 with deff = 2/(2σ − 1) should in- tersect at the same transition temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 2 shows the data for different system sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' We find the tempera- ture T ∗(N, 2N) at which the curves corresponding to the system sizes N and 2N intersect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' We then fit this data with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (17) to find the transition temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The exponent λ ≡ 5/3 − 2σ is known to equal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='467 in this case [17, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The result is displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 3, where the T ∗(N, 2N) data obtained from intersections of χSG are fitted against N −λ with a straight line for the largest 6 pairs of system sizes to give Tc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='8873 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The corresponding intersections of the ξSG data (omitting the two smallest system sizes) gives Tc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='8893 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0046.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The values of Tc obtained from χSG data and ξSG data 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='95 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='1 1 χSG/N 1/3 σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6 hr = 0 N = 128 N = 256 N = 512 N = 1024 N = 2048 N = 4096 N = 8192 N = 16384 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='9 T 10−3 10−2 10−1 100 101 ξSG/N deff /6 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The main figure shows the plot of χSG/N 1/3 as a function of the temperature T for different system sizes, for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6 with hr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The inset figure shows the corre- sponding data for ξSG/N deff/6, with deff = 2/(2σ − 1) in the mean-field regime (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (9)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The exponents of N are cho- sen according to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (14b) and (19b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Both the plots show that the curves for different system sizes intersect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The data for the intersection temperatures T ∗(N, 2N) between pairs of adjacent system sizes are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' are in agreement with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The mean-field pre- diction of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (7) is much higher, T MF c = √ 6/2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='2247.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Fluctuation effects not present in the SK limit must be responsible for this large difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' For hr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='1, the data is as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' When the T ∗(N, 2N) data obtained from χSG are fitted against N −λ with a straight line for the largest 4 pairs of system sizes we get TAT(hr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6735 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The corresponding ξSG data (omitting the two smallest system sizes) gives TAT(hr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6745 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Thus we have found that the AT line passes through the point (T, hr) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='674, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' To compare that with the predictions from the SK model, we use the zero-field transition temperature Tc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='887 obtained above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Then for hr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='1, the predicted value of the AT transition temperature ratio of the SK model would be TAT(hr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='1)/Tc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='74, while the Monte Carlo determined value at σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='7590 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (For the SK model, the Monte Carlo value of the ratio is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='7641 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0341).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Thus while the zero-field transition temperature at σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6 is not close to the mean-field value of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (7), the SK form of the AT line is a good approximation provided it is ex- pressed in terms of the renormalized zero-field transition temperature Tc (see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The AT line can be approached not only by reducing the temperature T but also by reducing the field at fixed T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 6 and 7 we have constructed the crossing plots 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='100 N −λ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 T ∗(N, 2N) σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6 hr = 0 χSG −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='72N −λ + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='89 ξSG −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='56N −λ + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='89 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' A plot of the intersection temperatures T ∗(N, 2N) for χSG/N 1/3 and ξSG/N deff/6 obtained from the data in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 2, as a function of N −λ, for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6 with hr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The value of the exponent λ is fixed to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='467 which is known exactly [17, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The fits give Tc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='8873 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0017 from χSG and Tc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='8893 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0046 from ξSG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6 χSG/N 1/3 σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6 hr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='1 N = 128 N = 256 N = 512 N = 1024 N = 2048 N = 4096 N = 8192 N = 16384 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='1 1 10 ξSG/N deff /6 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' A finite size scaling plot of χSG (main figure) and ξSG (inset figure), for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6 in a magnetic field of hr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Both the datasets clearly indicate that a phase transition occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The transition temperature in the thermodynamic limit is estimated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' as a function of hr for ξSG and χSG respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Analysis of the crossing plots of h∗(N, 2N) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 8 shows that the behavior is again consistent with the existence of an AT line at least at σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The same value of λ was used as when plotting T ∗(N, 2N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The h∗(N, 2N) data for all the pairs of system sizes are fitted against N −λ to give hAT(T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='1569 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0061 from χSG and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='100 N −λ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='68 T ∗(N, 2N) σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6 hr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='1 χSG −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='23N −λ + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='67 ξSG −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='58N −λ + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='67 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The intersection temperatures T ∗(N, 2N), for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6 with hr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='1 (look at Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Both the datasets are consistent with a spin glass transition temperature of TAT(hr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 10−2 10−1 100 hr 10−7 10−5 10−3 10−1 101 103 ξSG/N deff/6 σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6 T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6 N = 128 N = 256 N = 512 N = 1024 N = 2048 N = 4096 N = 8192 N = 16384 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' A finite size scaling plot of ξSG as a function of magnetic field hr, for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6 at a temperature of T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' hAT(T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='1571 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0067 from ξSG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' We found two points on the AT line, (T, hr) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='674, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='1) from T ∗(N, 2N), and (T, hr) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='157) from h∗(N, 2N) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' These points are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 1 for comparison with the exact AT line for the SK model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 The case σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 corresponds to the non-mean-field regime: the long-range diluted model for this value of σ is equivalent to a short-range model with d ≈ 4 dimen- 8 10−2 10−1 100 hr 10−2 10−1 100 χSG/N 1/3 σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6 T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6 N = 128 N = 256 N = 512 N = 1024 N = 2048 N = 4096 N = 8192 N = 16384 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' A finite size scaling plot of χSG as a function of magnetic field hr, for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6 at a temperature of T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='100 N −λ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='16 h∗(N, 2N) σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6 T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6 λ = 5 3 − 2σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='467 χSG −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='36N −λ + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='16 ξSG −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='61N −λ + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='16 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The intersection fields h∗(N, 2N), for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6 with T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Both the datasets are consistent with a spin glass transition at hAT(T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' In this regime, simulations of the corresponding Heisenberg model [17, 18] were thought consistent with an AT transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (14a), the data for χSG/N 2−η,where 2 − η = 2σ − 1, plotted for different system sizes should intersect at the transition temperature Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Similarly, ac- cording to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (19a), the curves of ξSG/N should also intersect at the transition temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The main plots of Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 9 and 11 show the finite-size-scaled data of χSG, and the corresponding inset plots show the finite-size- scaled data of ξSG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The curves for different system sizes show a clear tendency to intersect close to the same tem- perature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The data for T ∗(N, 2N) are then fitted with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='70 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='1 χSG/N 2−η σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 hr = 0 N = 128 N = 256 N = 512 N = 1024 N = 2048 N = 4096 N = 8192 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='7 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='1 1 ξSG/N Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The main figure shows data for χSG/N 2−η (with 2 − η = 2σ − 1) for different system sizes, for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 with hr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The inset shows the data for ξSG/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' According to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (14a) and (19a) the data should intersect at Tc, which is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='4 N −λ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='64 T ∗(N, 2N) σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 hr = 0 λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='1583 χSG −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='15N −λ + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='64 ξSG −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='15N −λ + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='64 Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' A plot of the intersection temperatures T ∗(N, 2N) obtained from the data in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 9, for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 with hr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Using the value of the scaling exponent λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='1583 (ob- tained from the h∗(N, 2N) data), the T ∗(N, 2N) data are fitted against N −λ using a straight line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The resulting values for the transition temperature are Tc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6397 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0051 from χSG and Tc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6440 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0154 from ξSG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (17) where the value of the exponent λ is not known in the non-mean-field regime and hence should be con- sidered as a fitting parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' If there were an AT transition there would be a unique value of λ, the same for both the ξSG and χSG intersec- tions corresponding to both h∗(N, 2N) and T ∗(N, 2N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The h∗(N, 2N) data obtained from χSG intersections in 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='50 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='4 χSG/N 2−η σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 hr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='05 N = 128 N = 256 N = 512 N = 1024 N = 2048 N = 4096 N = 8192 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='55 T 1 ξSG/N Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' A finite size scaling plot of χSG (main figure) and ξSG (inset figure) for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The magnetic field is hr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='4 N −λ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='5 T ∗(N, 2N) σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 hr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='05 λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='1583 χSG −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='35N −λ + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='48 ξSG 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='73N −λ + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='19 Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The intersection temperatures T ∗(N, 2N) for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 with hr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The data from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 11 did not fit well with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' So we used the value of λ obtained from the h∗(N, 2N) data and did a linear fitting which gives TAT(hr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='05) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='4832 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0394 from χSG and TAT(hr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='05) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='1894 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0383 from ξSG, and the values do not agree with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 14 (which is described later) are fitted with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (22) by considering λ, Tc and ˜A as fitting parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' This is a non-linear fitting procedure for which we use effi- cient methods like the Trusted Region Reflective (TRF) algorithm and the Levenberg-Marquardt(LM) algorithm (for which packages are available in python) to determine the fitting parameters, and we obtain λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='1583.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Since the exponent giving the leading correction to scaling λ is universal, we use the same value of λ with both intersec- 10−3 10−2 10−1 100 hr 10−4 10−3 10−2 10−1 100 ξSG/N σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='55 N = 128 N = 256 N = 512 N = 1024 N = 2048 N = 4096 N = 8192 N = 16384 N = 32768 10−3 10−2 10−1 100 hr 10−3 10−1 ξSG/N Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' A finite size scaling plot of ξSG as a function of magnetic field hr, for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 at a temperature of T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The inset shows our two largest system sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 10−3 10−2 10−1 100 hr 10−3 10−2 10−1 χSG/N 2−η σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='55 N = 128 N = 256 N = 512 N = 1024 N = 2048 N = 4096 N = 8192 N = 16384 N = 32768 Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' A finite size scaling plot of χSG as a function of magnetic field hr, for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 at a temperature of T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' tions h∗(N, 2N) and T ∗(N, 2N) obtained from χSG and ξSG data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' We substitute the value of λ obtained above in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (17) and fit the T ∗(N, 2N) data against N −λ with a straight line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 10, for hr = 0, the χSG fit (considering all the pairs of system sizes) gives Tc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6397 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0051.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The corresponding ξSG fit (omit- ting the smallest system size) gives Tc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6440±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0154.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' For hr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='05, the intersection temperatures data are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Omitting the smallest system 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='4 N −λ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='020 h∗(N, 2N) σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='55 λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='1583 χSG 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='03N −λ + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='00 ξSG −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='01N −λ + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='02 Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The intersection fields h∗(N, 2N) for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 with T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' size, the T ∗(N, 2N) data are fitted with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (17) to give TAT(hr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='05) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='4832 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0394 from χSG and TAT(hr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='05) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='1894 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0383 from ξSG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Compared to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 5 which gives the equivalent plot for the case with σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='60, the data in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 12 does not look like data which is converging to the same asymptotic limit when N is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' If the crossings were actually due to a genuine AT transition, then the asymptotic limit should be the same for both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' We have also studied ξSG and χSG at fixed T, but vary- ing hr and the finite size scaling plots for these are given in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 13 and 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' There appears to be good inter- sections in the curves, supporting therefore the possible existence of an AT transition at the temperature studied T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' A plot of h∗(N, 2N) versus 1/N λ is in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 15, using the same value of λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='1583.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' In the intersections of ξSG there is a clear rising trend of h∗(N, 2N) with in- creasing N until N = 1024, followed by decreasing values of h∗(N, 2N) for N > 2048.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' For the case of σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='60, where there is almost certainly a genuine AT transition, (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 8) only the rising trend is seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' It is as if for the smaller systems N < 2048 the system at σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 is behaving similarly to its mean-field cousin at σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Note that this change of trend cannot be attributed to the correction to scaling terms of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' These only apply in the limit N → ∞ with N 1/ν(hr −hAT(T)) fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' For a genuine AT transition the intersections h∗(N, 2N) from both ξSG and χSG should both extrapolate as N → ∞ to the same field hAT(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' It is hard to argue that Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 15 provides good evidence for this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' On the other hand, on the droplet picture, it would be expected that h∗(N, 2N) should extrapolate to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The evidence that is happen- ing is also weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='40 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='3 χSG/N 2−η σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 hr = 0 N = 128 N = 256 N = 512 N = 1024 N = 2048 N = 4096 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='325 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='375 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='4 T 1 ξSG/N Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' A finite size scaling plot of χSG (main figure) and ξSG (inset figure), for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 in a magnetic field of hr = 0 (with 2 − η = 2σ − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Both the datasets clearly indicate that a phase transition occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The transition temperature in the thermodynamic limit is estimated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='006 N −λ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='38 T ∗(N, 2N) σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 hr = 0 λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0315 χSG −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='93N −λ + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='33 ξSG 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='78N −λ + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='33 Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The intersection temperatures T ∗(N, 2N), for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 with hr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' A non-linear fit of the χSG data from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 16 with the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (17) using the Levenberg-Marquadt algorithm gives λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0315.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' A linear fit of the data using this value of λ gives Tc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='3336±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0013 from χSG and Tc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='3297±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0036 from ξSG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 For σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 we are further into the non-mean-field region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (9), σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 corresponds to a short-range model close to three dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' In this regime, simulations of the corresponding Heisenberg model [17, 18] did not find an AT line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='4 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='1 χSG/N 2−η σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 hr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='05 N = 128 N = 256 N = 512 N = 1024 N = 2048 Figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' A finite size scaling plot of χSG for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The magnetic field is hr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='4 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='2 χSG/N 2−η σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 hr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='02 N = 128 N = 256 N = 512 N = 1024 N = 2048 Figure 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' A finite size scaling plot of χSG, for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 with hr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The data do not intersect even at very low temperatures (much lower than the mean field value of TAT(hr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='02) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='2997 obtained using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (8)) indicating that there is no phase transition in this regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' For hr = 0, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 16 clearly shows that the curves for different system sizes are intersecting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The data for in- tersection temperatures are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Similar to the case of σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75, the T ∗(N, 2N) data obtained from χSG are fitted with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (17) by considering λ, Tc, and A as fitting parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' We obtain λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0315 from both TRF and LM methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The fit using the χSG data for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='4 T 1 ξSG/N σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 hr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='05 N = 128 N = 256 N = 512 N = 1024 N = 2048 Figure 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' A finite size scaling plot of ξSG, for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 with hr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The data show merging behavior at low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='4 T 100 101 ξSG/N σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 hr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='02 N = 128 N = 256 N = 512 N = 1024 N = 2048 Figure 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' A finite size scaling plot of ξSG, for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 with hr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The data show merging behavior at low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' all the pairs of system sizes gives Tc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='3336 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The corresponding ξSG fit (omitting the smallest system size) gives Tc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='3297 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0036.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The two values of Tc are quite close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' For hr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='05 the χSG/N 2−η data do not intersect as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Such a field could conceivably be above the largest AT field even at T = 0 so we also studied a smaller field: hr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='02 shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' There is no 12 10−3 10−2 10−1 100 hr 10−2 10−1 100 ξSG/N σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='3 N = 128 N = 256 N = 512 N = 1024 N = 2048 N = 4096 10−3 10−2 10−1 100 hr 10−2 10−1 100 ξSG/N Figure 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' A finite size scaling plot of ξSG as a function of magnetic field hr, for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 at a temperature of T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The inset shows our two largest system sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 10−3 10−2 10−1 100 hr 10−3 10−2 10−1 χSG/N 2−η σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='3 N = 128 N = 256 N = 512 N = 1024 N = 2048 N = 4096 Figure 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' A finite size scaling plot of χSG as a function of magnetic field hr, for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 at a temperature of T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' sign of any crossing at this field either!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='. The ξSG data is less clearcut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 20 shows there are no intersections at a field of hr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='05 while a merging behavior is seen for the larger systems at hr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='02, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' In our simulations we went to very low temperatures such as T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='1, which is small in comparison with the mean-field values of TAT for hr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='02 and hr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='05 using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (8), but we still could not find any clear intersections in the χSG or ξSG data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' This suggests that there is no phase 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='006 N −λ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='03 h∗(N, 2N) σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='3 λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0315 χSG 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='36N −λ + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='00 ξSG 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='96N −λ + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='00 Figure 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The intersection fields h∗(N, 2N) for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 with T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' We substitue the value of the exponent λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0315 obtained from the T ∗(N, 2N) data at hr = 0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (22) and fit h∗(N, 2N) data agianst N −λ to get hAT(T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='3) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0046±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0006 from χSG and hAT(T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='3) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0047±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0016 from ξSG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' transition in this regime in the presence of a magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Our data are consistent with the scenario where the external magnetic field destroys the phase transition, just as happens for a ferromagnet when a uniform field is turned on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Very similar features were seen for the Heisenberg version of this model [17, 18] and in the three dimensional Ising model [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Confusingly, intersections are seen at fixed T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='3 as hr is varied in the plots of ξSG/N in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 22 and of χSG/N 2−η in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The usual analysis of h∗(N, 2N) is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Thus in crossing the AT line along a trajectory of fixed T we have seen intersections, suggest- ing there might be an AT transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' However, the large N limit of h∗(N, 2N) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 24 in the case of σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85, suggests that hAT(T) might actually be zero, consistent with the droplet scaling picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' In the next section the dependence of ξSG and χSG on hr will be explained using the droplet scaling approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' DATA ANALYSES ON THE DROPLET PICTURE In this section we give the field dependence of ξSG and χSG according to the droplet picture [44–46], including also their finite size modifications, and compare these with our simulation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' In the droplet picture one uses an Imry-Ma argument [47] for the correlation length ξ and identifies it with the size of the region or domain within which the spins become re-oriented in the presence of the random field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The free energy gained from such a reorientation by the the random field is of order � q(T)hrξd/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The size of such domains ξ is determined by equating this free energy 13 10−3 10−2 10−1 100 101 hr 10−1 101 103 105 107 ξSG σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='55 N = 128 N = 256 N = 512 N = 1024 N = 2048 N = 4096 N = 8192 N = 16384 N = 32768 fit for N = 32768 (∼ h−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='71 r ) Figure 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' A plot of ξSG as a function of magnetic field hr, for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 at a temperature of T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 10−3 10−2 10−1 100 101 hr 100 102 104 106 ξSG σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='3 N = 128 N = 256 N = 512 N = 1024 N = 2048 N = 4096 fit for N = 4096 (∼ h−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='20 r ) Figure 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' A plot of ξSG as a function of magnetic field hr, for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 at a temperature of T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' to the free energy cost of the interface of this domain of re-ordered spins with the rest of the system, which is of the form Υ(T)ξθ [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Equating these two free energies gives ξ ∼ � Υ(T) � q(T)hr �1/(d/2−θ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (24) While there is a considerable literature on the depen- dence of the interface exponent θ on σ for the case of 10−2 10−1 100 101 hr 10−6 10−3 100 103 106 109 1012 ξSG σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6 T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6 N = 128 N = 256 N = 512 N = 1024 N = 2048 N = 4096 N = 8192 N = 16384 fit for N = 16384 (∼ h−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='88 r ) Figure 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' A plot of ξSG as a function of magnetic field hr, for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6 at a temperature of T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 10−1 101 N 1/xhr 10−6 10−4 10−2 100 ξSG/N σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='55 x = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='7077 N = 128 N = 256 N = 512 N = 1024 N = 2048 N = 4096 N = 8192 N = 16384 N = 32768 Figure 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' A complete finite size scaling plot of ξSG as a function of magnetic field hr, plotted on a log-log scale, for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 at a temperature of T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Ising spin glasses [49], we know of no equivalent studies for the case of the XY spin glass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (Our data suggests that its θ might be close to that of the Ising spin glass).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (24) shows that as hr → 0, the length scale be- comes infinite;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' ξ diverges as ξ ∼ 1/hx r, where x = 1 d/2 − θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (25) The exponent x is the analogue of ν at the AT transition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 14 10−1 101 N 1/xhr 10−5 10−4 10−3 10−2 10−1 100 ξSG/N σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='3 x = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='2019 N = 128 N = 256 N = 512 N = 1024 N = 2048 N = 4096 Figure 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' A complete finite size scaling plot of ξSG as a function of magnetic field hr, plotted on a log-log scale, for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 at a temperature of T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 10−1 100 101 102 N 1/xhr 10−6 10−4 10−2 100 ξSG/N σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='55 x = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='7077 N = 16384 N = 32768 Figure 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' A complete finite size scaling plot of ξSG as a function of magnetic field hr, for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 at a temperature of T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='55, showing our two largest system sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' it is as if the AT transition hAT(T) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' We would expect this formula to apply until finite size effects limit its growth, which will occur when ξ is of O(L) (or O(N) in our one-dimensional system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Identifying ξSG with ξ, Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 25 and 26 show that the Imry-Ma fit indeed works well at the larger fields;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' the data for the larger hr collapse nicely onto a power law form as predicted by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (24) for all sizes N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' It only departs from this formula when 10−1 100 101 102 N 1/xhr 10−5 10−4 10−3 10−2 10−1 100 ξSG/N σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='3 x = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='2019 N = 2048 N = 4096 Figure 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' A complete finite size scaling plot of ξSG as a function of magnetic field hr, for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 at a temperature of T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='3, showing our two largest system sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 10−1 101 N 1/xhr 10−5 10−4 10−3 10−2 10−1 χSG/N z σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='55 x = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='7077 xχ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6114 z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='5951 N = 128 N = 256 N = 512 N = 1024 N = 2048 N = 4096 N = 8192 N = 16384 N = 32768 Figure 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' A complete finite size scaling plot of χSG as a function of magnetic field hr, for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 at a temperature of T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' ξSG becomes of order N, when finite size corrections to the Imry-Ma formula are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Also TNT effects (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' VII) produce corrections to the Imry-Ma formula when ξSG is of O(N) unless N = L > L∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The crossover scale L∗ is thought to be large, especially as σ approaches 2/3 (or d → 6) [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' To allow for finite size effects on the Imry-Ma formula we use the analogue of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (21a) with hAT = 0 and ν = x 15 10−1 100 101 102 N 1/xhr 10−5 10−4 10−3 10−2 10−1 χSG/N z σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='55 x = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='7077 xχ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6114 z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='5951 N = 16384 N = 32768 Figure 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' A complete finite size scaling plot of χSG as a function of magnetic field hr, for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 at a temperature of T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='55, for our two largest system sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 10−1 101 N 1/xhr 10−6 10−5 10−4 10−3 10−2 10−1 χSG/N z σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='3 x = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='2019 xχ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='8919 z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='8592 N = 128 N = 256 N = 512 N = 1024 N = 2048 N = 4096 Figure 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' A complete finite size scaling plot of χSG as a function of magnetic field hr, for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 at a temperature of T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' to write: ξSG/N = X(N 1/xhr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (26) Our results for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 28 and for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' There are clearly finite size corrections to this formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' It is a formula which formally would be expected to hold in the scaling limit of N → ∞ with N 1/xhr fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The crossover function X(y) ∼ 1/yx 10−1 100 101 102 N 1/xhr 10−6 10−5 10−4 10−3 10−2 10−1 χSG/N z σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='3 x = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='2019 xχ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='8919 z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='8592 N = 2048 N = 4096 Figure 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' A complete finite size scaling plot of χSG as a function of magnetic field hr, for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 at a temperature of T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='3, for our two largest system sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='7 1/N z−(2σ−1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='8 h∗(N, 2N) N 1/x χSG(σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='7) ξSG(σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='7) χSG(σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75) ξSG(σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75) χSG(σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85) ξSG(σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85) Figure 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' A plot of h∗(N, 2N)N 1/x versus 1/N ω, for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='70, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The values of x and ω, which is obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (36) were taken from Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' when y is large, in order to recover Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' It goes to a constant when y → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' However, a closer look at our two largest system sizes N = 16384 and N = 32768 at σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 30) and our two largest system sizes at σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85, N = 2048 and N = 4096 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 31) shows that the finite size corrections are becoming small, and are smaller the further the system is away from the mean- field region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' If one moves in the other direction, towards 16 103 104 N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='36 R σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6, T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6 Figure 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' A plot of R = χSG(h∗(N, 2N), N)/N 1/3 versus N, for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6 at a temperature of T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 103 104 N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='34 R σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75, T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='55 Figure 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' A plot of R = χSG(h∗(N, 2N), N)/N 2σ−1 versus N, for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 at a temperature of T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' the start of the mean-field region σ = 2/3, the finite size corrections are larger, as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 40 for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The finite size scaling form for these corrections to the scaling of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (26) will be of the form ξSG N = X(N 1/xhr) + N −ωH(N 1/xhr), (27) where ω is the correction to scaling exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' However, TNT effects (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' VII) produce large further correc- tions to these asymptotic forms when L < L∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Since in 103 N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='2200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='2225 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='2250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='2275 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='2300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='2325 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='2350 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='2375 R σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85, T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='3 Figure 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' A plot of R = χSG(h∗(N, 2N), N)/N 2σ−1 versus N, for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 at a temperature of T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' our studies L∗ is probably larger than the length N of our system, at least for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75, the scaling form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (27) does not work in the region where ξSG is of or- der N (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' For σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 where L∗ is expected to be smaller, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 43 hints that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (27) might apply as the plots at adjacent sizes for the larger N values seem to be getting closer together as N is increased, which is a feature predicted by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 27 we show a similar plot to those in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 25 and 26 but for the case of σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Notice however that because of the AT transition at this value of σ, at which ξSG would diverge to infinity as N → ∞ at some finite field hr = hAT(T), a shoulder above the dashed line has started to appear which is the beginning of this divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Such a feature is absent in the figures for both σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 and at σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The spin glass susceptibility according to the droplet picture is a similar generalization of the finite size scaling form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (14a): χSG N z = C(N 1/xhr), (2/3 ≤ σ < 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (28) The crossover function C(y) ∼ 1/yxχ when y is large, so that then χSG ∼ ξz and becomes independent of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Its form is then χSG ∼ 1/hxχ r , (29) which implies that xχ = xz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' In the opposite limit as y → 0, C(y) goes to a finite constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The exponent z depends upon whether we are dealing with short-range interactions, (such as nearest-neighbor interactions) or with the long-range interactions employed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' For short-range interactions, the average value of χ2 ij falls 17 Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Values of the exponents xχ, x, and z obtained from our simulations for different values of σ and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' σ T xχ x z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='5747 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0009 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='3220 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0413 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='4740 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0062 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='55 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6114 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0005 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='7077 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0531 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='5951 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0119 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='8919 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0014 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='2019 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0152 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='8592 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0066 off with spin separation rij as χ2 ij ∼ q(T)2T Υ(T)rθ ij , (30) [45, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' This result applies in the zero-field spin glass state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Then as, χSG = 1 N N � i,j=1 χ2 ij, (31) so in d dimensions for the zero-field spin glass χSG ∼ Ld−θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Hence z = d − θ d , (32) in order to recover the result χSG → N z as N 1/xhr goes to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' We caution that this formula for z will only hold for short-range interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' With long-range interactions a “droplet” is not a single connected region but a set of isolated islands of flipped spins [49] and this will make the decay of χ2 ij with rij faster than in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' This is an effect which has not been studied before, and so in our problem the exponent z has to be determined by fitting the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The results of our determinations of the droplet exponents x, xχ and z for the different values of σ which we have studied are summarised in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The resulting excellent data collapse (at least when ξSG < N), is shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 32, 33, 34, and 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The value of z was determined from the observation that when N 1/xhr is large, χSG should be independent of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' It is remarkable that z determined at large values of N 1/xhr results in a decent collapse of the data in the opposite limit where N 1/xhr → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Nevertheless corrections to the Imry-Ma scaling form are visible in the figures (and are sizeable in the region where N 1/xhr is small when viewed in a linear plot rather than a log scale plot, (just as in the ξSG plots Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 28 and 29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' In the limit when N 1/xhr is held fixed with N → ∞ the leading correction to scaling will be χSG N z = C(N 1/xhr) + N −ωG(N 1/xhr), (33) where G(y) is an unknown scaling function and the cor- rection to scaling exponent ω is not known with any cer- tainty (but see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (36)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Let us suppose that the droplet picture is correct and that (say) the spin glass susceptibility χSG is described by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' This equation predicts that there will be a crossing in the plots of χSG/N 2σ−1 used in AT line criti- cal scaling studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (Note we are setting 2 − η = 2σ − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The correction to scaling term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (33) is not needed for this, but this correction does strongly influence where the crossings take place for the N values which are reached in our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The crossing arises as fol- lows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' At small values of hrN 1/x, the function C goes to a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' It turns out that z > (2σ − 1), so χSG/N 2σ−1 diverges as N is increased as N z−(2σ−1) as hr → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' On the other hand when hrN 1/x is large, χSG → 1/hz r, so χSG/N 2σ−1 → 1/N 2σ−1hz r → 0 as N goes to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Because at small fields, χSG/N 2σ−1 is larger for large N, but at bigger hr fields it is smaller at the larger N val- ues, so there must be a crossing point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' We shall denote the crossing value between the lines at N and 2N by h∗(N, 2N) = H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Then H is determined by the solution of the following χSG(H, N) N 2σ−1 = C(N 1/xH)N z−(2σ−1) = χSG(H, 2N) (2N)2σ−1 = C((2N)1/xH)(2N)z−(2σ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (34) Assuming C(y) → a − by, when y → 0, it is easy to show then that the N dependence of h∗(N, 2N) at very large N will be as 1/N 1/x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' In reality we have no data in this region of very large N where the corrections to scaling term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (33) can be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The corrections to scaling are numerically small but are very important in determining the values of h∗(N, 2N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' There is a similar crossing predicted in the plots of ξSG/N as a function of hr when Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (27) holds, using the analogue of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' In this case it is the scaling cor- rection which causes the curves to cross, (which requires H(0) to be negative), and for these curves the crossings h∗(N, 2N) at very large N will decrease as 1/N 1/x+ω, (compare with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (18b)) on taking χ(y) = c − dy and H(y) → constant as y → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Once again we have no data in this very large N regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 36 we have plotted h∗(N, 2N)N 1/x versus 1/N ω, assuming that ω is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Note that the size of the corrections to scaling ∼ 1/N ω is simply not small for the values of N which we can study, contrary to what was assumed in the above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' h∗(N, 2N)N 1/x should go to a constant as N goes to in- finity and it is only for the case of σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85, where the corrections to scaling are the smallest of the three cases studied, does that look remotely possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' For the case of σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='70 the corrections look to be very large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' We conclude that for the values of σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='70 and σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75, the crossing data on h∗(N, 2N) is not close to the large N asymptotic form predicted by the droplet picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' But the droplet picture does predict that the existence of such intersections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' If we only had information on the values of the cross- ing fields h∗(N, 2N) it would be difficult to really be sure whether the droplet picture or the RSB picture best de- scribed the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The results on h∗(N, 2N) alone are in- conclusive as regards both the AT transition line picture 18 100 101 102 N 1/xhr 10−7 10−5 10−3 10−1 101 ξSG/N σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='7 T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6 x = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='3220 N = 8192 N = 16384 Figure 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' A complete finite size scaling plot of ξSG as a function of magnetic field hr, for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='70 at a temperature of T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6 for our two largest system sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' and the droplet picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' While on the droplet picture h∗(N, 2N) are predicted to go to zero as N → ∞, the values of h∗(N, 2N) are not convincingly going to zero as N is increased (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Fortunately, there is another way of distinguishing the two approaches, which does not require us to reach the N values at which h∗(N, 2N) starts to approach zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' We define R ≡ χSG(h∗(N, 2N), N)/N 2σ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (35) (Because we only determine χSG(hr, N) at a finite num- ber of values of hr, we use linear interpolation to calcu- late χSG(h∗(N, 2N), N) using the χSG(hr, N) values at the two determined values of hr which lie on either side of h∗(N, 2N)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' On the phase transition picture, R should approach a finite constant as N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' On the droplet picture R should increase as N z−(2σ−1) as N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' For σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='60 where an AT line is expected R should go to a constant but at the N values studied it actually still appears to be decreasing (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 37) and has yet to be- come constant, presumably due to finite size effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' This indicates that trying to determine whether σ = 2/3 is the exact value at which the crossover to droplet scaling be- havior will also be challenging from the side below 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' However, for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 38 shows that R is clearly increasing with N for large N values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' But if we had had only data for system sizes < 2048 we might have indeed concluded that there was good evidence for an AT transi- tion in that R seemed to be an N independent constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' While at the sizes we can reach R is clearly increasing with N it has yet to reach its asymptotic form of in- crease as N z−(2σ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The quantity R also increases with N for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='70 and σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85, (see for example Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' In order for χSG to match as σ → 2/3 from either the 100 101 102 N 1/xhr 10−4 10−3 10−2 10−1 χSG/N z σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='7 T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6 x = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='3220 xχ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='5747 z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='4740 N = 8192 N = 16384 Figure 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' A complete finite size scaling plot of χSG a function of magnetic field hr, for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='70 at a temperature of T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6 for our two largest system sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' mean-field side, (where z = 1/3) with its value in the non-mean field region, we would expect that z should approach 1/3 as σ → 2/3 from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' At σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85, z ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='8409, at σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75, z ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6065, while at σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='70, we find z ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='4737.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Thus it seems quite plausible that z could approach 1/3 as σ → 2/3 from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Then the combination z − (2σ − 1) would approach zero in this limit, which means that the divergence of R with N will become harder and harder to see as σ approaches 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' We conclude that it will be challenging to do numerical work which shows that the AT line disappears at precisely σ = 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' On the mean-field side of 2/3 the correction to scaling exponent ω = 1/3 − (2σ − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' It therefore seems natural to expect that on the non-mean field regime ω = z − (2σ − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (36) If valid, this would imply that corrections to scaling should be larger at σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='70 than at σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75, and this is what we observed in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 40 and 41, in comparison with Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 30 and 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' In the presence of a genuine AT transition, as hr is reduced one would pass through three regions: first the paramagnetic state at larger values of hr, then the criti- cal region, then the low-temperature phase with RSB at smaller values of hr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The good data collapse for all val- ues of hr using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (26), and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (28) shows that at any finite value of hr there is just one region, the paramag- netic region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Studying “intersections” as in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' V is an attempt to find the critical region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' But the intersections at finite values of hr for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 and σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 are not signs of a genuine phase transition, but at least in the case of χSG these crossings are also just a consequence of droplet scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The behavior of h∗(N, 2N) as a function 19 10−2 10−1 100 N 1/xhr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0 ξSG/N σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='55 x = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='7077 N = 128 N = 256 N = 512 N = 1024 N = 2048 N = 4096 N = 8192 N = 16384 N = 32768 Figure 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' A finite size scaling plot of ξSG as a function of magnetic field hr, for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 at a temperature of T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' of N is greatly complicated by finite size effects and will only become clear at much larger N values than those which we have been able to study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Because on the droplet picture there is no AT line and so one is always in the paramagnetic phase at any non- zero field (just as in a ferromagnet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' However, length scales like ξSG become very large as hr → 0 for temper- atures T < Tc(hr = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Once they become comparable to the system dimensions L and one is in the regime hr < h∗(N, 2N), the system will have many of the fea- tures which might be associated with being in the bro- ken replica symmetric phase which is envisaged to exist below the AT line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' For physical systems in three dimen- sions the relevant length scale is not the linear dimension of the system L, but the linear dimension of a fully equi- librated region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' This may explain why both simulations and experiments have failed for many years to resolve the debate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Might it be possible to find by simulations whether the borderline between RSB ordering and droplet ordering is at σ = 2/3, which is the equivalent of d = 6 with short- range interactions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' To this end we looked at the case of σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' We found from studying the crossings of ξSG and χSG for the zero field case that the zero field transition temperature is ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='724.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 40 and 41 show our attempt to collapse the data with the droplet scaling forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Clearly the effects of corrections to scaling are larger than was the case at σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 30 and 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' This is in accord with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (36) which predicts that the correction to scaling exponent ω will go to zero as σ → 2/3 if also z → 1/3 as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' We conclude that it will be difficult to provide good numerical evidence that σ = 2/3 is the lower critical dimension of the AT transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 10−2 10−1 100 101 N 1/xhr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0 ξSG/N σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='3 x = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='2019 N = 128 N = 256 N = 512 N = 1024 N = 2048 N = 4096 Figure 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' A finite size scaling plot of ξSG as a function of magnetic field hr, for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 at a temperature of T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' TNT VERSUS THE DROPLET SCALING PICTURE Newman and Stein [50] (see also the recent review [51]), have suggested that the ordered phase of spin glasses in finite dimensions will fall into one of 4 cate- gories, (and which one might depend on the dimension- ality d of the system): The RSB state is one of these, and is somewhat similar to that envisaged by Parisi for the SK model, but there is also the chaotic pairs state picture of Newman and Stein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' In both of these pictures there is an AT transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The other two pictures are the so-called TNT picture of Krzakala and Martin [35] and Palassini and Young [36] and the droplet scaling pic- ture [44–46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' In neither the TNT picture nor the droplet scaling picture is there an AT transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' In the droplet picture the Parisi overlap function P(q) is trivial, consist- ing of two delta functions at ±qEA in zero field, whereas in the TNT picture the form of P(q) is quite similar to the non-trivial (NT) form which Parisi found for the SK model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The TNT picture accounts for the non-trivial form of the Parisi overlap function by postulating that there exist droplets of the linear size L of the system, which contain O(Ld) spins, and which do not have a free energy of order Lθ (as they would in the droplet scaling picture), but which have instead a free energy of O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' It is the presence of such droplets which makes P(q) non- trivial, which is a feature observed in all simulations of it to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' In a recent paper [3] one of us argued that once the linear dimension of the system became larger than a crossover length L∗ the non-trivial behavior observed in P(q) will change to the trivial form predicted by droplet scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Estimates of L∗ in d = 3 suggest it might be 20 large, of the order of several hundred lattice spacings and it is probably the case that to date the regime where L > L∗ has not been reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Furthermore it was sug- gested that as d → 6, L∗ would grow towards infinity, as the droplets of O(1) evolve to the O(1) excitations in the Parisi RSB solution, where the pure states have free energies which differ from each other by O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' In our one dimensional proxy system we would therefore expect to find that L∗ is much larger when σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 than it is when σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' In this paper there are TNT-like effects visible in the behavior of ξSG/N in the region where ξSG is of O(N) (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 28 and 29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' When ξSG is of order N the droplets which are important are those of size N and if L < L∗ some of these will have free energy of O(1) rather than Lθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' As a consequence the good scaling collapse of the data visible when ξSG/N ≪ 1 will be lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 42 and 43 we have plotted ξSG/N on a linear scale versus N 1/xhr focussing only on the region where ξSG/N is of O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' If the droplet scaling collapse had been good and of the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (27) then as N is increased the collapse should get better and better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' In fact due to TNT effects the data in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 42 for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 show the opposite trend, and the lines get further apart with increasing N in the region where ξSG is of O(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' However, for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 43 shows the lines seem to be getting closer with increasing N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' It suggests that for this value of σ we are getting into the region where L > L∗ when droplet scaling applies even when ξSG is of O(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Data at larger values of N than 4096 would be nice to confirm this trend but because these simulations have to be done at quite low temperatures compared to those for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 it will be challenging to do this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Despite this limitation on the size of N which can be reached for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85, there is evidence that for it, TNT and finite size scaling effects are less troublesome than for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75, despite the fact that much larger values of N can be studied at this σ value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' SUMMARY AND CONCLUSIONS In this paper, we have studied the phase transitions in the one-dimensional power-law diluted XY spin glass, both in the zero-field limit, and in the presence of a mag- netic field random in the component directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Whether or not an AT line exists for various values of the param- eter σ is a question of fundamental interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' To address this, we have performed large scale Monte-Carlo sim- ulations using a new heatbath algorithm, described in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' This algorithm hopefully speeds up equi- libration, so cutting computational costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' We certainly do gain some advantage in terms of computational time due to the smaller number of components of XY spins compared to those of the Heisenberg model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Alas, the heatbath algorithm for XY spins suffers from an intrin- sic disadvantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Because our algorithm has to generate two random numbers during each Monte Carlo step, the benefits of the smaller number of components are largely counterbalanced by the additional labor involved in the heatbath step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' We were unable to go to larger system sizes than in the corresponding work with Heisenberg spins [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The largest system sizes that we are able to simulate are: N = 16384 for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6, N = 32768 for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75, while the largest N for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 was 4096.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The total CPU time spent in generating all the data that we presented at fixed hr and varying T was 1183636.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='2 hrs, which is 135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='12 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The total CPU time consumed in generating the data at fixed T and varying hr was 96101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6 days which is 263.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='29 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Thus despite the algorithm not producing significant dividends, we are able to study fairly large system sizes owing to the expenditure of a large amount of computer time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The results from our work are broadly in accord with those for the corresponding Heisenberg spin glass model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' For σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6, which is in the mean-field regime, we find a phase transition in the absence of an external mag- netic field, and in the presence of a magnetic field, which indicates the existence of an AT line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The location of the AT line is close to the mean-field predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' For σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75, which is in the non-mean-field regime, the con- ventional data collapse suggests the existence of an AT line, but the behavior of the intersections as a function of N indicate that the data is not close to its large N asymptotic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The estimated location of the AT field based upon intersections that we get from our data at σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 is strikingly smaller than estimates based on the mean-field theory formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' For σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85, which is deep in the non-mean-field regime and corresponds to a space dimension of about 3, our data are consistent with the absence of an AT line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' In this case there is no crossing of the curves of χSG/N 2−η versus T at various N values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' But confusingly intersections h∗(N, 2N), as a function of hr, seem to exist, whereas intersections T ∗(N, 2N) are absent at least for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' However, for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 and for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 we found that the droplet picture provided a much better description of our data from that obtained assuming the existence of an AT transition line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The Imry-Ma formula for the field dependence of ξSG works well until ξSG becomes comparable to the system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' A similar behavior was reported for the Ising spin glass at σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' A finite-size scaling formulation was developed to treat the data at small fields when ξSG is comparable to the system size N, and with it an excellent collapse of all our data on ξSG and χSG was obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' We showed that droplet scaling predicts the existence of the intersections h∗(N, 2N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Our data unfortunately does not extend to values of N large enough to be in the asymptotic region where the N-dependence of h∗(N, 2N) is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Fortu- nately there exists a way of testing whether the intersec- tions are due to an AT transition or are just those pre- dicted by droplet scaling, which is to study the N depen- dence of R = χSG/N 2σ−1, calculated at h∗(N, 2N), and this test supports the droplet picture provided N > 1024 at σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Thus it is only for large systems that one 21 can obtain good evidence for the droplet picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' We now summarize our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The strongest evidence for droplet scaling is the success of the Imry-Ma formula for the field dependence of ξSG for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 and σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 30 and 31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' If droplet scaling works, then no AT line is to be expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' When ξSG ∼ N there are visible sizeable corrections to the Imry-Ma formula which are related to TNT effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' However for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 there is tentative evidence in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 43 that if even larger systems could be studied then the TNT effects might be absent, and so there could exist a length scale L∗ above which TNT effects become unimportant (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' If instead of droplet scaling one assumes that there is an AT phase transition then the usual finite size scaling plots used to determine hAT as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 15 for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 are unsatisfactory: for example the values of hAT which would be derived from the crossings of ξSG and χSG as N becomes large look to be significantly different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' In the equivalent data plot for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='60 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 8) they are in good agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Furthermore the quantity R of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (35) should approach a constant as N → ∞ if there is a genuine AT transition, but instead for the cases σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='75 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 38) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='85 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 39), it is increasing with N once N becomes large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The simulations of this paper provide numerical evi- dence that the AT line and hence RSB is absent in spin glasses below six dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' What is now needed is an explanation of why this might be the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Better still would be a rigorous proof that the lower critical dimen- sion for replica symmetry breaking is six.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Our work indi- cates that showing that σ = 2/3 is the precise value of the critical value of σ will be challenging using simulations as finite size effects are large in its vicinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' ACKNOWLEDGMENTS We are grateful to the High Performance Comput- ing (HPC) facility at IISER Bhopal, where large- scale calculations in this project were run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' We thank Peter Young and Dan Stein for helpful discus- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='V is grateful to the Council of Scientific and Industrial Research (CSIR), India, for his PhD fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='S acknowledges financial support from SERB via the grant (File Number: CRG/2019/003447), and from DST via the DST-INSPIRE Faculty Award [DST/INSPIRE/04/2014/002461].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Appendix A: The simulation method We now give some technical aspects of how the simula- tions are run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' In the simulations we start with a random initial configuration and allow it to evolve according to the prescription given in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' To incorporate par- allel tempering, we simultaneously simulate NT copies of the system over NT different temperatures ranging from Tmin ≡ T1 to Tmax ≡ TNT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' In order to facilitate the computation of the observables outlined in this section, it is convenient to simulate 4 sets of NT copies (2 for hr = 0), which we label (1),(2),(3), and (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' We perform overrelaxation, heatbath and parallel tempering sweeps over all these copies keeping track of the labels appropri- ately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' For every 10 overrelaxation sweeps we perform 1 heatbath and 1 parallel tempering sweep, since the overrelaxation sweep involves a significantly lower com- putational cost, and is known to speed up equilibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The parameters of the simulations are shown in Tables II and III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Once the system reaches equilibrium, we perform the same number of sweeps in the measurement phase, so Nsweep is the total number of sweeps over which the simulation is run, inclusive of both the equilibration and measurement phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The last column in the table shows the amount of computer time expended to generate the data corresponding to the parameters in that row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' In the measurement phase, we perform one measurement on the system for every 4 sweeps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The following sections contain the details of our Monte Carlo simulation pro- cedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' In order to equilibrate the system as quickly as possible, we perform three kinds of sweeps: overre- laxation or microcanonical sweeps, heatbath sweeps, and parallel tempering sweeps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Overrelaxation sweep We sweep sequentially through all the lattice sites and compute the local field Hi = � j JijSj+hi at a particular lattice site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The new spin direction S′ i at the ith lattice site is taken to be the mirror image of the vector Si about Hi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=', S′ i = −Si + 2Si · Hi H2 i Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (A1) Since S′ i · Hi = Si · Hi, the energy of the system does not change due to these sweeps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Hence these sweeps are also called microcanonical sweeps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' These sweeps help us in sampling out the microstates with the same energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The process of equilibration speeds up when we include overrelaxation sweeps along with the other sweeps [33, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Heatbath sweep The overrelaxation sweeps generate states with the same energy and hence they cannot directly equilibrate the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Therefore, we also perform a heatbath sweep for every 10 microcanonical sweeps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Similar to the micro- canonical case, we sweep sequentially through the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' To equilibrate the system, the angle θ between Hi and S ′ i should be sampled out from the Boltzmann distribu- tion given by fΘ(θ) = e−βEi Z = eβHiSi cos θ Z = ew cos θ Z , (A2) 22 where w = βHiSi and Z = π � −π eβHiSi cos θ dθ (A3) is the normalizing constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The simplest way to do this is to equate the cumulative density function (CDF) of θ, FΘ(θ), to that of a uniform distribution: FΘ(θ) = θ � −π fΘ(θ′) dθ′ = Π(r1) = r1, (A4) where r1 is a random variable sampled from a uniform distribution in the interval (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The value of θ can be obtained by simply inverting this function to get θ = F −1 Θ (r1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (A5) This method works well with the Heisenberg spins as fΘ(θ) is integrable, which gives an invertible CDF FΘ(θ) [18, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Since the probabililty density function (PDF) fΘ(θ) for the XY spin glasses given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (A2) is not exactly integrable, this method cannot be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' To overcome this problem and to sample out θ from the Boltzmann distribution (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (A2)) in as few a number of sweeps as possible, we develop a heatbath sweep based on the rejection method [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' We generate two random numbers r1 ∈ uniform(−π, π) and r2 ∈ uniform(0, fmax).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' If r2 < fθ(r1), we accept the move, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=', take θ = r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Else, we reject the move and generate another pair of random numbers (r1, r2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' This process is repeated until we find an acceptable value of r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' A graphical representation for this method is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The new spin direction S′ i in Cartesian co-ordinates is given by: S′ x = cos(θ + θH), (A6a) S′ y = sin(θ + θH), (A6b) where θH is the angle made by the Hi vector with the X- axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Since the generation of random numbers is involved, this sweep is computationally costlier than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Hence we perform more microcanonical sweeps than heatbath sweeps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Parallel tempering sweep Spin glasses have a complex free energy landscape due to which, at low temperatures, they tend to get stuck in- side metastable valleys, and true equilibration consumes a lot of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' At high temperatures, the system can eas- ily escape the valley due to thermal fluctuations, and so equilibration is quick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' To equilibrate the system in as small a number of moves as possible, we perform one parallel tempering sweep for every 10 overrelaxation sweeps [28, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' To benefit from the parallel tempering algorithm [54, 55], we simultaneously run the simulation −2 0 2 θ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='4 fΘ(θ) fmax (r1, r2) (r1, fΘ(r1)) (r1, r2) (r1, fΘ(r1)) w = βHS = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='5 Rejected Accepted Figure 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Graphical representation of the rejection method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' We randomly pick a point (r1, r2) within the rectangle from a uniform distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' If the point lies under the fΘ(θ) curve given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (A2), then the point is accepted, and θ is taken to be r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Otherwise, the point is rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' for NT copies of the system at NT different temperatures T1 < T2 < T3 < · · · < TNT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The minimum temperature T1 is the low temperature at which we are interested in studying the behavior of the system, and the maximum temperature TNT is high enough that the system equi- librates very fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' We perform overrelaxation and heat- bath sweeps separately on each of the NT copies of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' In the parallel tempering sweep, we compare the energies of two spin configurations at adjacent tempera- tures, Ti and Ti+1, starting from the smallest tempera- ture T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' We swap these two spin configurations such that the detailed balance condition is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The Metropo- lis probability for such a swap is P(T swap) = min{1, exp(∆β∆E)} (A7) = � exp(∆β∆E) (if ∆β∆E < 0), 1 (otherwise), (A8) where ∆β = 1/Ti − 1/Ti+1 and ∆E = Ei(Ti) − Ei+1(Ti+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' In this way, a given set of spins performs a random walk in temperature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Checks for equilibration In order to check whether the system has reached equi- librium, we have used a convenient test [56] which is pos- sible because of the Gaussian nature of the interactions and the onsite external magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The relation U = zJ2 2T (ql − qs) + h2 r T � q − |S|2� , (A9) 23 Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Parameters of the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Nsamp is the number of disorder samples, Nsweep is the number of over-relaxation Monte Carlo sweeps for a single disorder sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The system is equilibrated over the first half of the sweeps, and measurements are done over the 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0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='05 2048 3240 4194304 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='4 36 151503 is valid in equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Here U = 1 N [⟨H⟩]av = − 1 N � �� ⟨i,j⟩ ϵijJij ⟨Si · Sj⟩ + � i,µ hµ i ⟨Sµ i ⟩ � � av (A10) is the average energy per spin, q = 1 N � i [⟨Si⟩ · ⟨Si⟩]av is the Edwards-Anderson order parameter, ql = 1 Nb � ⟨i,j⟩ � ϵij ⟨Si · Sj⟩2� av is the “link overlap”, and qs = 1 Nb � ⟨i,j⟩ � ϵij � (Si · Sj)2�� av is the “spin overlap”, where 24 Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Parameters of the simulations done at fixed temperature T and varying field hr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' N(hr) is the number of values of field taken in the range hr(min,max).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The equilibration times are different for different values of the field hr, which lie in the range Nsweep(min,max).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The number of disorder samples for different fields lie in the range Nsamp(min,max).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' ttot is the total CPU time consumed in hours to generate data for a particular system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' σ T N hr(min,max) N(hr) Nsweep(min,max) Nsamp(min,max) ttot(hrs) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='6 128 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='010, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content='000) 32 (2048, 2048) (2000, 64000) 11.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The [· · · ]av in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (A10) is analytically evaluated by performing integration over Jij and hµ i [57] since they have Gaussian distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' On evaluating this integral using integration by parts, we get Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (A9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' As the system reaches equilibrium, the two sides of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (A9) approach their common equilibrium value from opposite directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' In simulations, we evaluate both sides of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (A9) for different number of Monte-Carlo sweeps (MCSs), which increase in an exponential manner, each value being twice the previous one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' The averaging is done over the last half of the sweeps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' We initially start with a random spin con- figuration, so the LHS of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (A9) is small and the RHS is very large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' As the system gets closer to equilibrium, these two values come closer to each other from opposite directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' When we notice that the averaged quanti- ties satisfy Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' (A9) within error bars, consistently for at least the last two points, we declare that our system has reached equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' Once the system reaches equi- librium, we perform the same number of sweeps in the measurement phase, where we evaluate different quanti- ties (given below) used to study the possible phase tran- sitions of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' [1] Marc M´ezard, Giorgio Parisi, and Miguel Angel Vira- soro, Spin glass theory and beyond: An Introduction to the Replica Method and Its Applications, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' 9 (World Scientific Publishing Company, 1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E2T4oBgHgl3EQfCga9/content/2301.03615v1.pdf'} +page_content=' [2] J R L de Almeida and D J Thouless, “Stability of the 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dating of planets in protoplanetary discs +Roman R. Rafikov1,2★, Nicolas P. Cimerman1 +1Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK +2Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +High-resolution sub-mm observations of some protoplanetary discs reveal non-asixymmetric features, which can often be +interpreted as dust concentrations in vortices that form at the edges of gaps carved out by the embedded planets. We use recent +results on the timescale for the planet-driven vortex development in low-viscosity discs to set constraints on the mass and age of +a planet producing the vortex. Knowledge of the age of the central star in a vortex-bearing protoplanetary disc system allows one +to set a lower limit on the planetary mass at the level of several tens of 𝑀⊕. Also, an independent upper limit on the planetary +mass would constrain the planetary age, although given the current direct imaging detection limits this constraint is not yet very +stringent (it is also sensitively dependent on the disc scale height). These results can be extended to account for the history of +planetary mass accretion if it is known. We apply our calculations to several protoplanetary discs harbouring vortex-like features +as revealed by ALMA and set limits of (30−50)𝑀⊕ (for disc aspect ratio of 0.1) on the minimum masses of putative planets that +could be responsible for these vortices. Our vortex-based method provides an independent way of constraining the properties of +embedded planets, complementary to other approaches. +Key words: hydrodynamics – instabilities – shock waves – accretion discs – planets and satellites: formation – methods: +numerical +1 INTRODUCTION +Observations of protoplanetary discs (PPDs) in dust continuum emis- +sion with ALMA revealed a variety of substructures (Andrews 2020), +including axisymmetric gaps and rings as well as non-axisymmetric +clumps and arcs. An intriguing possibility is that these features could +be produced by young embedded planets. In particular, gravitational +coupling between a massive planet and the disc is known to result in +formation of observable gaps around the planetary orbit (Papaloizou +& Lin 1984; Rafikov 2002b). The evolution of vortensity (potential +vorticity) at the edges of these gaps (Lin & Papaloizou 2010; Dong +et al. 2011; Cimerman & Rafikov 2021) due to shock dissipation of +the planet-driven density waves (Goodman & Rafikov 2001; Rafikov +2002a) can trigger the Rossby Wave Instability (RWI, Lovelace et al. +1999) resulting in the formation of fluid vortices at these locations. +Dust accumulation inside the vortices (Barge & Sommeria 1995) +naturally leads to observable non-axisymmetric arcs and lobes. +Formation of these structures does not necessarily require massive +(Jovian) planets. For example, it was shown (Dong et al. 2017; Bae +et al. 2017; Miranda & Rafikov 2019, 2020a,b) that multiple visible +gaps and rings in the dust distribution can result from nonlinear +damping of multiple spirals triggered by a single sub-Jovian mass +planet in a low viscosity disc. The mass of the planet 𝑀p can in fact be +below the so-called thermal mass defined as 𝑀th = �𝐻p/𝑅p +�3 𝑀★ = +ℎ3p 𝑀★, where 𝐻p is the disc scale height at the planetary distance 𝑅p, +ℎp = 𝐻p/𝑅p is the disc aspect ratio there, and 𝑀★ is the stellar mass. +Emergence of vortices at the edges of planetary gaps also does not +★ E-mail: rrr@damtp.cam.ac.uk (RRR) +require massive planets if the disc is almost inviscid (Hammer et al. +2021; Hallam & Paardekooper 2020), and sub-𝑀th mass planets can +easily trigger them (Cimerman & Rafikov 2023, hereafter CR23). +In this study we focus on non-axisymmetric disc features which +can be interpreted as planet-induced vortices (other ways to produce +vortices are mentioned in Section 2). Their development is not an +instantaneous process as the evolution of vortensity near the planetary +orbit towards the RWI takes a certain amount of time. As we show +in this work, this fact can be exploited to set useful constraints on +the mass and/or age of a putative planet responsible for production +of the observed vortices in a PPD. This method relies on the recent +calculation of CR23 who studied the development of vortices in +inviscid PPDs. In that work, we showed that the time it takes for +vortices to emerge at the edge of the gap carved out by a sub-𝑀th +mass planet can be approximated as +𝜏vrt ≈ 𝐴 𝑃p +� 𝑀p +𝑀th +� 𝛼 +ℎ𝛽 +p , +where +(1) +𝐴 ≈ 1.6, +𝛼 ≈ −2.7, +𝛽 ≈ −0.86, +(2) +and 𝑃p is the orbital period at 𝑅p. This result assumes 𝑀p to be fixed +in time. Interestingly, CR23 found 𝜏vrt to only weakly depend on the +radial profile of surface density in the disc. +We describe the general idea of our method in Section 2 and show +how it can constrain the mass and age of the planet in Section 3 and +Section 4, respectively. In Section 5 we extend our constraints to the +case of a planet accreting its mass over an extended period of time. +We apply our results to observed PPDs in Section 6 and discuss them +in Section 7. +© 2022 The Authors +arXiv:2301.01789v1 [astro-ph.EP] 4 Jan 2023 + +2 +R. R. Rafikov and N. P. Cimerman +2 GENERAL IDEA OF THE METHOD +Let us suppose that observations of dust continuum emission reveal +a gap in a PPD, together with a non-axisymmetric lobe or arc at +the gap edge, indicative of a vortex which traps dust grains at this +location (e.g. van der Marel et al. 2016; Kraus et al. 2017; Dong +et al. 2018; Pérez et al. 2018). We will interpret this observation by +assuming that a planet (not necessarily directly visible) is located +within the gap and is responsible for creating both the gap and the +vortex (via the RWI). The spatial association of a vortex with an +adjacent gap provides strong support to this interpretation and makes +some other possibilities for triggering vortices, e.g. global baroclinic +instability (Klahr & Bodenheimer 2003), convective overstability +(Teed & Latter 2021), vertical shear instability (Richard et al. 2016) +less attractive. +We assume the disc viscosity to be low (essentially inviscid), +consistent with many observations of PPDs (Pinte et al. 2016; Rafikov +2017; Flaherty et al. 2020). For now we will also assume that 𝑀p has +been constant ever since the planet appeared in the disc, a constraint +that we will relax in Section 5. With these conditions fulfilled, the +result (1)-(2) applies. We can then use the observation of the vortex +to set a constraint on a particular combination of the planetary mass +𝑀p and the planetary age 𝜏p — the time that has passed since the +planet has reached its final mass 𝑀p. +Indeed, the observation of a vortex at the gap edge implies that the +RWI had enough time to fully develop into the non-linear stage in +that region, i.e. that +𝜏p > 𝜏vrt. +(3) +Together with equation (1) this leads to the following combined +constraint on 𝜏p and 𝑀p: +𝜏p𝑀−𝛼 +p +> 𝐴𝑃p𝑀−𝛼 +th ℎ𝛽 +p . +(4) +With fit parameters (2) we can write this in physical units as +𝜏p +Myr +� +𝑀p +102𝑀⊕ +�2.7 +> 0.11 +� +𝑅p +50AU +�1.5 � 𝑀★ +𝑀⊙ +�2.2 � ℎp +0.1 +�7.2 +. +(5) +This condition must be fulfilled whenever a vortex is observed at the +gap edge. It is illustrated in Fig. 1 for several values of ℎp and 𝑅p. +In a similar vein, the absence of vortex-like structures at the edges +of a visible gap in a disc might be interpreted as meaning that 𝜏p < +𝜏vrt, i.e. that planet-driven accumulation of vortensity has not yet led +to RWI. If that were the case, the inequality in the constraint (4)-(5) +would change its sign. However, this possibility has an important +caveat as the absence of a vortex may also be interpreted differently: +it could have formed at the gap edge earlier but then got destroyed +through one of the processes that tend to destabilize vortices once +they evolve into the nonlinear regime: the elliptical instability (Lesur +& Papaloizou 2009), baroclinic effects (Rometsch et al. 2021; Fung & +Ono 2021), dust feedback (Fu et al. 2014), etc. Also, vortex formation +may have been delayed or suppressed altogether if disc viscosity +is sufficiently high (Hammer et al. 2017; Hallam & Paardekooper +2020). Thus, the lack of a vortex near a planetary gap cannot be +unambiguously interpreted as meaning that the embedded planet did +not get a chance to create it, i.e. that 𝜏p < 𝜏vrt. For that reason (and +unlike Hallam & Paardekooper 2020) in the following we will not +draw any conclusions from the absence of vortices at the edges of +putative planetary gaps found in sub-mm observations. +We will now show how equations (4) & (5) can be used to sepa- +rately constrain 𝑀p or 𝜏p. +10 +100 +1000 +Mp [M +] +0.01 +0.1 +1 +10 +p [Myr] +no vortex +hp = 0.07 +hp = 0.1 +hp = 0.15 +Rp = 100 AU +Figure 1. Combined constraint (5) on the planetary mass 𝑀p and age 𝜏p, +shown for different parameters of a system with 𝑀★ = 𝑀⊙. Solid lines are +for 𝑅p = 50 AU and ℎp = 0.07 (fuchsia), ℎp = 0.1 (blue), ℎp = 0.15 +(green). Blue dashed line is for 𝑅p = 100 AU, ℎp = 0.1. Grey shaded region +is excluded as no vortices should appear in this part of the parameter space +(bounded by ℎp = 0.07 curve for illustration). Arrows indicate 𝑀th calculated +using ℎp corresponding to the arrow color (same as in the legend). Constraint +(5) — solid curves — is strictly valid only for 𝑀p ≲ 𝑀th. +0.1 +1 +10 +sys [Myr] +10 +100 +1000 +Mp [M +] +no vortex +hp = 0.07 +hp = 0.1 +hp = 0.15 +Rp = 100 AU +Figure 2. Mass constraint (6), (7) as a function of the system (stellar) age +𝜏sys. Meaning of curves, arrows and shading are the same as in Fig. 1. +3 VORTEX WEIGHING OF PLANETS +Let us suppose that the age (time since formation) of the protostar- +disc system 𝜏sys is known, e.g. from isochrone fitting of the charac- +teristics of the central star. This is usually the case at some level of +accuracy. Since, obviously, the planet is younger than its parent star, +one must have 𝜏sys > 𝜏p. However, the presence of a vortex at the gap +edge means that the inequality (3) is also fulfilled, which necessarily +implies that 𝜏sys > 𝜏vrt. Using equation (4), this condition can be +converted into a lower limit on 𝑀p: +𝑀p > 𝑀vrt = 𝑀th +� +𝐴 ℎ𝛽 +p +𝑃p +𝜏sys +�−1/𝛼 +. +(6) +In physical units, +𝑀vrt ≈ 40𝑀⊕ +� 𝜏sys +Myr +�−0.37 � +𝑅p +50AU +�0.56 � ℎp +0.1 +�2.7 � 𝑀★ +𝑀⊙ +�0.81 +. +(7) +Note a strong dependence of 𝑀vrt on ℎp, but a rather weak scaling +with 𝜏sys. This constraint is illustrated in Fig. 2. +Note that for the mass constraint (6)-(7) to be valid, the timescale +fit (1) should be justified in the first place. For this to be the case, +the planetary mass must be in the sub-thermal mass regime. One can +MNRAS 000, 1–7 (2022) + +Vortex weighing and dating +3 +easily show that +𝑀vrt +𝑀th +≈ 0.13 +� 𝜏sys +Myr +�−0.37 � +𝑅p +50AU +�0.56 � ℎp +0.1 +�−0.32 � 𝑀★ +𝑀⊙ +�−0.19 +, +(8) +i.e. the condition 𝑀p ≲ 𝑀th should be not difficult to satisfy in +general (in Figs. 1,2 we illustrate the values of 𝑀th with arrows). +Thus, we expect 𝑀vrt to provide a lower limit on 𝑀p quite generally. +The constraint (6)-(7) can be improved (i.e. 𝑀vrt increased) if we +had some independent way to set an upper limit on 𝜏p, which is +lower than the system age 𝜏sys. In practice, however, such refined +information on 𝜏p may be difficult to obtain. +4 VORTEX DATING OF PLANETS +One can also turn the argument around and assume that, in addition +to observing a vortex adjacent to a gap, we also know the mass 𝑀p +of the gap-opening planet — either via atmospheric modelling if the +planet is visible, or through indirect dynamical measurements if it +has not been imaged. We can then use the presence of the vortex to +set a lower limit on the planetary age 𝜏p via equation (3), in which +𝜏vrt ≈ 105yr +� +𝑀p +102𝑀⊕ +�−2.7 � +𝑅p +50AU +�1.5 � ℎp +0.1 +�7.2 � 𝑀★ +𝑀⊙ +�2.2 +. +(9) +If only an upper limit 𝑀↓ on planetary mass is available to us, +𝑀p < 𝑀↓ (e.g. from non-detection of the planet through near-IR +imaging), then one should use 𝑀↓ instead of 𝑀p in (9). Solid and +dashed lines in Fig. 1 give 𝜏vrt (as a function of 𝑀p or 𝑀↓) for +different values of ℎp and 𝑅p. +A constraint on 𝜏p would be extremely useful for understanding the +timing of planet formation. It can also serve as a consistency check for +calculations of planetary evolution post-formation, since the present +day temperature and luminosity of the planet are themselves functions +of its age 𝜏p (e.g. Linder et al. 2019), see Section 7. Unfortunately, the +accuracy of the lower limit (3) & (9) may be somewhat compromised +by the uncertainties in the determination of various parameters that +enter it, e.g. 𝑀p and, especially, ℎp, given how steeply 𝜏vrt scales +with them. +5 ACCOUNTING FOR ACCRETION HISTORY OF A +PLANET +Our results (1) & (2) for 𝜏vrt have been obtained in CR23 for a +constant 𝑀p (not varying in time). This implicitly assumes that planet +has grown to its final 𝑀p very rapidly, having accreted its mass +almost instantaneously; this accretion history is illustrated in panel +(a) of Fig. 3. In panel (b) we also illustrate the corresponding growth +of the characteristic amplitude1 𝐴𝜁 of the planet-induced vortensity +perturbation 𝜁, which is the variable that eventually determines vortex +generation (CR23): very crudely, one may expect the RWI to set in +when 𝐴𝜁 reaches some threshold value (illustrated with red dotted +line). The growth rate of 𝐴𝜁 in panel (b) is constant since it sensitively +depends on 𝑀p and 𝑀p is fixed in this case. +One may consider other representative histories of planetary mass +evolution. For example, in Fig. 3c 𝑀p undergoes an initial period of +accretion and then stays at its final value until the RWI sets in. As +another example, in panel (e) the planetary mass increases steadily +1 E.g. a maximum or minimum value of 𝜁 as a function of radius, see CR23. +and the RWI gets triggered while 𝑀p is still growing. For these +growth histories, the increase of 𝐴𝜁 is no longer purely linear, see +panels (d) and (f), and using the final planetary mass2 in formula +(1) we would underestimate the true age of the planet 𝜏p (illustrated +in top panels), i.e. the time since its growth has started and until +the present day when the vortex has emerged. Instead, application of +equation (1) would give us some other time 𝜏0, which is illustrated +by the orange lines (based on the growth rate of 𝐴𝜁 at the time when +RWI sets in) in panels (d) & (f). Since growth of 𝜁 accelerates (quite +steeply) for higher 𝑀p, the growth rate of 𝐴𝜁 can only increase in +time, so that 𝜏p ≥ 𝜏0 always (with equality only for 𝑀p(𝑡) = const., +see Fig. 3a,b). +Very importantly, this complication does not affect the validity +of our time constraint, since 𝜏0 is given by our equation (1) and +we just saw that 𝜏p ≥ 𝜏0. However, in some scenarios, e.g. in the +continuous accretion case shown in panels (e),(f), 𝜏0 can be much +shorter than 𝜏p, making our time constraint (3) too conservative. +Thus, it is desirable to find ways to somehow account for the history +of accretion (provided that it is known) to improve limits on 𝜏p. +One way to do this has already been discussed in CR23 and +amounts to replacing 𝜏p𝑀−𝛼 +p +with +∫ 𝜏p +0 +� +𝑀p(𝑡) +�−𝛼 d𝑡 in equation +(4); this modification allows us to account for the evolution of the +vortensity (or 𝐴𝜁 ) growth rate, which is proportional to 𝑀−𝛼 +p +, as +𝑀p(𝑡) increases. Thus, we generalize the combined constraint on 𝜏p +and 𝑀p in the case of an accreting planet to +∫ 𝜏p +0 +� +𝑀p(𝑡) +�−𝛼 d𝑡 > 𝐴𝑃p𝑀−𝛼 +th ℎ𝛽 +p . +(10) +Since this constraint must reduce to the inequality (4), we will assume +all its parameters — 𝛼, 𝛽, 𝐴 — to be still given by the equation3 (2). +Then in physical units equation (10) becomes +(Myr)−1 +∫ 𝜏p +0 +� 𝑀p(𝑡) +102𝑀⊕ +�2.7 +d𝑡 > 0.11 +� +𝑅p +50AU +�1.5 +× +� 𝑀★ +𝑀⊙ +�2.2 � ℎp +0.1 +�7.2 +. +(11) +For 𝑀p(𝑡) = const this inequality reduces to (5). +We can apply this generalized criterion to the simulations of Hal- +lam & Paardekooper (2020) who considered planetary accretion his- +tory in the form 𝑀p(𝑡) = 𝑀f sin2 [(𝜋/2)(𝑡/𝑡G)] (where 𝑡G is the +growth time) and determined the values of the final planet mass 𝑀f +such that the RWI would marginally set in at 𝑡 = 𝑡G. In our notation +this means setting 𝑡G = 𝜏vrt. We can use our results and determine the +relation between such 𝑡G and 𝑀f by changing inequality to equality +in equation (10) and setting 𝜏p = 𝑡G. We find, using the definition of +𝑀th and introducing 𝑞f = 𝑀f/𝑀★, +𝑡G = 𝐴 𝜅−1𝑃p 𝑞𝛼 +f ℎ𝛽−3𝛼 +p +, +(12) +where, for a particular accretion history of Hallam & Paardekooper +(2020), 𝜅 = +∫ 1 +0 [sin(𝜋𝑥/2)]2𝛼𝑑𝑥 ≈ 0.33. +As these authors also included the effects of viscosity, which is +2 For simplicity we neglect the possible growth of 𝑀p after the vortex has +appeared and until the present time. +3 This assumption is only approximate since the non-trivial history of accre- +tion may modify the radial profile of 𝜁 , which determines the RWI stability +(Cimerman & Rafikov 2021). Also, the RWI threshold itself is not entirely +universal (CR23). But this approximation should not be too bad as the RWI +onset is mainly determined by the late-time behavior of 𝑀p(𝑡). Finally, note +that theoretical arguments suggest 𝛼 = 2.6, but the numerical results are +closer to 𝛼 ≈ 2.7 (CR23). +MNRAS 000, 1–7 (2022) + +4 +R. R. Rafikov and N. P. Cimerman +0 +Mp(t) +p +a +Instant accretion +p +c +Initial episode of accretion +p +e +Continuous accretion +t +0 +A (t) +0 +b +t +0 +d +t +0 +f +Figure 3. Illustration of the different representative planetary accretion histories: (left) very rapid (instant) initial accretion to the final mass, (centre) extended +initial interval of accretion, (right) continuous accretion. Top panels illustrate 𝑀p(𝑡) (blue) while the bottom panels show the corresponding growth of the +characteristic amplitude 𝐴𝜁 (green) of the planet-driven vortensity perturbation (this calculations assumes d𝐴𝜁 /d𝑡 ∝ [𝑀p(𝑡)]2.7, see text). Arrows in the top +panels indicate the planetary age 𝜏p (time since the start of its accretion), while in the bottom panels they show the "time to vortex formation" 𝜏0 calculated +using equation (1) and assuming 𝑀p given by its final value. The red dotted line indicates the critical value of 𝐴𝜁 when the vortices are expected to appear. The +key point illustrated here is that 𝜏0 ≤ 𝜏p always. +known to delay the onset of RWI (Hammer et al. 2017), we cannot +directly compare our results for 𝑡G with theirs. However, if we focus +on their smallest 𝑞f = 1.5×10−4 (since in their setup this corresponds +to the lowest value of viscosity, closer to our inviscid setup) and adopt +their ℎp = 0.05 and the fit parameters (2), we find the age of the planet +to satisfy 𝜏p ≳ 𝑡G ≈ 39𝑃p. This is comfortably below 𝑡G ≈ 200𝑃p +that Hallam & Paardekooper (2020) find for the same 𝑞f, consistent +with the viscosity-driven delay. Equally importantly, had we used the +equation (4), that assumes 𝑀p = const, instead of (10), we would +have found 𝜏p ≳ 13𝑃p (a factor of 𝜅 lower), far less constraining +than the result that we obtained accounting for the (known) accretion +history. +Given the steep dependence of the integrand in (11) on 𝑀p (reflect- +ing 𝑀p-dependence of the 𝜁 growth rate), we expect 𝐴𝜁 to increase +the most when 𝑀p(𝑡) is close to its final value. This is indeed what +we see in Fig. 3d, in which the initial accretion episode contributes +only weakly to the total increase of 𝐴𝜁 , despite its duration being +comparable to the time interval when 𝑀p stayed at its final value (our +calculation in this plot assumed d𝐴𝜁 /d𝑡 ∝ 𝑀2.7 +p +for compatibility +with equation (11), see CR23). Thus, it is the history of 𝑀p accre- +tion at late times that is most important for determining the age of a +putative planet in an observed vortex-hosting system. +6 APPLICATION TO OBSERVED DISCS +We apply the constraints derived above to several protostellar +systems observed by ALMA, for which the vortices have been +invoked as a possible explanation of the observed non-axisymmetric +features — arcs, clumps, etc. It is important to remember that +the features detected in continuum emission by ALMA are due to +thermal emission of dust grains, while our results on the emergence +of vortices apply to the gaseous component of the disc. However, +it has been shown by a number of authors (Barge & Sommeria +1995; Godon & Livio 1999; Fu et al. 2014) that vortices are very +efficient at trapping dust, providing support to our association of +the dust asymmetries with the gas vortices in PPDs. Since our +limits on 𝜏p and 𝑀p are highly sensitive to the disk aspect ratio ℎp, +which is poorly known in most cases, we will retain the scaling with +ℎ0.1 = ℎp/0.1 in our estimates. +HD 135344B (SAO 206462) +This 𝑀★ = 1.5𝑀⊙, 𝜏sys ≈ 9 Myr old (Asensio-Torres et al. 2021) +Herbig F star harbours a transitional disc. ALMA dust continuum +observations reveal an axisymmetric inner ring separated by a gap- +like structure (centered around 70 AU) from an (outer) arc that can +be interpreted as a vortex at the outer gap edge (van der Marel +et al. 2016; Cazzoletti et al. 2018). The possibility of a planetary +origin of these structures is supported by the near-IR scattered light +observations of a two-armed spiral (Muto et al. 2012), although a +unified model explaining all these features at once is lacking. We +will nevertheless assume that the gap and the outer vortex are due to +the (unseen) gap-opening planet at 𝑅p ≈ 70 AU, and the inner ring +reflects dust trapping at the pressure maximum at the inner gap edge. +These data and equations (6)-(7) allow us to constrain planetary mass +as 𝑀p ≳ 32ℎ2.7 +0.1𝑀⊕. +Direct imaging of HD 135344B with VLT/SPHERE sets an upper +limit of 𝑀↓ ≈ 4𝑀J on the mass of a planetary object at ∼ 102AU +scales (Asensio-Torres et al. 2021). Unfortunately, this 𝑀↓ is higher +than the thermal mass 𝑀th = 1.5ℎ3 +0.1𝑀J, which makes the use of the +timescale constraint (3),(9) unjustified (its blind application would +give 𝜏vrt ≈ 500ℎ7.2 +0.1 yr, comparable to the orbital period at the gap +location and not constraining 𝜏p effectively). +HD 36112 (MWC 758) +This 𝑀★ ≈ 1.8𝑀⊙, 𝜏sys ≈ 9 Myr old (Asensio-Torres et al. 2021) +star harbours two clumps on top of the two rings separated by a gap +in the outer disc (Dong et al. 2018). Neglecting the slight eccentricity +of the disc and assuming the rings with clumps to correspond to the +inner and outer edges of the gap carved by a planet, we will adopt +𝑅p ≈ 70 AU for the planetary orbit. Then equations (6)-(7) allow us +to set a mass constraint 𝑀p ≳ 37ℎ2.7 +0.1𝑀⊕. +Analysis of the direct imaging observations of this system by +Asensio-Torres et al. (2021) suggests that the upper limit on the +possible point source inside the assumed gap is ∼ 8𝑀J, significantly +higher than 𝑀th, precluding us from meaningfully constraining the +MNRAS 000, 1–7 (2022) + +Vortex weighing and dating +5 +age of the planet. +HD 143006 +This G-type T Tauri star with 𝑀★ = 1.8𝑀⊙ and an estimated age +of 𝜏sys ≈ 8 Myr harbours a disc rich in substructures (Pérez et al. +2018). In addition to a misaligned inner disc, it features two outer +rings separated by a gap centered around 52 AU, with an arc just +outside the outermost ring. Interpreting these features as produced +by an unseen planet inside the gap at 𝑅p = 52 AU, we get the mass +constraint 𝑀p ≳ 33ℎ2.7 +0.1𝑀⊕ from equations (6)-(7). +NaCo/VLT direct imaging does not provide a useful constraint on +the mass of a putative planet, with 𝑀↓ at the level of several tens of +𝑀J at the outer gap location (Jorquera et al. 2021). Thus, we cannot +set a useful lower limit on the planetary age. +V1247 Ori +V1247 Ori is a 𝜏sys = 7.5 Myr old, 𝑀★ ≈ 1.9𝑀⊙ star (Willson et al. +2019) harbouring a pre-transitional disc. ALMA dust continuum ob- +servations (Kraus et al. 2017) reveal an inner disc (or ring) separated +by a gap from the outer arc, which may be interpreted as a vortex at +the outer gap edge. Assuming a planet to be in the gap, at 𝑅p ≈ 90 +AU, one finds that the planetary mass must satisfy 𝑀p ≳ 48ℎ2.7 +0.1𝑀⊕. +While we could not find explicit limits on the mass of the putative +planet in V1247 Ori system from direct imaging observations, Kraus +et al. (2017) found that an 𝑀p = 3𝑀J planet can roughly match +the shape of the spiral observed in scattered light using HiCIAO. +Unfortunately, this 𝑀p is again above 𝑀th, not allowing the age of +the planet to be meaningfully constrained. +7 DISCUSSION +7.1 Combination of multiple constraints +Our limits on 𝑀p and 𝜏p based on the presence of vortices next +to gaps in PPDs become even more powerful when combined with +additional constraints on these key parameters. In particular, young +planets passively lose thermal energy that they have been endowed +with at formation, resulting in their luminosity decreasing with time. +As more massive planets retain more heat at formation, it takes +them longer to cool. Thus, if one can observationally constrain the +luminosity of a planet 𝐿p to lie below a certain limit (or determine +it in the case of direct detection), this would provide an additional +constraint on 𝑀p and 𝜏p. +We illustrate this approach in Fig. 4, where we show the vortex- +based constraints from Fig. 1 together with the constraint 𝐿p < +10−6𝐿⊙ (outside the pink shaded region to the right of the black +dotted curve) based on the work4 of Linder et al. (2019). We also +show the 𝐿p = 10−7𝐿⊙ curve (red dotted) which may be relevant +for future direct imaging experiments. In addition, we impose a con- +straint 𝜏p < 15 Myr (region below the orange dot-dashed line) since +protoplanetary discs usually do not survive for that long (similar to +the logic used in Section 3). There are other, complementary ways +of constraining planetary properties, for example gap width/depth +fitting (Dong & Fung 2017; Asensio-Torres et al. 2021) which can +provide model-dependent information on 𝑀p for individual systems; +we will not consider them here. +4 We use the tracks for the bolometric luminosity 𝐿p of a fixed mass planet +from Fig. 6 of Linder et al. (2019), which assume evolution with a cloud-free +atmosphere of solar metallicity and use the petitCODE grid. +10 +100 +1000 +Mp [M +] +0.01 +0.1 +1 +10 +p [Myr] +no vortex +hp = 0.07 +hp = 0.1 +hp = 0.15 +Rp = 100 AU +Lp = 10 +6L +Lp = 10 +7L +p = 15 Myr +Figure 4. Combination of the different constraints on the mass 𝑀p and age +𝜏p of a planet in a vortex-hosting PPD. Grey shaded region is excluded +(for ℎp = 0.07) as vortices have no time to develop in this part of the +parameter space, analogous to Fig. 1 (solid and dashed lines are the same as +in that figure). Pink shaded region is excluded as it corresponds to planetary +(bolometric) cooling luminosity 𝐿p exceeding 𝐿p = 10−6𝐿⊙ (black dotted +curve). We also show the curve 𝐿p = 10−7𝐿⊙ (red dotted curve); the 𝐿p +curves are based on Linder et al. (2019). The orange shaded region above +the orange dot-dashed curve excludes planetary ages above 15 Myr. Planets +satisfying all three constraints reside in the unshaded part of the parameter +space. +A combination of the three constraints — based on planetary lu- +minosity, age and presence of vortices — limits planetary 𝑀p and +𝜏p to lie within the unshaded region. This region shrinks for hotter +discs with larger ℎp (compare fuchsia and green solid curves), as +well as for larger 𝑅p (compare blue solid and dashed lines). Thus, the +vortex-based limits are more stringent in hotter discs and for more +distant planets. Also, the allowed region would shrink even further as +the upper limit on 𝐿p gets lowered in the future. It should also be re- +membered that the luminosity-based (dotted) curves assume that the +(possible ongoing) gas accretion provides insignificant contribution +to 𝐿p. If the planetary accretion luminosity is non-negligible, this +would additionally shift the dotted curves to the left, constraining +𝑀p and 𝜏p even further. +7.2 Utility of the vortex-based constraint +The sub-Jovian value of 𝑀vrt implied by the equation (7) and our +estimates in Section 6 is very relevant in light of the recent results +(Dong et al. 2017; Bae et al. 2017; Miranda & Rafikov 2019, 2020a,b) +showing that in a low viscosity disc a single sub-𝑀th planet can give +rise to a series of several prominent gaps and rings in the radial dust +distribution. For example, for the AS 209 system (𝑀★ = 0.83𝑀⊙, +𝜏sys ≈ 1 Myr, Andrews et al. 2018) imaged with ALMA Zhang et al. +(2018) have shown that a single planet with 𝑀p as low as 25𝑀⊕ orbit- +ing within the outer (primary) gap at 𝑅p ≈ 100AU can be responsible +for creating all five gaps observed in this disc. This possibility makes +the typical values of 𝑀p implied by the constraint (6)-(7) very inter- +esting for understanding the architecture of the underlying planetary +system. +Given the upper limits on 𝑀p based on direct imaging in several +systems covered in Section 6, we found our age constraint (3) & (9) +to be not very useful at present. However, things will improve as +𝑀↓ decreases in the future. Once 𝑀↓ is below 𝑀th, our constraint +(3) & (9) becomes valid and may provide useful information on the +MNRAS 000, 1–7 (2022) + +6 +R. R. Rafikov and N. P. Cimerman +planetary age. The decrease of 𝑀↓ may not necessarily come from +improved direct imaging capabilities. In particular, one may use the +technique of multiple gap fitting used by Zhang et al. (2018) for AS +209 to get a much better measurement of 𝑀p or 𝑀↓. +Just for illustration, let us imagine that AS 209 did possess a vortex +at the edge of its outermost gap (just outside 𝑅p = 100AU). Then +using 𝑀p ≈ 25𝑀⊕ (based on Zhang et al. 2018) equation (9) would +predict 𝜏p ≳ 8ℎ7.2 +0.1 Myr. This 𝜏p is much longer (for ℎp = 0.1) than +the age of the system 𝜏sys ≈ 1 Myr, and could have implied that either +the planetary mass is underestimated (by a factor of ∼ 2), or that the +stellar age is underestimated (by almost an order of magnitude), or +that the disc is somewhat colder — using ℎp = 0.075 (consistent +with Paneque-Carreño et al. 2022) in (9) would reconcile 𝜏p with its +estimated 𝜏sys. +The latter possibility represents a simple way to resolve the age +discrepancy for this imaginary AS 209-like system. It also highlights +the importance of good knowledge of the thermal state of the disc +near the planet, which sets ℎp. Indeed, 𝜏vrt depends very sensitively +on ℎp and a mis-estimate of ℎp by a factor of 2 would result in a +factor of ≈ 150 error in the determination of 𝜏vrt and the planetary +age. The situation is somewhat improved for the mass constraint +(6)-(7), in which variation of ℎp by a factor of 2 results in 𝑀vrt +changing by a factor of ≈ 6.5. In any case, good understanding +of disc thermodynamics is clearly needed when applying the age +constraint (3) & (9). Recent ALMA measurements of emission heights +of different molecular lines in PPDs (Law et al. 2021, 2022; Paneque- +Carreño et al. 2022) provide a (model-dependent) way to determine +disc aspect ratio at different radii, generally finding values in the +range ℎp ∼ (0.07 − 0.1) for 𝑅p ∼ (50 − 100) AU. +On the other hand, our constraints (5),(7) & (9) should be rather +insensitive to the radial profile of the disc surface density near the +planet. Indeed, CR23 showed that the parameters of the fit (1),(2) +show little variation when changing the slope of the surface density +profile near the planet. Also, the dependence of the vortex-based con- +straints on 𝑅p and 𝑀★ is not as steep as for ℎp, and the characteristic +accuracy with which these parameters can be measured is (10−20)% +or better. +7.3 Additional processes and further extensions +Since the constraints (3)-(6) are lower limits on 𝜏p and 𝑀p, respec- +tively, they do not change if the vortices we observe in discs now are +not the first generation vortices. It is possible that the vortices that +formed early on have then dissolved and what we are seeing now are +the second (or multiple) generation vortices (Hammer et al. 2021). +Nevertheless, even in this case the condition 𝜏p > 𝜏vrt would still +need to be fulfilled, definitely for the first generation of vortices, as +well as for the following generations, confirming the validity of the +constraints (3)-(6). +Similarly, dust trapped in vortices can maintain observable non- +axisymmetric distribution even after the vortices in the gaseous com- +ponent dissolve (Fu et al. 2014). Thus, when we see an asymmetry in +dust continuum observations, the original vortex that has led to it may +have already been gone. However, this would again not invalidate the +constraints obtained in Sections 3 & 4. +The fit (1),(2) for 𝜏vrt was derived by CR23 for discs which are +inviscid or have low viscosity, an assumption which is consistent +with observations of many systems (see Section 2). We can roughly +estimate the upper limit on the viscosity 𝜈 below which the inviscid +assumption should be valid by demanding the timescale on which the +vortensity structures produced by the planet get viscously diffused +away to be longer that the age of the system 𝜏sys. For the characteristic +radial scale of the vortensity structures 𝐿 ∼ 𝐻p(𝑀p/𝑀th)−0.4 (Dong +et al. 2011; CR23) this timescale is ∼ 𝐿2/𝜈 ∼ 𝑃p𝛼−1(𝑀p/𝑀th)−0.8, +where we adopted the 𝛼-ansatz for the viscosity 𝜈 = 𝛼Ωp𝐻2p (and +Ωp = 2𝜋𝑃−1 +p ). For this to exceed 𝜏sys for a sub-thermal mas planet +we require that roughly 𝛼 ≲ 𝑃p/𝜏sys ∼ 10−4, given the long orbital +periods at 𝑅p = 50 − 100 AU. A more refined estimate of the critical +𝛼 can be found in CR23. +However, even if the disc were sufficiently viscous (i.e. for +𝛼 ≳ 10−4), the RWI development would get only delayed (Hallam & +Paardekooper 2020) or the instability may be suppressed altogether, +see Hammer et al. (2017), CR23. Because of that, our inviscid esti- +mate for 𝜏vrt continues to provide a lower limit on 𝜏p in the presence +of a vortex, i.e. the equation (3) and all other constraints remain valid +(see Section 5 for application of this logic). +On the other hand, some other effects may accelerate vortex pro- +duction compared to the results of CR23. For example, this could +happen as a result of baroclinicity of the disc near the planet since +the RWI is sensitive to entropy gradients (Lovelace et al. 1999). +Under certain circumstances dust feedback can also promote vor- +tex production (Lin & Youdin 2017). These processes, if they are +important, may somewhat weaken our constraints on 𝑀p and 𝜏p. +We demonstrated in Section 5 how our constraints can be modified +to account for the evolution of planetary mass 𝑀p. Other relevant +parameters might change as well, for example 𝑅p can vary as a result +of planet migration, or ℎp can change as the disc evolves in time. +CR23 outlined ways in which one can account for these processes +to derive a new estimate for 𝜏vrt instead of (1),(2), thus providing a +pathway to modifying our constraints on 𝑀p and 𝜏p. +Of the four systems considered in Section 6, three show vortex-like +non-axisymmetries only at the outer edge of the putative planetary +gap, and only one, MWC 758, has them on both sides of the gap. +This is somewhat surprising, since the simulations of CR23 not only +show the emergence of vortices on both sides of the gap, but also +demonstrate that the time interval separating their production by +RWI is typically smaller than 𝜏vrt (see Table 1 in that work). Thus, +one would expect to see vortices on both sides of the gap more +often. It is not clear why this expectation fails. It could be that the +dust concentration is more efficient in the outer vortices5 or that it +tends to survive there considerably longer than in the inner ones. +Or that some physical processes neglected in our study suppress the +formation of the inner vortices. Expanding the sample of observed +discs with vortex-like asymmetries would help in resolving this issue +in the future. +8 SUMMARY +In this work we used the results of CR23 on the time it takes visible +gas vortices to appear next to a gap carved by a low-mass planet in a +low-viscosity PPD to set constraints on the masses 𝑀p and ages 𝜏p of +planets in PPDs with observed vortex-like structures. We found that +the presence of a vortex sets a lower limit on a particular combination +of 𝑀p and 𝜏p, with separate constraints on these variables possible +if some additional information (such as the system age 𝜏sys or the +upper limit on the planetary mass 𝑀↓) is available. These considera- +tions allowed us to constrain the masses of putative planets in several +vortex-bearing PPDs to be above several tens of 𝑀⊕. The limits +5 Outer vortices should form first in inviscid discs with radially decreasing +surface density (CR23). +MNRAS 000, 1–7 (2022) + +Vortex weighing and dating +7 +on the planetary age are not very constraining at the moment, but +they will improve as future observations lower 𝑀↓. Our constraints +can be extended to account for the non-trivial history of planetary +mass accretion, and we provide a recipe for doing that in Section +5. Finally, we showed the robustness of our constraints in light of +additional complications (e.g. non-zero disc viscosity, multiple gen- +eration of vortices, etc.) and demonstrated their useful synergy with +other types of constraints on 𝑀p and 𝜏p, e.g. based on the upper lim- +its on the planetary cooling luminosity coming from direct imaging +observations. +ACKNOWLEDGEMENTS +Software: Matplotlib (Hunter 2007). Authors are grateful to Ewine +van Dishoeck for illuminating discussions and to an anonymous ref- +eree for useful suggestions. R.R.R. acknowledges financial support +through the Science and Technology Facilities Council (STFC) grant +ST/T00049X/1 and Ambrose Monell Foundation. N.P.C. is funded +by a STFC and Isaac Newton studentship. +DATA AVAILABILITY +The data underlying this article will be shared on reasonable request +to the corresponding author. +REFERENCES +Andrews S. M., 2020, ARA&A, 58, 483 +Andrews S. 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N., 2021, MNRAS, 507, 5523 +Willson M., et al., 2019, A&A, 621, A7 +Zhang S., et al., 2018, ApJ, 869, L47 +van der Marel N., Cazzoletti P., Pinilla P., Garufi A., 2016, ApJ, 832, 178 +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–7 (2022) + diff --git a/TdAzT4oBgHgl3EQf0v6x/content/tmp_files/load_file.txt b/TdAzT4oBgHgl3EQf0v6x/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2d8611d82c05321b16e36de5add292cf90499326 --- /dev/null +++ b/TdAzT4oBgHgl3EQf0v6x/content/tmp_files/load_file.txt @@ -0,0 +1,646 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf,len=645 +page_content='MNRAS 000, 1–7 (2022) Preprint 6 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='0 Vortex weighing and dating of planets in protoplanetary discs Roman R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Rafikov1,2★, Nicolas P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Cimerman1 1Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK 2Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' in original form ZZZ ABSTRACT High-resolution sub-mm observations of some protoplanetary discs reveal non-asixymmetric features, which can often be interpreted as dust concentrations in vortices that form at the edges of gaps carved out by the embedded planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' We use recent results on the timescale for the planet-driven vortex development in low-viscosity discs to set constraints on the mass and age of a planet producing the vortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Knowledge of the age of the central star in a vortex-bearing protoplanetary disc system allows one to set a lower limit on the planetary mass at the level of several tens of 𝑀⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Also, an independent upper limit on the planetary mass would constrain the planetary age, although given the current direct imaging detection limits this constraint is not yet very stringent (it is also sensitively dependent on the disc scale height).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' These results can be extended to account for the history of planetary mass accretion if it is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' We apply our calculations to several protoplanetary discs harbouring vortex-like features as revealed by ALMA and set limits of (30−50)𝑀⊕ (for disc aspect ratio of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='1) on the minimum masses of putative planets that could be responsible for these vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Our vortex-based method provides an independent way of constraining the properties of embedded planets, complementary to other approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Key words: hydrodynamics – instabilities – shock waves – accretion discs – planets and satellites: formation – methods: numerical 1 INTRODUCTION Observations of protoplanetary discs (PPDs) in dust continuum emis- sion with ALMA revealed a variety of substructures (Andrews 2020), including axisymmetric gaps and rings as well as non-axisymmetric clumps and arcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' An intriguing possibility is that these features could be produced by young embedded planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' In particular, gravitational coupling between a massive planet and the disc is known to result in formation of observable gaps around the planetary orbit (Papaloizou & Lin 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Rafikov 2002b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' The evolution of vortensity (potential vorticity) at the edges of these gaps (Lin & Papaloizou 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Cimerman & Rafikov 2021) due to shock dissipation of the planet-driven density waves (Goodman & Rafikov 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Rafikov 2002a) can trigger the Rossby Wave Instability (RWI, Lovelace et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 1999) resulting in the formation of fluid vortices at these locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Dust accumulation inside the vortices (Barge & Sommeria 1995) naturally leads to observable non-axisymmetric arcs and lobes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Formation of these structures does not necessarily require massive (Jovian) planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' For example, it was shown (Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Bae et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Miranda & Rafikov 2019, 2020a,b) that multiple visible gaps and rings in the dust distribution can result from nonlinear damping of multiple spirals triggered by a single sub-Jovian mass planet in a low viscosity disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' The mass of the planet 𝑀p can in fact be below the so-called thermal mass defined as 𝑀th = �𝐻p/𝑅p �3 𝑀★ = ℎ3p 𝑀★, where 𝐻p is the disc scale height at the planetary distance 𝑅p, ℎp = 𝐻p/𝑅p is the disc aspect ratio there, and 𝑀★ is the stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Emergence of vortices at the edges of planetary gaps also does not ★ E-mail: rrr@damtp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='cam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='uk (RRR) require massive planets if the disc is almost inviscid (Hammer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Hallam & Paardekooper 2020), and sub-𝑀th mass planets can easily trigger them (Cimerman & Rafikov 2023, hereafter CR23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' In this study we focus on non-axisymmetric disc features which can be interpreted as planet-induced vortices (other ways to produce vortices are mentioned in Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Their development is not an instantaneous process as the evolution of vortensity near the planetary orbit towards the RWI takes a certain amount of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' As we show in this work, this fact can be exploited to set useful constraints on the mass and/or age of a putative planet responsible for production of the observed vortices in a PPD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' This method relies on the recent calculation of CR23 who studied the development of vortices in inviscid PPDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' In that work, we showed that the time it takes for vortices to emerge at the edge of the gap carved out by a sub-𝑀th mass planet can be approximated as 𝜏vrt ≈ 𝐴 𝑃p � 𝑀p 𝑀th � 𝛼 ℎ𝛽 p , where (1) 𝐴 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='6, 𝛼 ≈ −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='7, 𝛽 ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='86, (2) and 𝑃p is the orbital period at 𝑅p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' This result assumes 𝑀p to be fixed in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Interestingly, CR23 found 𝜏vrt to only weakly depend on the radial profile of surface density in the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' We describe the general idea of our method in Section 2 and show how it can constrain the mass and age of the planet in Section 3 and Section 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' In Section 5 we extend our constraints to the case of a planet accreting its mass over an extended period of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' We apply our results to observed PPDs in Section 6 and discuss them in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' © 2022 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='01789v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='EP] 4 Jan 2023 2 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Rafikov and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Cimerman 2 GENERAL IDEA OF THE METHOD Let us suppose that observations of dust continuum emission reveal a gap in a PPD, together with a non-axisymmetric lobe or arc at the gap edge, indicative of a vortex which traps dust grains at this location (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' van der Marel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Kraus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Pérez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' We will interpret this observation by assuming that a planet (not necessarily directly visible) is located within the gap and is responsible for creating both the gap and the vortex (via the RWI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' The spatial association of a vortex with an adjacent gap provides strong support to this interpretation and makes some other possibilities for triggering vortices, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' global baroclinic instability (Klahr & Bodenheimer 2003), convective overstability (Teed & Latter 2021), vertical shear instability (Richard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 2016) less attractive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' We assume the disc viscosity to be low (essentially inviscid), consistent with many observations of PPDs (Pinte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Rafikov 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Flaherty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' For now we will also assume that 𝑀p has been constant ever since the planet appeared in the disc, a constraint that we will relax in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' With these conditions fulfilled, the result (1)-(2) applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' We can then use the observation of the vortex to set a constraint on a particular combination of the planetary mass 𝑀p and the planetary age 𝜏p — the time that has passed since the planet has reached its final mass 𝑀p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Indeed, the observation of a vortex at the gap edge implies that the RWI had enough time to fully develop into the non-linear stage in that region, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' that 𝜏p > 𝜏vrt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' (3) Together with equation (1) this leads to the following combined constraint on 𝜏p and 𝑀p: 𝜏p𝑀−𝛼 p > 𝐴𝑃p𝑀−𝛼 th ℎ𝛽 p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' (4) With fit parameters (2) we can write this in physical units as 𝜏p Myr � 𝑀p 102𝑀⊕ �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='7 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='11 � 𝑅p 50AU �1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='5 � 𝑀★ 𝑀⊙ �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='2 � ℎp 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='1 �7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' (5) This condition must be fulfilled whenever a vortex is observed at the gap edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' It is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 1 for several values of ℎp and 𝑅p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' In a similar vein, the absence of vortex-like structures at the edges of a visible gap in a disc might be interpreted as meaning that 𝜏p < 𝜏vrt, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' that planet-driven accumulation of vortensity has not yet led to RWI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' If that were the case, the inequality in the constraint (4)-(5) would change its sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' However, this possibility has an important caveat as the absence of a vortex may also be interpreted differently: it could have formed at the gap edge earlier but then got destroyed through one of the processes that tend to destabilize vortices once they evolve into the nonlinear regime: the elliptical instability (Lesur & Papaloizou 2009), baroclinic effects (Rometsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Fung & Ono 2021), dust feedback (Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 2014), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Also, vortex formation may have been delayed or suppressed altogether if disc viscosity is sufficiently high (Hammer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Hallam & Paardekooper 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Thus, the lack of a vortex near a planetary gap cannot be unambiguously interpreted as meaning that the embedded planet did not get a chance to create it, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' that 𝜏p < 𝜏vrt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' For that reason (and unlike Hallam & Paardekooper 2020) in the following we will not draw any conclusions from the absence of vortices at the edges of putative planetary gaps found in sub-mm observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' We will now show how equations (4) & (5) can be used to sepa- rately constrain 𝑀p or 𝜏p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 10 100 1000 Mp [M ] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='1 1 10 p [Myr] no vortex hp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='07 hp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='1 hp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='15 Rp = 100 AU Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Combined constraint (5) on the planetary mass 𝑀p and age 𝜏p, shown for different parameters of a system with 𝑀★ = 𝑀⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Solid lines are for 𝑅p = 50 AU and ℎp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='07 (fuchsia), ℎp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='1 (blue), ℎp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='15 (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Blue dashed line is for 𝑅p = 100 AU, ℎp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Grey shaded region is excluded as no vortices should appear in this part of the parameter space (bounded by ℎp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='07 curve for illustration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Arrows indicate 𝑀th calculated using ℎp corresponding to the arrow color (same as in the legend).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Constraint (5) — solid curves — is strictly valid only for 𝑀p ≲ 𝑀th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='1 1 10 sys [Myr] 10 100 1000 Mp [M ] no vortex hp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='07 hp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='1 hp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='15 Rp = 100 AU Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Mass constraint (6), (7) as a function of the system (stellar) age 𝜏sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Meaning of curves, arrows and shading are the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 3 VORTEX WEIGHING OF PLANETS Let us suppose that the age (time since formation) of the protostar- disc system 𝜏sys is known, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' from isochrone fitting of the charac- teristics of the central star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' This is usually the case at some level of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Since, obviously, the planet is younger than its parent star, one must have 𝜏sys > 𝜏p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' However, the presence of a vortex at the gap edge means that the inequality (3) is also fulfilled, which necessarily implies that 𝜏sys > 𝜏vrt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Using equation (4), this condition can be converted into a lower limit on 𝑀p: 𝑀p > 𝑀vrt = 𝑀th � 𝐴 ℎ𝛽 p 𝑃p 𝜏sys �−1/𝛼 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' (6) In physical units, 𝑀vrt ≈ 40𝑀⊕ � 𝜏sys Myr �−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='37 � 𝑅p 50AU �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='56 � ℎp 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='1 �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='7 � 𝑀★ 𝑀⊙ �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='81 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' (7) Note a strong dependence of 𝑀vrt on ℎp, but a rather weak scaling with 𝜏sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' This constraint is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Note that for the mass constraint (6)-(7) to be valid, the timescale fit (1) should be justified in the first place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' For this to be the case, the planetary mass must be in the sub-thermal mass regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' One can MNRAS 000, 1–7 (2022) Vortex weighing and dating 3 easily show that 𝑀vrt 𝑀th ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='13 � 𝜏sys Myr �−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='37 � 𝑅p 50AU �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='56 � ℎp 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='1 �−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='32 � 𝑀★ 𝑀⊙ �−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='19 , (8) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' the condition 𝑀p ≲ 𝑀th should be not difficult to satisfy in general (in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 1,2 we illustrate the values of 𝑀th with arrows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Thus, we expect 𝑀vrt to provide a lower limit on 𝑀p quite generally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' The constraint (6)-(7) can be improved (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 𝑀vrt increased) if we had some independent way to set an upper limit on 𝜏p, which is lower than the system age 𝜏sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' In practice, however, such refined information on 𝜏p may be difficult to obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 4 VORTEX DATING OF PLANETS One can also turn the argument around and assume that, in addition to observing a vortex adjacent to a gap, we also know the mass 𝑀p of the gap-opening planet — either via atmospheric modelling if the planet is visible, or through indirect dynamical measurements if it has not been imaged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' We can then use the presence of the vortex to set a lower limit on the planetary age 𝜏p via equation (3), in which 𝜏vrt ≈ 105yr � 𝑀p 102𝑀⊕ �−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='7 � 𝑅p 50AU �1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='5 � ℎp 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='1 �7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='2 � 𝑀★ 𝑀⊙ �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' (9) If only an upper limit 𝑀↓ on planetary mass is available to us, 𝑀p < 𝑀↓ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' from non-detection of the planet through near-IR imaging), then one should use 𝑀↓ instead of 𝑀p in (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Solid and dashed lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 1 give 𝜏vrt (as a function of 𝑀p or 𝑀↓) for different values of ℎp and 𝑅p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' A constraint on 𝜏p would be extremely useful for understanding the timing of planet formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' It can also serve as a consistency check for calculations of planetary evolution post-formation, since the present day temperature and luminosity of the planet are themselves functions of its age 𝜏p (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Linder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 2019), see Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Unfortunately, the accuracy of the lower limit (3) & (9) may be somewhat compromised by the uncertainties in the determination of various parameters that enter it, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 𝑀p and, especially, ℎp, given how steeply 𝜏vrt scales with them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 5 ACCOUNTING FOR ACCRETION HISTORY OF A PLANET Our results (1) & (2) for 𝜏vrt have been obtained in CR23 for a constant 𝑀p (not varying in time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' This implicitly assumes that planet has grown to its final 𝑀p very rapidly, having accreted its mass almost instantaneously;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' this accretion history is illustrated in panel (a) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' In panel (b) we also illustrate the corresponding growth of the characteristic amplitude1 𝐴𝜁 of the planet-induced vortensity perturbation 𝜁, which is the variable that eventually determines vortex generation (CR23): very crudely, one may expect the RWI to set in when 𝐴𝜁 reaches some threshold value (illustrated with red dotted line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' The growth rate of 𝐴𝜁 in panel (b) is constant since it sensitively depends on 𝑀p and 𝑀p is fixed in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' One may consider other representative histories of planetary mass evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' For example, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 3c 𝑀p undergoes an initial period of accretion and then stays at its final value until the RWI sets in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' As another example, in panel (e) the planetary mass increases steadily 1 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' a maximum or minimum value of 𝜁 as a function of radius, see CR23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' and the RWI gets triggered while 𝑀p is still growing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' For these growth histories, the increase of 𝐴𝜁 is no longer purely linear, see panels (d) and (f), and using the final planetary mass2 in formula (1) we would underestimate the true age of the planet 𝜏p (illustrated in top panels), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' the time since its growth has started and until the present day when the vortex has emerged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Instead, application of equation (1) would give us some other time 𝜏0, which is illustrated by the orange lines (based on the growth rate of 𝐴𝜁 at the time when RWI sets in) in panels (d) & (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Since growth of 𝜁 accelerates (quite steeply) for higher 𝑀p, the growth rate of 𝐴𝜁 can only increase in time, so that 𝜏p ≥ 𝜏0 always (with equality only for 𝑀p(𝑡) = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=', see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 3a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Very importantly, this complication does not affect the validity of our time constraint, since 𝜏0 is given by our equation (1) and we just saw that 𝜏p ≥ 𝜏0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' However, in some scenarios, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' in the continuous accretion case shown in panels (e),(f), 𝜏0 can be much shorter than 𝜏p, making our time constraint (3) too conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Thus, it is desirable to find ways to somehow account for the history of accretion (provided that it is known) to improve limits on 𝜏p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' One way to do this has already been discussed in CR23 and amounts to replacing 𝜏p𝑀−𝛼 p with ∫ 𝜏p 0 � 𝑀p(𝑡) �−𝛼 d𝑡 in equation (4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' this modification allows us to account for the evolution of the vortensity (or 𝐴𝜁 ) growth rate, which is proportional to 𝑀−𝛼 p , as 𝑀p(𝑡) increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Thus, we generalize the combined constraint on 𝜏p and 𝑀p in the case of an accreting planet to ∫ 𝜏p 0 � 𝑀p(𝑡) �−𝛼 d𝑡 > 𝐴𝑃p𝑀−𝛼 th ℎ𝛽 p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' (10) Since this constraint must reduce to the inequality (4), we will assume all its parameters — 𝛼, 𝛽, 𝐴 — to be still given by the equation3 (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Then in physical units equation (10) becomes (Myr)−1 ∫ 𝜏p 0 � 𝑀p(𝑡) 102𝑀⊕ �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='7 d𝑡 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='11 � 𝑅p 50AU �1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='5 × � 𝑀★ 𝑀⊙ �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='2 � ℎp 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='1 �7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' (11) For 𝑀p(𝑡) = const this inequality reduces to (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' We can apply this generalized criterion to the simulations of Hal- lam & Paardekooper (2020) who considered planetary accretion his- tory in the form 𝑀p(𝑡) = 𝑀f sin2 [(𝜋/2)(𝑡/𝑡G)] (where 𝑡G is the growth time) and determined the values of the final planet mass 𝑀f such that the RWI would marginally set in at 𝑡 = 𝑡G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' In our notation this means setting 𝑡G = 𝜏vrt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' We can use our results and determine the relation between such 𝑡G and 𝑀f by changing inequality to equality in equation (10) and setting 𝜏p = 𝑡G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' We find, using the definition of 𝑀th and introducing 𝑞f = 𝑀f/𝑀★, 𝑡G = 𝐴 𝜅−1𝑃p 𝑞𝛼 f ℎ𝛽−3𝛼 p , (12) where, for a particular accretion history of Hallam & Paardekooper (2020), 𝜅 = ∫ 1 0 [sin(𝜋𝑥/2)]2𝛼𝑑𝑥 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' As these authors also included the effects of viscosity, which is 2 For simplicity we neglect the possible growth of 𝑀p after the vortex has appeared and until the present time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 3 This assumption is only approximate since the non-trivial history of accre- tion may modify the radial profile of 𝜁 , which determines the RWI stability (Cimerman & Rafikov 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Also, the RWI threshold itself is not entirely universal (CR23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' But this approximation should not be too bad as the RWI onset is mainly determined by the late-time behavior of 𝑀p(𝑡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Finally, note that theoretical arguments suggest 𝛼 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='6, but the numerical results are closer to 𝛼 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='7 (CR23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' MNRAS 000, 1–7 (2022) 4 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Rafikov and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Cimerman 0 Mp(t) p a Instant accretion p c Initial episode of accretion p e Continuous accretion t 0 A (t) 0 b t 0 d t 0 f Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Illustration of the different representative planetary accretion histories: (left) very rapid (instant) initial accretion to the final mass, (centre) extended initial interval of accretion, (right) continuous accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Top panels illustrate 𝑀p(𝑡) (blue) while the bottom panels show the corresponding growth of the characteristic amplitude 𝐴𝜁 (green) of the planet-driven vortensity perturbation (this calculations assumes d𝐴𝜁 /d𝑡 ∝ [𝑀p(𝑡)]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='7, see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Arrows in the top panels indicate the planetary age 𝜏p (time since the start of its accretion), while in the bottom panels they show the "time to vortex formation" 𝜏0 calculated using equation (1) and assuming 𝑀p given by its final value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' The red dotted line indicates the critical value of 𝐴𝜁 when the vortices are expected to appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' The key point illustrated here is that 𝜏0 ≤ 𝜏p always.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' known to delay the onset of RWI (Hammer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 2017), we cannot directly compare our results for 𝑡G with theirs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' However, if we focus on their smallest 𝑞f = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='5×10−4 (since in their setup this corresponds to the lowest value of viscosity, closer to our inviscid setup) and adopt their ℎp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='05 and the fit parameters (2), we find the age of the planet to satisfy 𝜏p ≳ 𝑡G ≈ 39𝑃p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' This is comfortably below 𝑡G ≈ 200𝑃p that Hallam & Paardekooper (2020) find for the same 𝑞f, consistent with the viscosity-driven delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Equally importantly, had we used the equation (4), that assumes 𝑀p = const, instead of (10), we would have found 𝜏p ≳ 13𝑃p (a factor of 𝜅 lower), far less constraining than the result that we obtained accounting for the (known) accretion history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Given the steep dependence of the integrand in (11) on 𝑀p (reflect- ing 𝑀p-dependence of the 𝜁 growth rate), we expect 𝐴𝜁 to increase the most when 𝑀p(𝑡) is close to its final value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' This is indeed what we see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 3d, in which the initial accretion episode contributes only weakly to the total increase of 𝐴𝜁 , despite its duration being comparable to the time interval when 𝑀p stayed at its final value (our calculation in this plot assumed d𝐴𝜁 /d𝑡 ∝ 𝑀2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='7 p for compatibility with equation (11), see CR23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Thus, it is the history of 𝑀p accre- tion at late times that is most important for determining the age of a putative planet in an observed vortex-hosting system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 6 APPLICATION TO OBSERVED DISCS We apply the constraints derived above to several protostellar systems observed by ALMA, for which the vortices have been invoked as a possible explanation of the observed non-axisymmetric features — arcs, clumps, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' It is important to remember that the features detected in continuum emission by ALMA are due to thermal emission of dust grains, while our results on the emergence of vortices apply to the gaseous component of the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' However, it has been shown by a number of authors (Barge & Sommeria 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Godon & Livio 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 2014) that vortices are very efficient at trapping dust, providing support to our association of the dust asymmetries with the gas vortices in PPDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Since our limits on 𝜏p and 𝑀p are highly sensitive to the disk aspect ratio ℎp, which is poorly known in most cases, we will retain the scaling with ℎ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='1 = ℎp/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='1 in our estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' HD 135344B (SAO 206462) This 𝑀★ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='5𝑀⊙, 𝜏sys ≈ 9 Myr old (Asensio-Torres et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 2021) Herbig F star harbours a transitional disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' ALMA dust continuum observations reveal an axisymmetric inner ring separated by a gap- like structure (centered around 70 AU) from an (outer) arc that can be interpreted as a vortex at the outer gap edge (van der Marel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Cazzoletti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' The possibility of a planetary origin of these structures is supported by the near-IR scattered light observations of a two-armed spiral (Muto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 2012), although a unified model explaining all these features at once is lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' We will nevertheless assume that the gap and the outer vortex are due to the (unseen) gap-opening planet at 𝑅p ≈ 70 AU, and the inner ring reflects dust trapping at the pressure maximum at the inner gap edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' These data and equations (6)-(7) allow us to constrain planetary mass as 𝑀p ≳ 32ℎ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='1𝑀⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Direct imaging of HD 135344B with VLT/SPHERE sets an upper limit of 𝑀↓ ≈ 4𝑀J on the mass of a planetary object at ∼ 102AU scales (Asensio-Torres et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Unfortunately, this 𝑀↓ is higher than the thermal mass 𝑀th = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='5ℎ3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='1𝑀J, which makes the use of the timescale constraint (3),(9) unjustified (its blind application would give 𝜏vrt ≈ 500ℎ7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='1 yr, comparable to the orbital period at the gap location and not constraining 𝜏p effectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' HD 36112 (MWC 758) This 𝑀★ ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='8𝑀⊙, 𝜏sys ≈ 9 Myr old (Asensio-Torres et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 2021) star harbours two clumps on top of the two rings separated by a gap in the outer disc (Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Neglecting the slight eccentricity of the disc and assuming the rings with clumps to correspond to the inner and outer edges of the gap carved by a planet, we will adopt 𝑅p ≈ 70 AU for the planetary orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Then equations (6)-(7) allow us to set a mass constraint 𝑀p ≳ 37ℎ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='1𝑀⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Analysis of the direct imaging observations of this system by Asensio-Torres et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' (2021) suggests that the upper limit on the possible point source inside the assumed gap is ∼ 8𝑀J, significantly higher than 𝑀th, precluding us from meaningfully constraining the MNRAS 000, 1–7 (2022) Vortex weighing and dating 5 age of the planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' HD 143006 This G-type T Tauri star with 𝑀★ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='8𝑀⊙ and an estimated age of 𝜏sys ≈ 8 Myr harbours a disc rich in substructures (Pérez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' In addition to a misaligned inner disc, it features two outer rings separated by a gap centered around 52 AU, with an arc just outside the outermost ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Interpreting these features as produced by an unseen planet inside the gap at 𝑅p = 52 AU, we get the mass constraint 𝑀p ≳ 33ℎ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='1𝑀⊕ from equations (6)-(7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' NaCo/VLT direct imaging does not provide a useful constraint on the mass of a putative planet, with 𝑀↓ at the level of several tens of 𝑀J at the outer gap location (Jorquera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Thus, we cannot set a useful lower limit on the planetary age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' V1247 Ori V1247 Ori is a 𝜏sys = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='5 Myr old, 𝑀★ ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='9𝑀⊙ star (Willson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 2019) harbouring a pre-transitional disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' ALMA dust continuum ob- servations (Kraus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 2017) reveal an inner disc (or ring) separated by a gap from the outer arc, which may be interpreted as a vortex at the outer gap edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Assuming a planet to be in the gap, at 𝑅p ≈ 90 AU, one finds that the planetary mass must satisfy 𝑀p ≳ 48ℎ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='1𝑀⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' While we could not find explicit limits on the mass of the putative planet in V1247 Ori system from direct imaging observations, Kraus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' (2017) found that an 𝑀p = 3𝑀J planet can roughly match the shape of the spiral observed in scattered light using HiCIAO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Unfortunately, this 𝑀p is again above 𝑀th, not allowing the age of the planet to be meaningfully constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 7 DISCUSSION 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='1 Combination of multiple constraints Our limits on 𝑀p and 𝜏p based on the presence of vortices next to gaps in PPDs become even more powerful when combined with additional constraints on these key parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' In particular, young planets passively lose thermal energy that they have been endowed with at formation, resulting in their luminosity decreasing with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' As more massive planets retain more heat at formation, it takes them longer to cool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Thus, if one can observationally constrain the luminosity of a planet 𝐿p to lie below a certain limit (or determine it in the case of direct detection), this would provide an additional constraint on 𝑀p and 𝜏p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' We illustrate this approach in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 4, where we show the vortex- based constraints from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 1 together with the constraint 𝐿p < 10−6𝐿⊙ (outside the pink shaded region to the right of the black dotted curve) based on the work4 of Linder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' We also show the 𝐿p = 10−7𝐿⊙ curve (red dotted) which may be relevant for future direct imaging experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' In addition, we impose a con- straint 𝜏p < 15 Myr (region below the orange dot-dashed line) since protoplanetary discs usually do not survive for that long (similar to the logic used in Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' There are other, complementary ways of constraining planetary properties, for example gap width/depth fitting (Dong & Fung 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Asensio-Torres et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 2021) which can provide model-dependent information on 𝑀p for individual systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' we will not consider them here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 4 We use the tracks for the bolometric luminosity 𝐿p of a fixed mass planet from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 6 of Linder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' (2019), which assume evolution with a cloud-free atmosphere of solar metallicity and use the petitCODE grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 10 100 1000 Mp [M ] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='1 1 10 p [Myr] no vortex hp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='07 hp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='1 hp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='15 Rp = 100 AU Lp = 10 6L Lp = 10 7L p = 15 Myr Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Combination of the different constraints on the mass 𝑀p and age 𝜏p of a planet in a vortex-hosting PPD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Grey shaded region is excluded (for ℎp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='07) as vortices have no time to develop in this part of the parameter space, analogous to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 1 (solid and dashed lines are the same as in that figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Pink shaded region is excluded as it corresponds to planetary (bolometric) cooling luminosity 𝐿p exceeding 𝐿p = 10−6𝐿⊙ (black dotted curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' We also show the curve 𝐿p = 10−7𝐿⊙ (red dotted curve);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' the 𝐿p curves are based on Linder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' The orange shaded region above the orange dot-dashed curve excludes planetary ages above 15 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Planets satisfying all three constraints reside in the unshaded part of the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' A combination of the three constraints — based on planetary lu- minosity, age and presence of vortices — limits planetary 𝑀p and 𝜏p to lie within the unshaded region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' This region shrinks for hotter discs with larger ℎp (compare fuchsia and green solid curves), as well as for larger 𝑅p (compare blue solid and dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Thus, the vortex-based limits are more stringent in hotter discs and for more distant planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Also, the allowed region would shrink even further as the upper limit on 𝐿p gets lowered in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' It should also be re- membered that the luminosity-based (dotted) curves assume that the (possible ongoing) gas accretion provides insignificant contribution to 𝐿p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' If the planetary accretion luminosity is non-negligible, this would additionally shift the dotted curves to the left, constraining 𝑀p and 𝜏p even further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='2 Utility of the vortex-based constraint The sub-Jovian value of 𝑀vrt implied by the equation (7) and our estimates in Section 6 is very relevant in light of the recent results (Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Bae et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Miranda & Rafikov 2019, 2020a,b) showing that in a low viscosity disc a single sub-𝑀th planet can give rise to a series of several prominent gaps and rings in the radial dust distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' For example, for the AS 209 system (𝑀★ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='83𝑀⊙, 𝜏sys ≈ 1 Myr, Andrews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 2018) imaged with ALMA Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' (2018) have shown that a single planet with 𝑀p as low as 25𝑀⊕ orbit- ing within the outer (primary) gap at 𝑅p ≈ 100AU can be responsible for creating all five gaps observed in this disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' This possibility makes the typical values of 𝑀p implied by the constraint (6)-(7) very inter- esting for understanding the architecture of the underlying planetary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Given the upper limits on 𝑀p based on direct imaging in several systems covered in Section 6, we found our age constraint (3) & (9) to be not very useful at present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' However, things will improve as 𝑀↓ decreases in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Once 𝑀↓ is below 𝑀th, our constraint (3) & (9) becomes valid and may provide useful information on the MNRAS 000, 1–7 (2022) 6 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Rafikov and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Cimerman planetary age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' The decrease of 𝑀↓ may not necessarily come from improved direct imaging capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' In particular, one may use the technique of multiple gap fitting used by Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' (2018) for AS 209 to get a much better measurement of 𝑀p or 𝑀↓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Just for illustration, let us imagine that AS 209 did possess a vortex at the edge of its outermost gap (just outside 𝑅p = 100AU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Then using 𝑀p ≈ 25𝑀⊕ (based on Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 2018) equation (9) would predict 𝜏p ≳ 8ℎ7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='1 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' This 𝜏p is much longer (for ℎp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='1) than the age of the system 𝜏sys ≈ 1 Myr, and could have implied that either the planetary mass is underestimated (by a factor of ∼ 2), or that the stellar age is underestimated (by almost an order of magnitude), or that the disc is somewhat colder — using ℎp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='075 (consistent with Paneque-Carreño et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 2022) in (9) would reconcile 𝜏p with its estimated 𝜏sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' The latter possibility represents a simple way to resolve the age discrepancy for this imaginary AS 209-like system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' It also highlights the importance of good knowledge of the thermal state of the disc near the planet, which sets ℎp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Indeed, 𝜏vrt depends very sensitively on ℎp and a mis-estimate of ℎp by a factor of 2 would result in a factor of ≈ 150 error in the determination of 𝜏vrt and the planetary age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' The situation is somewhat improved for the mass constraint (6)-(7), in which variation of ℎp by a factor of 2 results in 𝑀vrt changing by a factor of ≈ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' In any case, good understanding of disc thermodynamics is clearly needed when applying the age constraint (3) & (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Recent ALMA measurements of emission heights of different molecular lines in PPDs (Law et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 2021, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Paneque- Carreño et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 2022) provide a (model-dependent) way to determine disc aspect ratio at different radii, generally finding values in the range ℎp ∼ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='07 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='1) for 𝑅p ∼ (50 − 100) AU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' On the other hand, our constraints (5),(7) & (9) should be rather insensitive to the radial profile of the disc surface density near the planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Indeed, CR23 showed that the parameters of the fit (1),(2) show little variation when changing the slope of the surface density profile near the planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Also, the dependence of the vortex-based con- straints on 𝑅p and 𝑀★ is not as steep as for ℎp, and the characteristic accuracy with which these parameters can be measured is (10−20)% or better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='3 Additional processes and further extensions Since the constraints (3)-(6) are lower limits on 𝜏p and 𝑀p, respec- tively, they do not change if the vortices we observe in discs now are not the first generation vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' It is possible that the vortices that formed early on have then dissolved and what we are seeing now are the second (or multiple) generation vortices (Hammer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Nevertheless, even in this case the condition 𝜏p > 𝜏vrt would still need to be fulfilled, definitely for the first generation of vortices, as well as for the following generations, confirming the validity of the constraints (3)-(6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Similarly, dust trapped in vortices can maintain observable non- axisymmetric distribution even after the vortices in the gaseous com- ponent dissolve (Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Thus, when we see an asymmetry in dust continuum observations, the original vortex that has led to it may have already been gone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' However, this would again not invalidate the constraints obtained in Sections 3 & 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' The fit (1),(2) for 𝜏vrt was derived by CR23 for discs which are inviscid or have low viscosity, an assumption which is consistent with observations of many systems (see Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' We can roughly estimate the upper limit on the viscosity 𝜈 below which the inviscid assumption should be valid by demanding the timescale on which the vortensity structures produced by the planet get viscously diffused away to be longer that the age of the system 𝜏sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' For the characteristic radial scale of the vortensity structures 𝐿 ∼ 𝐻p(𝑀p/𝑀th)−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='4 (Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' CR23) this timescale is ∼ 𝐿2/𝜈 ∼ 𝑃p𝛼−1(𝑀p/𝑀th)−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='8, where we adopted the 𝛼-ansatz for the viscosity 𝜈 = 𝛼Ωp𝐻2p (and Ωp = 2𝜋𝑃−1 p ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' For this to exceed 𝜏sys for a sub-thermal mas planet we require that roughly 𝛼 ≲ 𝑃p/𝜏sys ∼ 10−4, given the long orbital periods at 𝑅p = 50 − 100 AU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' A more refined estimate of the critical 𝛼 can be found in CR23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' However, even if the disc were sufficiently viscous (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' for 𝛼 ≳ 10−4), the RWI development would get only delayed (Hallam & Paardekooper 2020) or the instability may be suppressed altogether, see Hammer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' (2017), CR23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Because of that, our inviscid esti- mate for 𝜏vrt continues to provide a lower limit on 𝜏p in the presence of a vortex, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' the equation (3) and all other constraints remain valid (see Section 5 for application of this logic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' On the other hand, some other effects may accelerate vortex pro- duction compared to the results of CR23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' For example, this could happen as a result of baroclinicity of the disc near the planet since the RWI is sensitive to entropy gradients (Lovelace et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Under certain circumstances dust feedback can also promote vor- tex production (Lin & Youdin 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' These processes, if they are important, may somewhat weaken our constraints on 𝑀p and 𝜏p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' We demonstrated in Section 5 how our constraints can be modified to account for the evolution of planetary mass 𝑀p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Other relevant parameters might change as well, for example 𝑅p can vary as a result of planet migration, or ℎp can change as the disc evolves in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' CR23 outlined ways in which one can account for these processes to derive a new estimate for 𝜏vrt instead of (1),(2), thus providing a pathway to modifying our constraints on 𝑀p and 𝜏p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Of the four systems considered in Section 6, three show vortex-like non-axisymmetries only at the outer edge of the putative planetary gap, and only one, MWC 758, has them on both sides of the gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' This is somewhat surprising, since the simulations of CR23 not only show the emergence of vortices on both sides of the gap, but also demonstrate that the time interval separating their production by RWI is typically smaller than 𝜏vrt (see Table 1 in that work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Thus, one would expect to see vortices on both sides of the gap more often.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' It is not clear why this expectation fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' It could be that the dust concentration is more efficient in the outer vortices5 or that it tends to survive there considerably longer than in the inner ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Or that some physical processes neglected in our study suppress the formation of the inner vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Expanding the sample of observed discs with vortex-like asymmetries would help in resolving this issue in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' 8 SUMMARY In this work we used the results of CR23 on the time it takes visible gas vortices to appear next to a gap carved by a low-mass planet in a low-viscosity PPD to set constraints on the masses 𝑀p and ages 𝜏p of planets in PPDs with observed vortex-like structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' We found that the presence of a vortex sets a lower limit on a particular combination of 𝑀p and 𝜏p, with separate constraints on these variables possible if some additional information (such as the system age 𝜏sys or the upper limit on the planetary mass 𝑀↓) is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' These considera- tions allowed us to constrain the masses of putative planets in several vortex-bearing PPDs to be above several tens of 𝑀⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' The limits 5 Outer vortices should form first in inviscid discs with radially decreasing surface density (CR23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' MNRAS 000, 1–7 (2022) Vortex weighing and dating 7 on the planetary age are not very constraining at the moment, but they will improve as future observations lower 𝑀↓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Our constraints can be extended to account for the non-trivial history of planetary mass accretion, and we provide a recipe for doing that in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Finally, we showed the robustness of our constraints in light of additional complications (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' non-zero disc viscosity, multiple gen- eration of vortices, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=') and demonstrated their useful synergy with other types of constraints on 𝑀p and 𝜏p, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' based on the upper lim- its on the planetary cooling luminosity coming from direct imaging observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' ACKNOWLEDGEMENTS Software: Matplotlib (Hunter 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' Authors are grateful to Ewine van Dishoeck for illuminating discussions and to an anonymous ref- eree for useful suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' acknowledges financial support through the Science and Technology Facilities Council (STFC) grant ST/T00049X/1 and Ambrose Monell Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf'} +page_content=' is funded by a STFC and Isaac Newton studentship.' 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manuscript no. ges +© ESO 2023 +January 23, 2023 +Gaia-ESO Survey: massive stars in the Carina Nebula +I. A new census of OB stars +S. R. Berlanas1,2, J. Ma´ız Apell´aniz3, A. Herrero4,5, L. Mahy6, R. Blomme6, I. Negueruela1, R. Dorda1,7, +F. Comer´on8, E. Gosset9, M. Pantaleoni Gonz´alez3,10, J. A. Molina Lera11, A. Sota12, T. Furst6,9, +E. J. Alfaro12, M. Bergemann13,14, G. Carraro15, J. E. Drew16, L. Morbidelli17, and J. S. Vink18 +1 Departamento de F´ısica Aplicada, Universidad de Alicante, E-3690, San Vicente del Raspeig, Alicante, Spain +2 Astrophysics Group, Keele University, Keele ST5 5BG, Staffordshire, UK +3 Centro de Astrobiolog´ıa (CAB), CSIC-INTA, Campus ESAC, E-28 692 Villanueva de la Ca˜nada, Madrid, Spain +4 Instituto de Astrof´ısica de Canarias, E-38 200 La Laguna, Tenerife, Spain +5 Departamento de Astrof´ısica, Universidad de La Laguna, E-38 205 La Laguna, Tenerife, Spain +6 Royal Observatory of Belgium, Ringlaan 3, 1180 Brussels, Belgium +7 School of Architecture, Universidad Europea de Canarias, Tenerife, Spain +8 ESO, Karl-Schwarzschild-Strasse 2, 85 748 Garching bei M¨unchen, Germany +9 Space Sciences, Technologies and Astrophysics Research (STAR) Institute, Universit´e de Li`ege, All´ee du 6 Aoˆut, 19c, Bˆat B5c, +4000 Li`ege, Belgium +10 Departamento de Astrof´ısica y F´ısica de la Atm´osfera, Universidad Complutense de Madrid. E-28 040 Madrid, Spain +11 Instituto de Astronom´ıa y F´ısica del Espacio, UBA-CONICET. CC 67, Suc. 28, 1428 Buenos Aires, Argentina +12 Instituto de Astrof´ısica de Andaluc´ıa (IAA), CSIC. Glorieta de la Astronom´ıa s/n. E-18 008 Granada, Spain +13 Max Planck Institute for Astronomy, K¨onigstuhl 17, 69117, Heidelberg, Germany +14 Niels Bohr International Academy, Niels Bohr Institute, University of Copenhagen Blegdamsvej 17, DK-2100 Copenhagen, +Denmark +15 Dipartimento di Fisica e Astronomia Galileo Galilei, Universit´a di Padova, Vicolo Osservatorio 3, I-35122, Padova, Italy +16 Department of Physics & Astronomy, University College London, Gower Street, London, WC1E 6BT, UK +17 INAF - Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, 50125 Florence, Italy +18 Armagh Observatory and Planetarium, College Hill, Armagh BT61 9DG, N. Ireland +Received month day, year; accepted month day, year +ABSTRACT +Context. The Carina Nebula is one of the major massive star-forming regions in the Galaxy. Its relatively nearby distance (2.35 kpc) +makes it an ideal laboratory for the study of massive star formation, structure and evolution, both for individual stars and stellar sys- +tems. Thanks to the high-quality spectra provided by Gaia-ESO survey and the LiLiMaRlin library, as well as Gaia EDR3 astrometry, +a detailed and homogeneous spectroscopic characterization of its massive stellar content can be carried out. +Aims. Our main objective is to spectroscopically characterize all massive members of the Carina Nebula in the Gaia-ESO survey +footprint to provide an updated census of massive stars in the region and an updated estimate of the binary fraction of O stars. +Methods. We perform accurate spectral classification by using an interactive code that compares spectra with spectral libraries of OB +standards, as well as line-based classic methods. Membership is calculated using our own algorithm based on Gaia EDR3 astrometry. +To check the correlation between the spectroscopic n-qualifier and the rotational velocity, we use the semi-automated tool for the +line-broadening characterization of OB stars which is based on a combined Fourier Transform and Goodness-of-fit methodology. +Results. The Gaia-ESO survey sample of massive OB stars in the Carina Nebula consists of 234 stars. The addition of brighter sources +from the Galactic O-Star Spectroscopic Survey and additional sources from the literature allows us to create the most complete census +of massive OB stars done so far in the region. It contains a total of 316 stars, being 18 of them in the background and four in the +foreground. Of the 294 stellar systems in Car OB1, 74 are of O type, 214 are of non-supergiant B type and 6 are of WR or non-O +supergiant (II to Ia) spectral class. We identify 20 spectroscopic binary systems with an O-star primary, of which 6 are reported for the +first time, and another 18 with a B-star primary, of which 13 are new detections. The average observed double-lined binary fraction +of O-type stars in the surveyed region is 0.35, which represents a lower limit. We find a good correlation between the spectroscopic +n-qualifier and the projected rotational velocity of the stars. The fraction of candidate runaways among the stars with and without the +n-qualifier is 4.4% and 2.4%, respectively, although non resolved double-lined binaries can be contaminating the fast rotators sample. +Key words. stars: massive – stars: early-type – stars: rotation – binaries: spectroscopic – proper motions – open clusters and associa- +tions: individual: Carina Nebula +1. Introduction +The Gaia-ESO Large Public Spectroscopic Survey (GES, +Gilmore et al. 2022; Randich et al. 2022) has obtained high qual- +ity spectra of ∼105 stars in our Galaxy using FLAMES at the +Very Large Telescope (VLT) with its high-resolution UVES and +its intermediate-resolution GIRAFFE spectrographs. GES has +systematically covered all the major components of the Milky +Way, providing an homogeneous and unique overview of the +kinematics, chemical composition, formation history, and evo- +lution of young, mature and ancient Galactic populations. Open +1 +arXiv:2301.08310v1 [astro-ph.SR] 19 Jan 2023 + +Berlanas et al.: Massive stars in the Carina Nebula +clusters are useful tools for this aim, where it is possible to +study stellar populations of different ages in different evolution- +ary stages (see Bragaglia et al. 2022). +Numerous spectroscopic studies of massive stars have been +carried out in Galactic young stellar clusters and OB associ- +ations, the most extensive to date being the Galactic O-Star +Spectroscopic Survey (GOSSS, Ma´ız Apell´aniz et al. (2011)). +As some examples of such studies, Figer (2005) determined +the upper mass limit of the Initial Mass Function (IMF) in the +Arches Cluster, a result that was later challenged by studies in +R136 (Crowther et al. 2010). Other examples are the determina- +tion of the chemical composition of stars in Orion (Sim´on-D´ıaz +2010); the membership, chemical and stellar parameter determi- +nation studies in Cygnus OB2 (Berlanas et al. 2018a,b, 2020); +the characterization of very massive obscured clusters in the +Milky Way like Westerlund 1 (Clark et al. 2005; Negueruela +et al. 2010, 2022); and the analysis of the multiplicity of massive +stars in clusters (De Becker et al. 2004, 2006; Mahy et al. 2009; +Sana & Evans 2011; Mahy et al. 2013; Banyard et al. 2022) and +in the whole northern hemisphere (Ma´ız Apell´aniz et al. 2019b; +Trigueros P´aez et al. 2021; Mahy et al. 2022). Outside the Milky +Way, the most thorough analysis is that of the many papers1 +published by the VLT-FLAMES Tarantula Survey collaboration +(VFTS, Evans et al. 2011). +The Carina Nebula complex consists of several stellar +groups, some bound and some not, immersed in the Car OB1 +association (Ma´ız Apell´aniz et al. 2020, 2022a, from now on +Villafranca I and II, respectively, and references therein). It +represents a unique region to study Galactic massive stars +with FLAMES since it contains a large number of O-type +stars (Walborn 1972, 1973b, 1982b; Levato & Malaroda 1982; +Morrell et al. 1988; Sota et al. 2014; Ma´ız Apell´aniz et al. +2016; Alexander et al. 2016; Berlanas et al. 2017; Mohr-Smith +et al. 2017). It is the most massive star-forming region within +3 kpc of the Sun. The distance to its most famous member, +η Car, was geometrically determined with excellent precision +to be 2.35±0.05 kpc by Smith (2006b). The recent Gaia EDR3 +(Brown et al. 2021) analysis in Villafranca I+II has not only con- +firmed that value but has also found that there are little distance +variations between at least Trumpler 14, Trumpler 16 W, and +Trumpler 16 E, three of the stellar groups in the complex. In a +new installment of the series (Villafranca III, Molina Lera et al. +in preparation) the authors show that those distance variations +are still small when including other stellar groups in Car OB1. +Even though the Carina Nebula harbors hundreds of massive +stars, there is no systematic spectroscopic analysis of its early- +type members. Thanks to the high-quality spectra provided by +GES and astrometry by Gaia EDR3, a detailed and homoge- +neous spectroscopic study of its massive stellar content can be +carried out. The analysis of the Carina massive stellar popula- +tion will be highly relevant for problems like the initial mass +function (IMF, Crowther et al. 2010), the chemical composition, +rotation and internal mixing (Meynet & Maeder 2000; Ram´ırez- +Agudelo et al. 2013; Sim´on-D´ıaz & Herrero 2014; Herrero 2016; +Holgado et al. 2022), or the stellar multiplicity of massive stars +(see Langer 2012; Sana et al. 2012; Sota et al. 2014; de Mink +et al. 2014). In particular, although Sana & Evans (2011) and +Sana (2017) quote fractions of binary systems in excess of 0.5 +for the O-type star population in the Milky Way, the former au- +thors give a null fraction in a cluster like Trumpler 14, making +clear the need for a systematic survey in the region. In addition, +binarity may be the origin of fast rotating and runaway stars (e.g. +1 https://www.roe.ac.uk/˜cje/tarantula/f2-pubs.html. +Table 1. Wavelength range and resolving power of the GES +spectra obtained with different gratings. +Grating +Wavelength +Resolving +Data release +range (Å) +power +GIRAFFE +HR03 +4033 − 4201 +24 800 +iDR3-6 +HR04 +4188 − 4392 +20 350 +iDR3-6 +HR05A +4340 − 4587 +18 470 +iDR5-6 +HR06 +4538 − 4759 +20 350 +iDR3-6 +HR14A +6308 − 6701 +17 740 +iDR3-6 +UVES +520 +4140 − 6210 +47 000 +iDR3-6 +580 +4760 − 6840 +47 000 +iDR5-6 +de Mink et al. 2013, 2014; Mahy et al. 2020; Holgado et al. 2022) +by ejecting stars that have gained mass and angular momentum +from the binary system after the explosion of the primary as su- +pernova. The Carina region, containing a large number of mas- +sive stars at a relatively nearby distance, is an ideal place to test +the theories of massive star evolution. +As a first step this work focuses on the creation of the most +complete to date census of massive stars and the identification of +double-lined spectroscopic binaries (SB2) in Car OB1. It is orga- +nized as follows. In Section 2 we describe how we have obtained +our spectroscopy, compiled our spectral types, and used Gaia to +determine the distances. In Section 3 we present our census of +massive stars in the central part of the Carina Nebula. We dis- +cuss the results in Sect. 4, where we explore the completeness of +the census, determine the binary fraction of OB stars and inves- +tigate the correlation between the n spectroscopic qualifier, the +projected rotational velocity and the runaway status. Finally, we +summarize the conclusions in Sect. 5. +2. Data and methods +2.1. GES strategy and spectroscopy +GES spectroscopic data for hot stars were obtained using the +FLAMES intermediate-resolution (R∼20 000) GIRAFFE and +the high-resolution (R∼47 000) UVES spectrographs on the Very +Large Telescope (VLT). See Blomme et al. (2022) for further +details on the analysis of GES hot stars and Table 1 for the +wavelength range covered by each of the setups. In the rest of +this subsection we present the aspects that are more relevant to +the Carina Nebula GES data set. A previous GES paper on the +Carina Nebula (Damiani et al. 2017) used a different data set and +concentrated on stars of lower mass than the ones analyzed here. +The central part of the Carina Nebula can be divided into +six stellar groups: Trumpler 14, Trumpler 15, Trumpler 16 W, +Trumpler 16 E, Collinder 228, and Collinder 232 (Walborn +1995; Smith 2006a;Villafranca I+II+III). Of those stellar +groups, only Trumpler 14 and Trumpler 15 and possibly +Trumpler 16 E appear to be real bound clusters, with the rest +being parts of the association defined by (apparent or real) struc- +tures seen in the stellar distribution and nebulosity (e.g. the sepa- +ration of Collinder 228 from the other groups likely originates in +the prominent V-shaped dust lane that crosses the H ii region). +Other stellar groups farther away from the central region but +likely members of the Car OB1 association include NGC 3293 +(see Morel et al. 2022), NGC 3324, Bochum 10, Bochum 11, +Loden 153, IC 2581, Ruprecht 90, and ASCC 62. Given the large +2 + +Berlanas et al.: Massive stars in the Carina Nebula +O-002 +O-002 +Tr 14 +Tr 14 +O-003 +O-003 +Tr 16W +Tr 16W +O-025 +O-025 +Tr 16E +Tr 16E +O-027 +O-027 +Tr 15 +Tr 15 +O-028 +O-028 +Coll 228 +Coll 228 +O-029 +O-029 +Coll 232 +Coll 232 +O-030 +O-030 +Bochum 11 +Bochum 11 +Carina_Gendler.jpg +15’ +1.14´� x 52.24’ +N +E +Powered by Aladin +Fig. 1. Negative image of the Great Carina Nebula by Robert Gendler and Stephane Guisard showing the location of the whole census +of massive stars in the GES surveyed area presented in this work. Yellow and cyan colors indicate O and B-type stars, respectively. +Green, red, purple and pink colors have been used to represent the sdO, LBV, WR and RSG stars, respectively. Small filled-circles +refer to the GES sample while rhombuses and squares refer to stars from GOSSS/LiLiMarlin and other works (Smith 2006a; +Alexander et al. 2016; Preibisch et al. 2021) not present in GES, respectively. Red circles indicate the observing GES pointings +while the blue ones indicate the Villafranca groups: O-002 (Trumpler 14), O-003 (Trumpler 16 W), O-025 (Trumpler 16 E), O- +027 (Trumpler 15), O-028 (Collinder 228), O-029 (Collinder 232), and O-030 (Bochum 11). The V-shaped extinction lane that +dominates the appearance of the nebula is clearly seen crossing the image from top to bottom. +size of the nebula and the high stellar density in some regions +(e.g. the core of Trumpler 14), four different GIRAFFE+UVES +pointings were needed to cover a substantial fraction of the +massive stars in the six central groups and part of those in +Bochum 11 (see Fig. 1). The four pointings are centered at +RA+δ J2000 coordinates (161.10,−59.430), (161.07,−59.695), +(161.42,−59.835), and (160.79,−60.020), respectively, with the +values expressed in degrees. The sample selection was done by +compiling the available spectroscopic and photometric informa- +tion at the time of the survey design (e.g. the Galactic O-Star +Catalog, GOSSS, Ma´ız Apell´aniz et al. 2004; Sota et al. 2008) +but, of course, no Gaia data existed back then. For that reason, +we complemented our spectroscopy with GOSSS data and we +have to evaluate the completeness of our sample (see below for +both). +One important difference between this one and most of the +other GES data sets is the existence of a significant nebulosity in +the region. Furthermore, the nebular Balmer and He i emission +lines not only are strong but they are placed on top of important +diagnostic stellar absorption lines. For that reason, we devised a +specific strategy to eliminate or at least mitigate their influence +when we prepared these observations. Each of the four point- +ings was divided into two subpointings (for a total of eight) with +half of the fibers dedicated to stars and the other half to nebulos- +ity. Each of the two subpointings within a given pointing have +identical fiber configurations but the field center is displaced by +10′′ between the two of them. In that way, each star observed +in a given subpointing has a nebular counterpart 10′′ away ob- +served in the other subpointing. One of us (JMA) wrote an IDL +code to manually review each star/nebulosity spectral pair and +use the second one to subtract the nebular contribution from the +stellar fiber. This strategy is the best possible one given the lim- +itations of the observational setup, but it is not ideal, as in some +cases nebular emission can change substantially in scales smaller +than 10′′. This is one of the advantages of long-slit spectroscopy +(such as that obtained by GOSSS) over its fiber-fed alternative, +as the former allows a sampling of nebular emission closer to +the target and at two different locations with respect to the star. +3 + +Berlanas et al.: Massive stars in the Carina Nebula +In practical terms, the issue is significative only for faint stars at +Hα, as the nebular contribution for bright stars and other relevant +lines is usually small. +We examined all of the spectra in our GES datasets to iden- +tify OB massive stars (B2 or earlier for dwarfs, B5 or earlier for +giants, and all B subtypes for supergiants) and obtained a sam- +ple of 234 objects, 18 of which were observed with FLAMES- +UVES and 216 with FLAMES-GIRAFFE. The spectrograms are +shown in Figs. A.1 and A.2, in the first case at the original 47 000 +spectral resolution and in the second case at the 2500 spectral +resolution used for spectral classification. +2.2. GOSSS spectroscopy +The GOSSS project was born a decade and a half ago with the +idea of obtaining mid-low resolution (R ∼ 2500) blue-violet +spectroscopy of any optically-accessible Galactic object that had +ever been classified as an O star to confirm its nature and to pro- +vide homogeneous spectral classifications for the whole sample. +While doing that, GOSSS managed not only to discover a siz- +able number of new O-type stars but also to reject quite a number +of them as being of B type (or, more egregiously, of even later +types) and to obtain good-quality spectroscopy of several thou- +sands of other early-type stars. In the first three major papers, +Sota et al. (2011) or GOSSS I, Sota et al. (2014) or GOSSS II, +and Ma´ız Apell´aniz et al. (2016) or GOSSS III, GOSSS pub- +lished spectra for 590 O-type stars and for a few later-type ob- +jects. Since that time, GOSSS has collected a large number of +new spectra, some of them in the Carina Nebula. Of those, eight +new GOSSS spectra for O-type stars will appear in the fourth +major installment of the project (GOSSS IV, Ma´ız Apell´aniz +et al. in preparation) but the spectral classifications are already +listed here in Table A.2. In this paper we also present GOSSS +spectra for one Wolf-Rayet and 17 early-B stars in Fig. A.3 as a +complement to the GES data. +2.3. Spectral classifications +We obtained the spectral classifications using the MGB tool +(Ma´ız Apell´aniz et al. 2012, 2015), which compares the ob- +served spectra with a standard library of OB stars (in this case +the GOSSS library, see GOSSS III+IV). This interactive soft- +ware allows us to vary the spectral subtype, luminosity class, +line broadening, and spectral resolving power of the standard +spectrum until we obtain the best match. In addition, it also al- +lows us to combine two standard spectra (with different veloc- +ities and flux fractions) to fit SB2 systems. The spectral classi- +fication was performed for the three types of spectroscopic data +(UVES, GIRAFFE, and GOSSS) at the same spectral resolu- +tion of the GOSSS library, 2500. The spectral classifications are +given in Table A.2. +There is one specific issue with GIRAFFE spectra and spec- +troscopic binaries that needs to be discussed. In general, each +grating was observed at a different epoch and this generates a +problem for spectroscopic binaries, as different lines of the same +ion may be at different velocities. We have dealt with this issue +on a case by case basis but in some we are only able to provide a +poor-quality spectral classification. See subsection 3.2 for some +examples. +2.4. Spectral types from other sources and cataloguing +In addition to those from GES and GOSSS, in Table A.2 we +give spectral types from other further sources. The first one +is LiLiMaRlin (Library of Libraries of Massive-star High- +Resolution Spectra, Ma´ız Apell´aniz et al. 2019a) that is collect- +ing multi-epoch high-resolution optical+NIR spectra of massive +stars, with over 60 000 epochs to date. For the case of the Carina +Nebula, the library currently has FEROS, UVES, and HARPS +spectra. LiLiMaRlin is especially useful for the analysis of SB2 +and SB3 systems, where finding the right epoch is usually neces- +sary to separate the different components in velocity. One of the +LiLiMaRlin spectral types had appeared before in Villafranca I +but there are also nine whose spectra will appear in GOSSS IV +and another 16 whose spectra will appear in Villafranca III. For +the two latter papers, those spectral types are listed here for the +first time. +Multi-epoch high-resolution spectroscopy such as that from +LiLiMaRlin can be used to separate in velocity spectroscopic +binaries. If one wants to spatially separate close visual binaries, +then what is needed is the combination of high-spatial resolution +with spectroscopy, either from the ground (Ma´ız Apell´aniz et al. +2018, 2021a) or from space (Ma´ız Apell´aniz & Barb´a 2020). +For the case of the Carina Nebula, we give in Table A.2 the spa- +tially resolved spectral types for HD 93 129 Aa,Ab, one of the +most massive systems in the region, obtained with STIS/HST +(Ma´ız Apell´aniz et al. 2017). +Another source is the already mentioned GOSC, which is a +catalog that compiles information about massive stars (with an +emphasis on O stars) from different sources. GOSC has a private +and a (increasingly growing) public version, which will be heav- +ily updated after GOSSS IV is published. Here we have used the +private version of GOSC to search for additional massive stars in +the region of interest and provide spectral types. In particular, we +have included the results from Smith (2006a), a previous census +of the massive stars in the Carina Nebula, from Alexander et al. +(2016), a spectroscopic survey of the region, and from Preibisch +et al. (2021), a near-infrared spectroscopy survey to identify ob- +scured OB stars. +Besides the flow of information from GOSC to this pa- +per mentioned in the previous paragraph, there will be a flow +of information in the opposite direction, as the spectral types +here will be included in GOSC. In addition to GOSC, these +spectral types will be used to update the Alma Luminous Star +Catalog (ALS, Reed 2003), a compilation of (originally) photo- +metric and spectroscopic information for Galactic OB stars. In +Pantaleoni Gonz´alez et al. (2021), the original ALS catalog was +cross-matched with Gaia DR2 to eliminate the many misidenti- +fications and duplicates present and to provide astrometric infor- +mation. In a soon-to-be submitted third paper, the cross-match +will be revised with Gaia DR3 information and the catalog will +be expanded with new information, such as the one in this paper. +2.5. Gaia EDR3 data +We have searched the Gaia EDR3 archive (Brown et al. 2021) +for the astrometric and photometric information of the sample in +the paper. Gaia EDR3 parallaxes, ϖ, have a zero point, ZEDR3 +(Lindegren et al. 2021), that needs to be applied to yield cor- +rected parallaxes, ϖc. Furthermore, the internal parallax uncer- +tainties are underestimated and have to be converted into exter- +nal (or true) uncertainties. Here we follow the procedure outlined +in Ma´ız Apell´aniz et al. (2021b) and Ma´ız Apell´aniz (2022) to +list the corrected parallaxes with their external uncertainties in +4 + +Berlanas et al.: Massive stars in the Carina Nebula +Table A.1. We also list there the membership of each star to a +foreground or background population according to its parallax +(see Appendix A for details). +As the inverse of the parallax is a biased estimator of the +distance (Lutz & Kelker 1973), one has to do a proper estima- +tion of the distance involving a prior. The prior depends on the +analyzed population itself: notoriously, OB stars do not follow +the same spatial distribution in the Milky Way compared to its +older populations Here we use the thin disk model and prior +of Ma´ız Apell´aniz (2001, 2005) updated with the parameters of +Ma´ız Apell´aniz et al. (2008) to calculate the distances and uncer- +tainties of individual stars. See Pantaleoni Gonz´alez et al. (2021) +for a comparison among different distance estimates to OB stars +using Gaia parallaxes. +Gaia EDR3 provides photometry in three bands, G3, GBP3, +and GRP3, with the two last bands being actually the result +of integrating spectrophotometry in the wavelength direction. +The analysis of previous Gaia data releases (Ma´ız Apell´aniz +2017; Ma´ız Apell´aniz & Weiler 2018) revealed that the sensi- +tivity curves of the Gaia instrument change with time, leading +to slightly different intrinsic photometric values between data +releases (each being an average over different time frames). +Furthermore, in some cases the processing introduces small +trends and artifacts in the published magnitudes that require cor- +rections. In a paper that will be submitted soon, a team that in- +cludes some of us have computed such an analysis for G3, lead- +ing to a corrected value G′ +3. In Table A.1 we list the G′ +3 and +GBP3 − GRP3 values for our sample. +3. Census +Here we present the new census of massive stars in the central +region of the Carina Nebula and discuss some individual stars of +interest, especially if they have received little or no attention be- +fore. The census itself is presented in two tables in the Appendix +already introduced in the previous section. Table A.1 lists the star +identifications and coordinates, Gaia EDR3 corrected photome- +try and parallaxes, and the group identification (see previous sec- +tion and Fig. 1). Table A.2 gives the spectral classifications from +different sources. Disagreements between spectral classifications +are sometimes attributable to the nature of spectroscopic binaries +caught in different orbital phases but in other cases they are due +to differences in data quality (e.g. wavelength range, S/N, uncor- +rected artifacts) or classification criteria (e.g. choice of lines for +classification, standards used, consideration of line broadening). +When in doubt, one should consult the published spectrograms, +not the spectral types themselves. That is the reason for publish- +ing long appendices with figures such as the one here. +3.1. Overall properties +The resulting census of stars presented in this work contains +316 massive stars2. Note that, by definition and as stated be- +fore, massive OB stars include all O-types and those B2-types or +earlier for dwarfs, B5-types or earlier for giants, and all B sub- +types for supergiants (I or II luminosity classes). Red supergiants +(RSGs), Wolf-Rayet (WR) stars, and some B subtypes close to +the OB-star limit (e.g. B2.5 V) are also included in the census for +completeness. We have separated stars with distances compati- +ble with Car OB1 from those in the foreground and background, +2 Note that this number does not distinguish between single and bi- +nary or multiple stars, so hereinafter we refer to stellar systems when +both types are included in the statistics. +finding four systems in the foreground (one RSG, two B dwarfs +and one sdO) and 18 in the background (two O stars, five B su- +pergiants and 11 B non-supergiants). These systems are listed in +Table A.3. +Of the 294 stellar systems in our census in Car OB1, 74 are +of O type, 214 are of non-supergiant B type and 6 are of WR +or non-O supergiant (II to Ia) spectral class (they are listed in +Table A.4). Note that other WR stars in Car OB1 fall outside +the surveyed area (WR 22, WR 23 and WR 27). Compared to +the previous census of the massive stars in the Carina Nebula +by Smith (2006a) we have significantly increased the content of +known OB stars in the region. Considering only the area sur- +veyed in this work, the number of 105 OB stellar systems with +spectral types as late as B2 reported by Smith (2006a) has been +increased by a factor of 2.8. +There are three RSGs in the field of view: HD 93 420, +HD 93 281, and HDE 303 310 (= RT Car), all of them included +in the study of Humphreys et al. (1972). They are the second, +fifth, and sixth G′ +3 brightest sources, as the bolometric correc- +tion in that photometric band is significantly lower for RSGs +than for O-type and WR stars. η Car, of course, is in a differ- +ent luminosity category and is almost two magnitudes brighter +in G′ +3 than the brightest RSG. HD 93 420 is three sigmas3 closer +to us in parallax than 0.44 mas, the limit we are using to in- +clude a star in Car OB1, and that places it in the foreground (but +closer to Car OB1 than to us). The other two stars have parallaxes +compatible with being in Car OB1, HD 93 281 in Villafranca O- +028 (Collinder 228) and HDE 303 310 in Villafranca O-029 +(Trumpler 15). We list in this paper their new spectral classi- +fications from Villafranca III, derived from recently obtained +FEROS spectra. The classification for HD 93 420 is identical +to that of Humphreys et al. (1972) but the other two are of +slightly later type, with HDE 303 310 at M3 Iab and HD 93 281 +at M1.5 Iab. We see no sign of the alleged B-type companion for +HD 93 281 (see Humphreys et al. 1972) other than the strong Hα +emission. +The three RSGs are not the only sign of the existence of +previous generations of massive-star formation in Car OB1 and +its immediate foreground. We also find in our sample evolved +B stars such as HDE 305 535, HDE 305 452, CPD −58 2605, +CPD −59 2469, and CPD −59 2504. We find only one B- +supergiant of luminosity class I, HDE 305 530, at the distance of +Car OB1 in the footprint of this paper (the two stars by Damiani +et al. (2017) classified as B I, 2MASS J10440384−5934344 and +2MASS J10452875−5930037, are classified here as B2V and +B1.5Vp, respectively), but a number of B and later-type su- +pergiants are observed in its vicinity (Villafranca III). All of +this establishes the existence not only of those older massive +stars but also of the supernova explosions associated to those +star-formation episodes. It has been known for a long time that +the gas in the foreground of some of the OB stars in Carina +shows the most complex kinematics in any Galactic sightline +(Walborn & Hesser 1975; Walborn 1982a; Walborn et al. 2002a), +with up to 26 individual components and a range of velocities +between −388 km/s and +127 km/s. Those components must +have been produced by supernova explosions whose progenitors +were evolved massive stars. The remaining three RSGs and the +evolved B stars must be just the tip of the iceberg of the previous +massive populations. Those complex kinematics are the main +reason why the interstellar lines present in the spectra of the +3 The external parallax uncertainty for such a bright star is much +larger than the internal uncertainty (Ma´ız Apell´aniz 2022), so the dis- +tance in sigmas would be also larger if we were to use the second one. +5 + +Berlanas et al.: Massive stars in the Carina Nebula +Carina OB stars are so strong (Penad´es Ordaz et al. 2011, 2013), +as the spread in velocity yields a more advantageous curve of +growth. Due to the additional Routly-Spitzer effect (Routly & +Spitzer 1951; Routly & Spitzer 1952) the Ca ii H+K lines are +especially strong for these stars, making them deviate strongly +from the relationship between extinction and their EW derived +from other sightlines. +3.2. Individual stars +The Carina Nebula field has a large number of interesting stars, +starting with η Car, that have been analyzed in the past (see, e.g., +Damineli et al. 2000, 2008; Iping et al. 2005). Our goal in this +subsection is not to discuss such objects per se but to present +new interesting objects that have received little or no attention in +the past or new aspects of old objects that are mentioned for the +first time. +QZ Car Aa,Ac. This complex system (S´anchez-Berm´udez et al. +2017; Rainot et al. 2020) is the brightest of O-type in the +Carina Nebula. Mayer et al. (2001) identified it as an SB1E+SB1 +system and measured the two periods as 5.991 d (eclips- +ing) and 20.735 96 (non-eclipsing). In GOSSS II the system +was classified as O9.7 Ibn with no resolved components4. In +GOSSS IV the system is determined to be O9.7 Ib + O9 II: using +LiLiMaRlin but the authors note that the secondary luminosity +class is poorly determined, possibly as the result of contamina- +tion by one of the additional stars. The high luminosity of the +two primaries coupled with the smaller contribution of the sec- +ondaries explains why this system seats at the top of the optical- +luminosity food chain of the O-type stars in the Carina Nebula. +The Gaia EDR3 parallax uncertainty is quite large. +HD 93 129 Aa,Ab. This system was spatially resolved by +Ma´ız Apell´aniz et al. (2017) using HST/STIS and determined to +be composed of two O2 If* stars, with one of them having a com- +panion in a tight orbit, likely a late-O star. The orbit is highly ec- +centric and passed though periastron in 2018.70+0.22 +−0.12 (del Palacio +et al. 2020). Given that the periastron took place at a 3-D sepa- +ration of just 18.6±1.0 AU (when the system was first spatially +resolved in 1996 it was ∼375 AU), an order of magnitude (or +even less) smaller than the expected semi-major axis of the in- +ner orbit, and the high eccentricity, it is possible that the system +has transitioned from an elliptic orbit to a hyperbolic trajectory +and a possible ejection from Villafranca O-002 (Trumpler 14). If +that had happened, this could be another example of an orphan +cluster where the most (in this case, two) massive stars of a clus- +ter are ejected through a dynamical interaction (Ma´ız Apell´aniz +et al. 2022b). Further observations are needed, especially with +HST/STIS later in this decade (if it is still operational) when +Aa and Ab are expected to reach plane-of-the-sky separations of +∼40 mas. +HD 93 403. Rauw et al. (2000) classified this SB2 system as +O5.5 I + O7 V. In GOSSS II the two components could not +be resolved and it received a classification of O5.5 III(fc) var. +In GOSSS IV it is now kinematically resolved and classified as +O5 Ifc + O7.5 V using either GOSSS or LiLiMaRlin data, that +4 Some papers quote spectral types for the four components but these +are estimates: to our knowledge both spectroscopic binaries are still +SB1 and no resolved (spatially or kinematically) spectral types have +been determined. +is, the primary is slightly earlier and the secondary slightly later +compared to Rauw et al. (2000). The Gaia EDR3 parallax un- +certainty is quite large. +HDE 305 520. Alexander et al. (2016) classified this system as +B1 Ia. In Villafranca III we use LiLiMaRlin data to reclassify it +as B0.7 Iab. This Villafranca O-028 object is the only B super- +giant at the distance of Car OB1 in our sample. +V572 Car. Rauw et al. (2001) classified this SB3 system +in Villafranca O-025 (Trumpler 16 E) as composed by an +O7 V + O9.5 V inner eclipsing binary and an outer B0.2 IV +star. In GOSSS III only two components were seen and received +an O7.5 V(n) + B0 V(n) classification. With the new data we +now detect the system as an SB3: O6.5 Vz + B0 V + B0.2 V in +LiLiMaRlin data in GOSSS IV and O6.5 Vz + B0 V + B0.5: V +in UVES. As it happened with HD 93 403, the primary is slightly +earlier and the secondary slightly later compared to the original +classification. The outer star has been detected in NIR Long- +Baseline Interferometry and is currently further monitored (see +Gosset et al. 2014). +CPD −59 2554. This system was classified as O9.5 IV in +GOSSS II. Using LiLiMaRlin in GOSSS IV and UVES here it is +now found to be an SB2. In both cases the spectral classification +is O9.2 V + B1: V. +HD 93 342. This object was considered as a Villafranca O-027 +(Trumpler 15) member by Smith (2006a), where it received a +classification as O9 III. Alexander et al. (2016), on the other +hand, classified it as B1 Ia and in Villafranca III we reclas- +sify it as B1.5 Ib using LiLiMaRlin data, confirming it is a B- +type supergiant and not an O star. Its Gaia EDR3 distance is +3.58+0.41 +−0.33 kpc, placing it beyond Car OB1, something that it is +consistent with its red color (it is the brightest OB star in our +sample with GBP3 − GRP3 > 1.0). +HD 93 056. Alexander et al. (2016) classified this system as +O9 V + B2 V. In Villafranca III we use LiLiMaRlin data to re- +classify it as B1: V:n. Furthermore, in the UVES data no He ii +is detected (either 4542, 4686, or 5412), as it should in an SB2 +system composed of a late-O and an early-B stars even if caught +at a disfavorable phase. +HD 93 501. Alexander et al. (2016) classified this system as +B0 V. In Villafranca III we use LiLiMaRlin data to reclassify it +as B1.5: III:(n)e. Its Gaia EDR3 distance is 1.87+0.12 +−0.11 kpc, plac- +ing it in the foreground. +CPD −59 2592. Alexander et al. (2016) classified this object +as B1 Ib. In Villafranca III we use LiLiMaRlin data to reclassify +it as B2.5 Ia. Its Gaia EDR3 distance is 4.71+0.69 +−0.53 kpc, placing it +beyond Car OB1, something that is consistent with its red color +(it is the brightest OB star in our sample with GBP3 − GRP3 > +1.2). +HDE 305 439 A,B. With GIRAFFE data we classify the A com- +ponent as B0 Ia. Its Gaia EDR3 distance is 4.48+0.54 +−0.44 kpc, placing +it beyond Car OB1, something that is consistent with its moder- +6 + +Berlanas et al.: Massive stars in the Carina Nebula +ately red color (it is the fourth brightest OB star in our sample +with GBP3 − GRP3 > 0.7). In Villafranca III we use LiLiMaRlin +data to classify the B component, located 3.′′7 away, as B0.7 Ib. +Its parallax is consistent with being at the same distance, making +the system a likely pair of B supergiants. +HDE 305 535. This object was classified as B2.5 V by +Alexander et al. (2016). Here we derive a classification of +B4 III(n) from UVES data, which leads to an absolute magni- +tude more consistent with its spectral type and low extinction. +HD 93 343. Rauw et al. (2009) classified this SB2 system +as O7-8.5 + O8. In GOSSS III the two components could +not be resolved and it received a classification of O8 V. In +GOSSS IV it is now kinematically resolved and classified as +O7.5 Vz + O7.5: V(n). +CPD −59 2636 A,B. This system is a visual binary with a 0.′′3 +separation and ∆m = 0.6 mag in which both components are +spectroscopic binaries (Albacete Colombo et al. 2002): A (A+B +in Albacete Colombo et al. 2002) is an SB2 with a 3.6284 d pe- +riod and B (C in Albacete Colombo et al. 2002) is an SB1 with +a 5.034 d period. Those authors gave a spectral classification of +O7 V + O8 V to A and of O9 V to B. In GOSSS II, the authors +were only able to give two spectral types as O8 V + O8 V but +with GES we are able to see the three components in an UVES +single epoch and derive spectral types of O7.5 V + O8 V + O8 V. +Further epochs are needed to solve the small discrepancies with +the Albacete Colombo et al. (2002) classification. Gaia EDR3 +does not provide a parallax for CPD −59 2636 A,B, which +is common for a visual binary of this separation and magni- +tude difference, but it is a likely member of Villafranca O-025 +(Trumpler 16 E). +HDE 305 534. Alexander et al. (2016) identified this system +as a spectroscopic binary and classified it as B0 V + B0 V. In +Villafranca III we use LiLiMaRlin data to confirm it is an SB2 +and reclassify it as B0 V + B1: V. +HDE 305 543. Gagn´e et al. (2011) identified this system as +a spectroscopic binary and classified it as B0 V + B0 V. In +Villafranca III we use LiLiMaRlin data to confirm it is an SB2 +and reclassify it as B0.2 V(n) + B1: V(n). +HDE 303 312. We detect this object as a SB2 for the first time +with GIRAFFE and assign it spectral types O9.5 III + B0.5: V. +In GOSSS II, where it was likely caught at a disfavorable phase, +it had received the intermediate type O9.7 IV. It was already +known to be an eclipsing binary with a 9.4109 d period (Otero +2006). +CPD −58 2649 A. We classify this system as an SB2 with spec- +tral types O9.7 III: +B0: V in GOSSS IV with GOSSS data. +With GIRAFFE data, we can only give a poorer classification +of O9.5: + B0: due to the different phases in each grating, but in +any case both components are clearly later than the O7 V + O8 V +of Alexander et al. (2016). There is a visual companion detected +in Gaia EDR3 with a separation of 1.′′2 that, though relatively +weak, may contaminate the GOSSS and GES spectra. +ALS 15 860. This object was considered as a Villafranca O- +027 (Trumpler 15) member by Smith (2006a), where it re- +ceived a classification as O9 I-II. Using either the GOSSS or +GES data here we classify it as B1 Iab. Its Gaia EDR3 dis- +tance is 3.31+0.23 +−0.20 kpc, placing it beyond Car OB1, consistent +with its red color (it is the brightest OB star in our sample with +GBP3 − GRP3 > 1.8). +CPD −58 2634. We classify this object as B1.5 V using +GIRAFFE data. Its Gaia EDR3 distance is 1.869+0.077 +−0.071 kpc, plac- +ing it in the foreground. Its parallax is consistent with being at +the same distance as HD 93 501. +CPD +−59 +2591. This +system +in +Villafranca +O-028 +(Collinder 228) was classified as an SB2 with spectral +types O8 Vz + B0.5: V both in GOSSS III using GOSSS +spectroscopy and in GOSSS IV using LiLiMaRlin data. Here it +is seen as SB2 but the classification is of poorer quality due to +the multiple epochs of the GIRAFFE data. +CPD −59 2535. This system was classified as B2 V by +Alexander et al. (2016) but there is no GES, GOSSS, or +LiLiMaRlin data. Its Gaia EDR3 distance is 3.16+0.23 +−0.20 kpc, plac- +ing it in the background. +2MASS J10424476−6005020. A GIRAFFE spectrum is used +to identify this system as an SB2 with the spectral types +B0.2 V + B0.2 V. In a GOSSS spectrum no double lines are +seen, likely due to an unfavorable epoch, and the resulting spec- +tral classification is a poorer B0: IV. Its Gaia EDR3 distance is +4.02+0.39 +−0.33 kpc, placing it in the background. Its red color is con- +sistent with the measured distance. +2MASS J10460477−5949217. This object in Villafranca O- +030 (Bochum 11) is classified as an O star for the first time +with GIRAFFE data. It has a moderately high extinction and a +spectral classification of O9.7 V(n). The spectrum could be a +composite of a late-O and an early-B stars but more epochs are +needed to test that hypothesis. +2MASS J10444803−5954297. This object is an Oe star with +strong Balmer emission and no previous classification as O type. +As usual with Oe stars, spectral classification is of poor quality +and we can only give O7: Ve using GIRAFFE and O8: Ve in +GOSSS IV using GOSSS. The Gaia EDR3 parallax indicates a +background object at a distance of 4.71+0.79 +−0.59 kpc. +CPD −59 2618. Alexander et al. (2016) classified this system +as B2 V. A GIRAFFE spectrum indicates it is of an earlier sub- +type and with an anomalous composition, yielding a classifica- +tion of B1: V(n)p He rich. In Villafranca III we use LiLiMaRlin +data to conform the helium enrichment and to further discover it +is an SB2, classifying it as B0.7: V(n)p He rich + B1: V. +ALS 15 225. We identify this star in Villafranca O-028 +(Collinder 228) as a He-rich B star for the first time using both +GIRAFFE and GOSSS spectroscopy here. +7 + +Berlanas et al.: Massive stars in the Carina Nebula +V662 Car. Niemel¨a et al. (2006) identified this system as an +O5.5 Vz + O9.5 V SB2 and an eclipsing binary with a period of +1.41355 d. In GOSSS III it was classified as O5 V(n)z + B0: V. +The GIRAFFE data shows there are two separate components in +He ii and both narrow, with a third component clearly separated +in He i, making it an SB3. However, given the multiple epochs +in the GIRAFFE data, we can only classify it as O+O+B. Both +O stars have narrow lines, so the GOSSS III (n) suffix is likely +due to combination of two O stars. The second O star is likely +a third light not participating in the orbit and appears to be of +mid-O subtype. The first O star has He ii 4542 > He ii 4471 and +should be close to O5. This system needs further high-resolution +spectroscopy covering the whole classification range in a single +epoch at large velocity separation. +ALS 15 203 A,B. In Villafranca II this Villafranca O-002 +(Trumpler 14) object was identified as an SB3 with a classifica- +tion of B0 V + B + B. In GIRAFFE we see some double lines but +He ii lines are very weak, invalidating the Vijapurkar & Drilling +(1993) classification as O7 V. As Gaia EDR3 detects two sources +of similar magnitude separated by 1.′′2 (confirmed by HST imag- +ing), we reanalyzed the GOSSS long slit with the best seeing +and proper orientation and we were able to spatially separate the +two visual components. ALS 15 203 A is an SB2 with a spectral +classification of B0.5 V + B1: V, which corresponds to those of +the secondary and tertiary in Villafranca II, and ALS 15 203 B +has a classification of B0 V, which corresponds to the primary in +Villafranca II. There is a hint of emission at the bottom of Hβ for +ALS 15 203 B but it is unclear whether it is of stellar origin or +is due to an incorrect nebular subtraction. In any case, this SB3 +system is now a SB2+Cas following the SBS nomenclature of +Ma´ız Apell´aniz et al. (2019b) +2MASS +J10435902−5933196. We +identify +this +star +in +Villafranca O-002 (Trumpler 14) as a He-rich B star for the first +time using GIRAFFE data. +2MASS +J10441829−5942296. We +identify +this +star +in +Villafranca O-003 (Trumpler 16 W) as a He-rich B star for the +first time using GIRAFFE data. +2MASS J10453807−5944095. This object in Villafranca O- +025 (Trumpler 16 E) is identified as an O star for the first time +here using GIRAFFE data. It has an O8 Vz spectral classification +and a high extinction. +2MASS J10440744−5916399. This is a background object +caught as an SB2 but with an uncertain GIRAFFE classification +of O9.7: + B0.5:. If confirmed, it would be another new O star. +The derived Gaia EDR3 distance is d = 4.45+0.56 +−0.45 kpc. +[ESK2003] 148 = [S87b] IRS 41. This system was first iden- +tified as an O-type candidate at the distance of Car OB1 by +Damiani et al. (2017). The identification was based on a pho- +tometric analysis with CHORIZOS (Ma´ız Apell´aniz 2004) and +resulted in values of Teff = 42.4 ± 4.4 kK, E(4405 − 5495) = +1.351 ± 0.020 mag, and R5495 = 4.92 ± 0.09, which indicates an +O star with both large color excess and anomalous extinction. +It is classified as O9.2 V(n) both in GOSSS IV and here using +GIRAFFE, so the Teff appears to be slightly lower than the value +measured with CHORIZOS. It is highly reddened but with a po- +sition and parallax consistent with being in Villafranca O-025 +(Trumpler 16 E), likely slightly behind the rest of the cluster and +immersed in the molecular cloud. +2MASS J10471498−5953374. A GIRAFFE spectrum yields +the spectral classification B6: IIIe, with a double-peaked emis- +sion line in Hα but no emission in Hγ (no other Balmer lines are +covered by the GIRAFFE data). The derived Gaia EDR3 dis- +tance is d = 3.14+0.29 +−0.25 kpc, placing it in the background. +2MASS J10443089−5914461. This object is classified as a +highly extinguished supergiant with spectral type O7.5 II(f) us- +ing either GOSSS data in GOSSS IV or GIRAFFE data here. +Alexander et al. (2016) classified it as O8 V but it is clearly not +a dwarf. It has a large parallax uncertainty: it could not be dis- +carded as being in Car OB1 but it is more likely a background +object. +2MASS J10453185−6000293. This highly extinguished O +star in Villafranca O-030 (Bochum 11) is identified as an O star +for the first time. It receives a spectral classification of O7.5 V in +GOSSS IV using GOSSS data and a slightly later one of O8.5 V +here using GIRAFFE data. The latter is rather noisy but some +lines show signs of asymmetry, indicating a possible spectro- +scopic binary. +[ARV2008] 217 = [S87b] IRS 42. This object is one of the +most interesting discoveries in this paper. Using GOSSS data +in GOSSS IV the authors give it an O3: III: spectral classifica- +tion and using GIRAFFE here we arrive at the same classifica- +tion but with an (n) suffix. In both cases the O3: classification is +based on the apparent absence of He i 4471 but the two spectra +are too noisy to provide a more accurate classification based on +the N lines. Therefore, it is a new member of the limited family +of Galactic O stars with spectral types earlier than O4. It was +first identified as an O-type candidate at the distance of Car OB1 +by Damiani et al. (2017). The CHORIZOS analysis there gives +Teff = 42.0 ± 4.2 kK, E(4405 − 5495) = 1.932 ± 0.021 mag, +and R5495 = 4.60 ± 0.06, that is, an early-type O star with both +large color excess and anomalous extinction. That analysis is in +good agreement with the spectral classification. That object po- +sition and parallax are consistent with [ARV2008] 217 being in +Villafranca O-025 (Trumpler 16 E), making it the earliest O-type +star there. +2MASS J10431945−5944488. The Gaia EDR3 parallax for +this object yields d = 794+28 +−26 pc, clearly making it a foreground +(and very blue) object. The existence of broad Hγ and He ii lines +indicate that the spectrum is dominated by an sdO. However, +He ii 4542 and He ii 4686, observed at different epochs, have dif- +ferent velocities and some lines appear to originate in a later-type +star. Therefore, the system is a spectroscopic binary. +2MASS J10431945−5944488. The Gaia EDR3 parallax for +this object yields d = 794+28 +−26 pc, clearly making it a foreground +(and very blue) object. The existence of broad Hγ and He ii lines +indicate that the spectrum is dominated by an sdO. However, +He ii 4542 and He ii 4686, observed at different epochs, have dif- +8 + +Berlanas et al.: Massive stars in the Carina Nebula +ferent velocities and some lines appear to originate in a later-type +star. Therefore, the system is a spectroscopic binary. +High-extinction population of Preibisch et al. (2021). That pa- +per lists several stars that were too faint to be observed with +GIRAFFE in the blue-violet region. Regarding their Gaia EDR3 +parallaxes, most of them are similar to or smaller than that +of Car OB1 but with larger uncertainties. There is only one +with negative parallax, so it is likely a background object: +2MASS J10452648−5946188 (=[HSB2012] 3994), which was +already identified as a highly-extincted B star by Damiani et al. +(2017) using CHORIZOS. +4. Results and discussion +4.1. The observed CMD and completeness +We first discuss the observed Gaia EDR3 CMD for the sample +of 316 objects in this paper, which is plotted on the left panel of +Fig. 2. Of those, only four are located in the foreground but two +of them are in distinct regions of the CMD: HD 93 420, a RSG +in the upper right, and 2MASS J10431945−5944488, a sdO in +the lower left. The main group, the 294 objects in Car OB1, do +not follow the typical isochrone of a cluster or association be- +cause of the strong differential extinction present in the region. +The majority concentrates between the extinguished isochrones +that correspond to E(4405 − 5495) of 0.3 and 0.6 (assuming an +R5495 of 4.5) but some are significantly more extinguished than +that, including the four Preibisch et al. (2021) objects outside +the frame towards the lower right. The 18 background objects +have, on average, a higher extinction than the Car OB1 popu- +lation, an expected effect of the extinction associated with the +Carina Nebula. They appear mixed in the vertical direction with +the Car OB1 population but one should consider that if we plot- +ted absolute magnitude on the vertical axis, they would move up. +For example, five of the 18 objects are B supergiants. +The majority of the stars lie between the R5495 = 4.5 extinc- +tion tracks for average MS stars of Teff = 20 kK and 52.5 kK +stars. A significant fraction lies below the track of Teff = 20 kK, +due to a combination of different effects: an average-age B2.5 V +star can have a Teff somewhat lower than 20 kK, ZAMS stars +should be lower in the CMD than average-age ones, and the ex- +tinction tracks for R5495 > 4.5 (which is known to be appropriate +for some stars in the Carina Nebula, see Ma´ız Apell´aniz & Barb´a +2018) are steeper than the plotted ones. Above the extinction +track of Teff = 52.5 kK we find ten Car OB1 stars: η Car, the two +RSGs, WR 24, four O supergiants, and HD 93 250 A.B (a close +binary with two very early type components, see Le Bouquin +et al. 2017), that is, all of them objects that are expected to be +there. +The most notorious feature of the left panel of Fig. 2 is how +well the Car OB1 O and B stars are separated in the CMD, +with the O stars mostly above the average-age extinction track +of Teff = 30 kK for R5495 = 4.5 and the B stars below it. This +is an indirect confirmation of the quality of the spectral clas- +sifications. The separation is not perfect but it is not expected +to be for several reasons: B giants and supergiants (plus some +early B + early B binaries) are expected to be above the average- +age extinction track of Teff = 30 kK for R5495 = 4.5 and late +O-dwarfs near the ZAMS below that track. In addition, varia- +tions in R5495 among sightlines should produce some mixing, as +low values of R5495 can move B stars into the O-star territory +and high values of R5495 can move O stars into the B-star terri- +tory. Examples of the latter possibility are two of the stars from +the Preibisch et al. (2021) sample, 2MASS J10454595−5949075 +and 2MASS J10452013−5950104. If they are confirmed to be +normal O dwarfs, their value of R5495 should be high. +When building a census, one of the most important ques- +tions that have to be addressed is how complete it is and that +is especially important when the sample is built from multiple +sources such as in this paper. To answer that question, we have +plotted in the right panel of Fig. 2 all the Gaia EDR3 sources +found within the footprint that have positive corrected parallaxes +consistent with being at the distance of Car OB1 and that have +catalog values of GBP3 − GRP3. The right panel shows that the +left panel is just the tip of the iceberg in terms of a moderately +extinguished well-populated main sequence. In addition to that +main sequence, a significant population of red stars is present. +By comparison with the left panel, some of those are extin- +guished OB stars but a comparison with other Galactic sight- +lines indicates that most of them must be intrinsically red stars. +For example, the diagonal series of stars that follows the ex- +tinction track of Teff = 30 kK around GBP3 − GRP3 ∼ 1.5 is the +red-clump extinction sequence, ubiquitously seen in the Galactic +plane when plotting absolute magnitude in the vertical axis (as +we are effectively doing here by selecting a population consistent +with being at the same distance). That sequence starts around +GBP3 − GRP3 ∼ 1.1 for zero extinction and here we are just see- +ing it with an extinction distribution not too different from that +of the OB stars in Car OB1. +Given the dominance of the late-type population for red col- +ors (something that needs to be addressed with additional data +such as NIR photometry), we do not have the means to deter- +mine how complete the sample is for high extinctions. Indeed, +that is why a paper as recent as Preibisch et al. (2021) was able +to find several new O stars in Car OB1: thick dust clouds can eas- +ily hide OB stars if one does not have access to IR data and even +in that case finding the hot needle in the cool haystack is not al- +ways straightforward. Therefore, we concentrate on the low- and +moderate-extinction part of the sample, defined as those OB stars +with GBP3 − GRP3 < 1.0 (just to the left of the point where the +first red clump stars are expected to appear). Also, as for fainter +stars one expects any sample to be less complete, we restrict the +completeness analysis to the region above the extinction track +of Teff = 20 kK in Fig. 2. In other words, we are assessing how +complete the sample is regarding low/moderate extinction O and +early-B stars. +We cross-matched the two samples (the one used through- +out this paper and the full Gaia EDR3 one) inside that area and +found 154 coincidences. Three objects in the main sample are +not present in the Gaia EDR3 sample either because they lack +parallaxes (CPD −59 2636 A,B and ALS 19 740) or because +they are completely absent (HD 93 129 Ab), note that η Car +would be also absent if it were inside that area. Gaia EDR3 +is quite complete barring a few small-separation binaries. As +for the other way around, 19 systems in the Gaia EDR3 sam- +ple are not present in the main one (green stars in the right +panel). Of those, five have large external parallax uncertainties +(> 0.1 mas), so chances are they are not real Car OB1 mem- +bers. Therefore, we estimate that our sample is around 90% +complete for low/moderate extinction O and early B systems. +Furthermore, the location of the 19 green stars in the right panel +of Fig. 2, all of them below the extinction track of Teff = 30 kK, +suggests that those missing objects are likely of early-B type. +Therefore, we conclude that we are missing very few or even no +low/moderate O-type systems in Car OB1 within our footprint +in our sample. As mentioned above, objects with high extinc- +tion may be another story. In any case, the 74 Car OB1 O-type +9 + +Berlanas et al.: Massive stars in the Carina Nebula +0.0 +0.5 +1.0 +1.5 +2.0 +52.5 +30 +20 +−0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +GBP3−GRP3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +16 + G3’ +foreground +background +Car OB1 WR or (non−O) sg +Car OB1 O +Car OB1 other +Fig. 2. First panel, see next page for the second one: Gaia EDR3 CMD for the stars with spectral types in this paper. Different +symbols and colors are used to represent stars with parallaxes compatible with being (or otherwise assumed to be) in the foreground +(4), in the background (18), or in Car OB1 (294). Of the Car OB1 stars, 6 are of Wolf-Rayet or non-O supergiant (II to Ia) spectral +class, 74 are of O type, and 214 are of non-supergiant B type. Four of the Preibisch et al. (2021) stars are outside the frame towards +the lower right due to their high extinction. Black lines show the average main sequence at a distance of 2.35 kpc with no extinction +and with values of E(4405 − 5495) of 0.5, 1.0, 1.5, and 2.0 (labelled) using the extinction law of Ma´ız Apell´aniz et al. (2014) with +a value of R5495 of 4.5, which is typical of the region but with a large dispersion (Ma´ız Apell´aniz & Barb´a 2018). Solid orange lines +show the R5495 = 4.5 extinction tracks for average MS stars of Teff of 52.5 kK, 30 kK, and 20 kK (labelled), respectively. The dotted +orange line shows the R5495 = 3.0 extinction track for Teff = 30 kK. +systems in this paper are the largest nearly complete sample of +objects of that spectral type in any part of a Galactic OB associ- +ation. +4.2. Binary fraction +It is well known that multiplicity among massive stars is ubiq- +uitous. Commonly, multiplicity is divided into that which is de- +tected through spectroscopy (velocity changes and differences) +and imaging (or visual multiplicity) and is important to indicate +which one is being used, as some previous studies have conflated +them and caused confusion. In GOSSS II we analyzed the pop- +ulation of Galactic southern stars and found out that 65-91% of +them are multiple stars of one type or another, with the values for +spectroscopic and visual multiples being 50-60% and 30-76%, +respectively. One consequence of those numbers is that a signif- +icant fraction (at least 15%) are at the same time spectroscopic +and visual multiples, and most of those involved three stars, as +in 2014 the number of pairs detected simultaneously with spec- +troscopy and imaging was quite low. An analysis of known mul- +tiple O stars in the northern hemisphere (Ma´ız Apell´aniz et al. +2019a) confirmed the trend towards systems of three or more +stars and revealed that simple binaries are a minority once spec- +troscopic and visual multiples are included. For example, hier- +archical triples composed of a short-period (less than 1 month) +system orbited by a companion in a long-period (years or more) +orbit are quite common. +In the census presented here we find 20 spectroscopic binary +systems containing at least one O-type star (listed in Table A.5), +one of them located in the background. There are six new sys- +tems reported for the first time in this work, either from GES +and/or GOSSS IV observations. Excluding the background sys- +10 + +Berlanas et al.: Massive stars in the Carina Nebula +0.0 +0.5 +1.0 +1.5 +2.0 +52.5 +30 +20 +−0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +GBP3−GRP3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +16 + G3’ +outside triangle +inside triangle, matched +inside triangle, unmatched +Fig. 2. Second panel, see previous page for the first one: Equivalent plot but for all Gaia EDR3 stars in the region of interest with +corrected parallaxes that are compatible with the distance to Car OB1 and positive and with catalog values of GBP3 − GRP3. The +plotted objects are classified according to whether they are located inside or outside of the area limited by GBP3 − GRP3 = 1.0 and +the R5495 = 4.5 extinction track for average MS stars with Teff = 20 kK. Stars inside that area are further divided into those matched +with objects in the left panel (154) and those unmatched (19). Note that an additional three stars inside the above mentioned area in +the left panel (HD 93 129 Ab, CPD −59 2636 A,B, and ALS 19 740) plus η Car outside the that area are not shown either because +they are not included in Gaia EDR3 or have no parallaxes there. +tem, the total number of O stars in the 19 systems of Car OB1 +is 30. This represents a fraction of 0.35 (30 out of a total of +85 O-stars, see Table 2 where binary statistics and fractions for +the spectroscopic systems containing at least one O-type star in +the different Villafranca groups are summarized). This number is +still far to those reported in GOSSS II and also to the 0.44 frac- +tion of O-type stars in binaries quoted by Sana & Evans (2011) +or even the somewhat more than 0.50 indicated by Sana (2017). +This indicates that there is a significant number of binaries still +to be identified in the region. We highlight that the multiplicity +statistics reported in this work is, however, incomplete. We only +report double-line spectroscopic binaries. Visual binaries are not +considered and only a small fraction of the sample has signif- +icant multi-epoch coverage for single-line spectroscopic binary +detection. In spite of this, we significantly increase the fractions +quoted by Sana & Evans (2011) in Trumpler 14 and Trumpler +16, for which these authors quoted fractions of zero and 0.48, +respectively (see Table 2). +In addition, we found 18 spectroscopic binary systems +formed by early B-type stars, one of them a background sys- +tem and 13 new binary detections from GES, GOSSS and +LiLiMaRlin observations (see Table A.6). In Fig. 3 we show an +example of the new spectroscopic binary detections from GES +spectra at their original resolution. The Si III triplet at λ4552-68- +75 Å is shown for binary late-O (a and b panels) and early-B (c +and d panels) systems. See figures in Appendix A for full spectra +details. +4.3. The spectroscopic n-qualifier as an indicator for rotation +As stated in Sect. 2.3, the MGB tool has been used for the spec- +tral classification of GES data. This tool allows the user to obtain +not only the spectral subtype and luminosity classification, but +also spectral peculiarities and the rotation index. Broadening is +denoted by (n), n, nn and nnn indexes, progressing from some- +what to more and even more broadened lines. Therefore, this +11 + +Berlanas et al.: Massive stars in the Carina Nebula +Wavelength (A) +0.90 +0.95 +1.00 +1.05 +Normalized flux +HDE 303 312 (O9.5 III + B0.5: V) +(a) +I +P +I +S +I +P +I +S +HDE 303 312 (O9.5 III + B0.5: V) +(a) +I +P +I +S +I +P +I +S +HDE 303 312 (O9.5 III + B0.5: V) +(a) +I +P +I +S +I +P +I +S +Wavelength (A) +CPD -58 2649 (O9.5: + B0) +(b) +I +P +I +S +I +P +I +S +CPD -58 2649 (O9.5: + B0) +(b) +I +P +I +S +I +P +I +S +CPD -58 2649 (O9.5: + B0) +(b) +I +P +I +S +I +P +I +S +4550 +4555 +4560 +4565 +4570 +4575 +Wavelength (A) +0.90 +0.95 +1.00 +1.05 +Normalized flux +CPD -59 2661 (B0: + B) +(c) +I +S +I +P +I +S +I +P +CPD -59 2661 (B0: + B) +(c) +I +S +I +P +I +S +I +P +CPD -59 2661 (B0: + B) +(c) +I +S +I +P +I +S +I +P +4550 +4555 +4560 +4565 +4570 +4575 +Wavelength (A) +HD 93 128 B (B0.2: V + B1: V) +(d) +I +P +I +S +I +P +I +S +HD 93 128 B (B0.2: V + B1: V) +(d) +I +P +I +S +I +P +I +S +HD 93 128 B (B0.2: V + B1: V) +(d) +I +P +I +S +I +P +I +S +Fig. 3. Example of new spectroscopic binary systems reported in this work: Si iii line profiles from GES spectra shown at their +original resolution. For reference, P and S letters indicate the position of the primary and secondary component, respectively. See +figures on Appendix A for full spectra details. +Table 2. Binary statistics and fractions for the spectroscopic systems containing at least one O-type star present in the census. +Group +Single +Binary +O stars +Fraction +O-stars +O-systems +in binaries +O-002 (Trumpler 14) +10 +3 +5 +0.33 +O-003 (Trumpler 16 W) +2 +2 +3 +0.60 +O-025 (Trumpler 16 E) +8 +6 +10 +0.55 +O-027 (Trumpler 15) +3 +- +- +0 +O-028 (Collinder 228) +16 +3 +4 +0.15 +O-029 (Collinder 232) +2 +- +- +0 +O-030 (Bochum 11) +4 +1 +2 +0.33 +Car OB1 +10 +4 +6 +0.37 +Whole sample +55 +19 +30 +0.35 +Notes. Background and foreground members are excluded from the statistics. We +separate numbers considering the different Villafranca groups of Carina. Car OB1 +group refers to the stars just falling in the gaps between defined Villafranca groups +(see Appendix A) for further details. +qualifier has been traditionally interpreted as a sign for high ro- +tational velocity. As a consistency check, we have determined +the projected rotational velocity of stars with this qualifier, in +order to know whether there is a 1:1 relationship between both. +To that aim we used iacob-broad, a user-friendly tool +for the line-broadening characterization of OB stars (Sim´on- +D´ıaz & Herrero 2007, 2014). It is based on a combined Fourier +Transform (FT) and the Goodness-of-fit (GOF) method that al- +lows us to determine easily the stellar projected rotational ve- +locity (v sin i) and the amount of extra broadening (vmac) from a +specific diagnostic line. The FT technique is based on the iden- +tification of the first zero in the Fourier transform of a given +line profile (Gray 2008; Sim´on-D´ıaz & Herrero 2007). The GOF +technique is based on a comparison between the observed line +profile and a synthetic one that is convolved with different values +of v sin i and vmac to obtain the best-fit by means of a χ2 optimiza- +tion. The main advantage of this methodology is that we obtain +two independent measurements of the v sin i (resulting from ei- +ther the FT or the GOF analysis) whose comparison is used as +a consistency check and to better understand problematic cases. +Since metallic lines do not suffer from strong Stark broaden- +ing or nebular contamination, they are best suited for obtaining +accurate v sin i values. GIRAFFE set-ups cover the Si iii λ4552 +diagnostic line, while UVES/FEROS/HARPS set-ups also cover +the O iii λ5592 diagnostic line. In case none of them are present +or are too weak, we then use the nebular free or weakly contam- +inated He i lines (He i λ4387, λ4471, λ4713). +Fig. 4 presents the v sin i histogram of those OB stars in- +cluded in our census with any broadening index in their spec- +tral classification (see Table A.2). Mean v sin i values for (n), +12 + +Berlanas et al.: Massive stars in the Carina Nebula +50 +100 +150 +200 +250 +300 +350 +400 +450 +500 +550 +v sini (km/s) +0 +5 +10 +15 +20 +25 +30 +35 +N +(n) +n +nn +nnn +Fig. 4. v sin i histogram of those OB stars with any rotation index +in their spectral classification. Broadening is denoted by (n), n, +nn and nnn indexes, progressing from somewhat to more broad- +ened lines. Points and horizontal lines on the top of the figure +indicate the mean v sin i value for each rotating group and the +corresponding dispersion, respectively. +n, nn and nnn are 206, 265, 317 and 392 km s−1, respectively, +which confirms the trend that the higher the rotation index the +higher the projected rotational velocity. However, we find a sig- +nificant overlap in the ranges of projected rotational velocities. +For example, the (n)- and n-type stars peak at the same bin: be- +tween 200 and 250 km s−1; and the fastest n-star rotates 88 km +s−1faster than the slowest nn-star (but the distribution of (n)-stars +is slanted towards the left from that point while that of n-stars +is slanted towards the right). There are two likely explanations +for the overlap. On the one hand, a non-negligible fraction of +these stars could end up being actually spectroscopic binaries +since such broadened lines may prevent us from detecting bi- +nary line profiles when multi-epoch observations are not avail- +able. On the other hand, the sample is dominated by B1-B2.5 +dwarfs, which have few intrinsically deep and narrow lines in +the analyzed wavelength range (which is only a part of the stan- +dard blue-violet classification range), making the n-type indexes +more unreliable than for e.g. O stars or B supergiants. +4.4. Runaway candidates +Benefiting from the high-precision astrometry that Gaia EDR3 +provides in Carina, we have investigated the proper motions +of the stars in our census to identify bona-fide runaways as a +first step for future studies. Following the ideas of de Mink +et al. (2013) (who propose that fast rotating stars are the product +of post-interacting binaries and therefore could also have been +ejected from binary systems in which the mass donor exploded +as supernova) we are interested in exploring whether there is a +connection between O and B-type stars with the spectroscopic +n-qualifier5 and the runaway status. +5 as a proxy for fast rotation, although we emphasize that, even if +there is a good correlation with the average v sin i, not all stars with this +qualifier have a high projected rotational velocity, as shown in Sect. 4.3. +The proper motion distribution for each Villafranca group is +shown in Fig. 5. We define group centers through an iterative +process assuming the average values of each group members, +but excluding detected binaries, objects with a RUWE (renor- +malised unit weight error) > 1.4 and those stars that do not com- +ply with the proper motion constraint described below. For ref- +erence, group proper motions in α∗ and δ derived in this work +and those from the Villafranca II and III works are shown in +Table 3. As in Villafranca II, we find the proper motion of O- +025 not identical to that of O-003, indicating that both groups in +Trumpler 16 are well separated. We iteratively filter stars with +proper motions larger than the mean values for each group by +more than three sigma. To this aim, we calculated for each group +σg = +� +σ2µα∗ + σ2µδ, deriving a final mean σg of 0.342 mas a−1 +and thus a three sigma value of 1.03 mas a−1. We find four +stars with the n-qualifier in their spectral classification that do +not meet the imposed constraint (those stars falling outside the +circles in Fig. 5). Three of them (2MASS J10440866-5933488 +in O-002, CPD -59 2541 in O-028 and 2MASS J10451588- +5929563 in O-029) can be considered firm runaway candidates. +We note, however, that the large RUWE value for CPD -58 2657 +(in the O-027 group) indicates inaccurate astrometric measure- +ments (see Table A.7). The rest of stars with the n-qualifier +are homogeneously distributed around the core motion of each +group. Interestingly, the two extreme very fast B rotators of our +sample, ALS 15 248 and 2MASS J10433865-5934444 rotating +both at v sin i > 450 km s−1, do not show peculiar proper motions, +and so are consistent with the main values of each group. We +also find four further runaway candidates that are not included in +the group with the n-qualifier but show peculiar proper motions. +Two of them are RSG stars. We note that two stars identified in +Villafranca I as possible runaway stars ejected from Trumpler 14 +(HDE 303 313 and ALS 16 078) spatially fall in the gaps of the +redefined Villafranca groups. Therefore, they have been labelled +as just Car OB1 members and are not discussed here6. +Thus we have four out of 90 stars with the n-qualifier iden- +tified as candidate runaway objects and another four out of 168 +without the n-qualifier, which means fractions of 4.4% and 2.4%, +respectively 7. This points to a connection between runaways +and fast rotators, as pointed out by other works (see f.e. de Mink +et al. 2013; Holgado et al. 2022, and references therein) particu- +larly if we consider that the viewing angle may be affecting the +projected rotational velocities, resulting in less broadened lines. +However, given the limitations of our work, further research on +this topic (in particular, a distribution of projected rotational ve- +locities and a more detailed study of the runaway condition) is +needed in order to obtain a firm conclusion. +Finally, we remark that the distribution of binary systems in +the proper motion diagram (crosses in Fig 5), contrary to what +might be expected, is homogeneously distributed. A similar pat- +tern was found in the Cygnus OB2 association (Berlanas et al. +2020), implying that these systems may still keep their original +velocities. +6 a direct comparison between the runaway candidates identified in +both works must be done with caution since different methods have +been used. Note that in Villafranca works, stars are selected as candi- +date runaway/walkaway objects when their proper motion points in the +opposite direction to that of the center of the group (within some mar- +gins, see Villafranca I-II for further details). +7 Note that detected spectroscopic binary systems, with or without +the n-qualifier, have been excluded from the statistics +13 + +Berlanas et al.: Massive stars in the Carina Nebula +8 +7 +6 +5 +* (mas a +1) +1 +2 +3 +4 + (mas a +1) +O-002 +8 +7 +6 +5 +* (mas a +1) +O-003 +8 +7 +6 +5 +* (mas a +1) +O-025 +8 +7 +6 +5 +* (mas a +1) +O-027 +8 +7 +6 +5 +* (mas a +1) +1 +2 +3 +4 + (mas a +1) +O-028 +8 +7 +6 +5 +* (mas a +1) +O-029 +8 +7 +6 +5 +* (mas a +1) +O-030 +Fig. 5. Proper motion distribution from Gaia EDR3 astrometry for all stars of our census in each assigned Villafranca group. Orange +squares indicate those OB stars analyzed in this work that are rotating at v sin i ≥ 200 km s−1. Blue crosses represent identified +binary systems. Circles represent group proper motion constraints, whose centers µα∗,g and µδ,g are those shown in Table 3 in the +central columns. For comparison, red plus symbols indicate group centers from Villafranca II and III works. Note that stars labelled +as Car OB1 members are not included in the panels. +Table 3. Group proper motions in α and δ derived in this work and those from the Villafranca II and III works, all based on Gaia +EDR3 astrometry. +This work +Villafranca II-III +Group +N +µα∗,g +µδ,g +µα∗,g +µδ,g +O-002 (Trumpler 14) +32 +-6.580 ± 0.240 +2.089 ± 0.246 +-6.534 ± 0.023 +2.076 ± 0.023 +O-003 (Trumpler 16 W) +8 +-7.179 ± 0.102 +2.730 ± 0.103 +-7.128 ± 0.024 +2.670 ± 0.024 +O-025 (Trumpler 16 E) +47 +-6.894 ± 0.214 +2.647 ± 0.177 +-6.877 ± 0.023 +2.596 ± 0.023 +O-027 (Trumpler 15) +17 +-6.172 ± 0.164 +2.224 ± 0.175 +-6.282 ± 0.023 +2.131 ± 0.023 +O-028 (Collinder 228) +53 +-6.896 ± 0.334 +2.332 ± 0.396 +-6.713 ± 0.021 +2.070 ± 0.021 +O-029 (Collinder 232) +12 +-6.667 ± 0.414 +2.238 ± 0.294 +-6.552 ± 0.023 +2.142 ± 0.023 +O-030 (Bochum 11) +19 +-6.559 ± 0.204 +2.328 ± 0.307 +-6.635 ± 0.021 +2.279 ± 0.021 +Note. Group uncertainties reported in this work refer to the standard deviation of the selected OB stars while those in +Villafranca II-III correspond to the standard deviation of the mean (with the angular covariance term included) of all +the stars identified as group members, a much larger number than the one used in this work. +5. Conclusions +We present a new census of massive stars in the central part +of Carina, Car OB1, based on high-quality spectroscopic data +provided by GES, GOSSS, LiLiMaRlin and additional sources +from the literature. It contains a total of 316 massive stars. We +separated stars with distances compatible with Car OB1 (assign- +ing group membership) from those in the foreground and back- +ground, finding four systems in the foreground and 18 in the +background. Of the 294 stellar systems in Car OB1, 74 are of +O type, 214 are of non-supergiant B type and six are of WR or +non-O supergiant (II to Ia) spectral class. We estimate that our +sample is around 90% complete for low/moderate extinction O +and early B systems, missing very few or even no O stars within +our footprint. The 74 Car OB1 O-type systems quoted in this +paper are the largest nearly complete sample of objects of that +spectral type in any part of a Galactic OB association. +Among the stellar census, we identified 20 spectroscopic bi- +nary systems that contain at least one O-type star. Six of them +are new identifications and one is located in the background. The +observed binary fraction of O stars found in the Car OB1 region +is 0.35, although this number only refers to double-lined spec- +troscopic binaries and represents, therefore, a lower limit. Visual +binaries are not considered and only a small fraction of the sam- +14 + +Berlanas et al.: Massive stars in the Carina Nebula +ple has significant multi-epoch coverage for single-lined spectro- +scopic binary detection. Thus, this number should be considered +as a lower limit. In addition, we found another 18 spectroscopic +binary systems with a B-star primary, one of them being a back- +ground system and 13 of them new binary detections from GES, +GOSSS and LiLiMaRlin observations. +We explore the correlation between the spectroscopic rota- +tion index, n, and the actual projected rotational velocities of +the stars. We find a good correlation of the average v sin i values +with the qualitative classification of each group ((n), n, nn, nnn). +However, there is a significant overlap in their v sin i ranges. We +note that it is possible that a non-negligible fraction of these stars +are actually spectroscopic binaries contaminating the fast rota- +tors sample. +Finally, we investigated the proper motion distribution for +the sample of those O and B-type stars with a spectroscopic n- +qualifier. Our results indicate a connection between runaways +and fast rotators. Furthermore, the distribution of binary systems +in the proper motion diagram is homogeneously distributed, im- +plying that these systems may still keep their original velocities. +Acknowledgements. This paper is based mainly on data products from spec- +troscopic observations made with ESO Telescopes at the Paranal Observatory +under programme ID 188.B-3002. These data products have been processed by +the Cambridge Astronomy Survey Unit (CASU) at the Institute of Astronomy, +University of Cambridge, and by the FLAMES/UVES reduction team at +INAF/Osservatorio Astrofisico di Arcetri. These data have been obtained from +the Gaia-ESO Survey Data Archive, prepared and hosted by the Wide Field +Astronomy Unit, Institute for Astronomy, University of Edinburgh, which is +funded by the UK Science and Technology Facilities Council. This work was +partly supported by the European Union FP7 programme through ERC grant +number 320360 and by the Leverhulme Trust through grant RPG-2012-541. +We acknowledge the support from INAF and Ministero dell’ Istruzione, dell’ +Universit`a’ e della Ricerca (MIUR) in the form of the grant ’Premiale VLT +2012’. The results presented here benefit from discussions held during the Gaia- +ESO workshops and conferences supported by the ESF (European Science +Foundation) through the GREAT Research Network Programme. Additional +spectra were obtained using the 2.5 m du Pont Telescope at the Observatorio de +Las Campanas (LCO) and the 2.2 m MPG/ESO Telescope at the Observatorio +de La Silla (LSO). +This work has made use of data from the European Space Agency (ESA) +mission Gaia, processed by the Gaia Data Processing and Analysis Consortium +(DPAC). Funding for the DPAC has been provided by national institutions, in +particular the institutions participating in the Gaia Multilateral Agreement. +This +research +is +partially +funded +by +the +Spanish +Government +Ministerio de Ciencia e Innovaci´on and Agencia Estatal de Investigaci´on +(MCIN/AEI/10.130 39/501 100 011 033/FEDER, UE) through grants PGC2018- +93 741-B-C21/C22, PGC2018-95 049-B-C21/C22 and PID2021-122 397NB- +C21/C22. SRB also acknowledges funding by MCIN under the Juan de la +Cierva - Formaci´on grant (contract FJC 2020-45 785-I) and NextGeneration +EU/PRTR and MIU (UNI/551/2021) through grant Margarita Salas-ULL. +A.H. also acknowledges support by the Severo Ochoa Program through +CEX2019-000920-S. E.J.A also acknowledges financial support from the +State Agency for Research of the Spanish MCIU through the “Center of +Excellence Severo Ochoa” award to the Instituto de Astrof´ısica de Andaluc´ıa +(SEV-2017-0709). MB is supported through the Lise Meitner grant from the +Max Planck Society. We acknowledge support by the Collaborative Research +centre SFB 881 (projects A5, A10), Heidelberg University, of the Deutsche +Forschungsgemeinschaft (DFG, German Research Foundation). This project +has received funding from the European Research Council (ERC) under the +European Union’s Horizon 2020 research and innovation programme (Grant +agreement No. 949173). +References +Albacete Colombo, J. F., Morrell, N. I., Rauw, G., et al. 2002, MNRAS, 336, +1099 +Alexander, M. J., Hanes, R. J., Povich, M. S., & McSwain, M. V. 2016, AJ, 152, +190 +Banyard, G., Mahy, L., Sana, H., et al. 2022, arXiv e-prints, arXiv:2210.07149 +Berlanas, S. R., Herrero, A., Comer´on, F., et al. 2018a, A&A, 612, A50 +Berlanas, S. R., Herrero, A., Comer´on, F., et al. 2018b, A&A, 620, A56 +Berlanas, S. R., Herrero, A., Comer´on, F., et al. 2020, A&A, 642, A168 +Berlanas, S. 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V., et al. 2011, ApJS, 194, 12 +Appendix A: Tables and spectrograms +In this Appendix we present the tables with the information for +the stars in the field of the Carina Nebula analyzed in this paper +and the figures with the spectrograms that have not appeared in +previous papers. +Table A.1 lists the basic information for the stars: name, co- +ordinates, identifications, G′ +3 magnitude, Gaia EDR3 parallax +and group membership. Regarding the latter, the following al- +gorithm is used: +– All stars are initially labelled as Car OB1. +– A search is done to see if the star is located inside the re- +gion of the sky defined by the center and radius of one of +the groups defined in Villafranca II or III (Villafranca O- +002 = Trumpler 14, Villafranca O-003 = Trumpler 16 W, +Villafranca O-025 = Trumpler 16 E, Villafranca O- +027 = Trumpler 15, Villafranca O-028 = Collinder 228, +Villafranca O-029 = Collinder 232, and Villafranca O- +030 = Bochum 11). If found, then the membership is +changed to that group. Note that, as mentioned in the +Villafranca papers, the traditional division into such groups +is to some point arbitrary: Villafranca O-002, Villafranca O- +025, and Villafranca O-027 are likely real clusters while +the rest of the groups are just subassociations of the larger +Car OB1. As the apertures in the Villafranca papers are cir- +cular, some stars just fall in the gaps and remain labelled as +Car OB1. +– Stars without a parallax or with corrected parallax uncertain- +ties greater than 0.1 mas are given in bold face, to indicate +that the Gaia EDR3 information is not sufficient to determine +their distances. Note that many of those objects are known +to be located in Car OB1 for other reasons (e.g. η Car or +HD 93 129 Ab). +– Stars whose parallax is larger than 0.44 mas by more than +three sigmas are labelled as foreground. +– Stars whose parallax is smaller than 0.41 mas by more than +three sigmas or is negative are labelled as background. The +limits here and in the previous steps are determined from +Villafranca II and III. +Table A.2 lists the spectral types for the stars in this pa- +per. Three types of spectral types are given: those derived from +Gaia-ESO spectra (all new), those derived from GOSSS spectra +(most from previous papers and from GOSSS IV but some new, +marked as TW), and those derived from LiLiMaRlin and STIS +spectra as well as from the literature (all previously published). +Tables A.3 and A.4 list those stars of the census identified in +the foreground or in the background, and those of WR or non-O +supergiant (II to Ia) spectral class, respectively. +Tables A.5 and A.6 list spectroscopic binary systems iden- +tified in our census containing at least one O-type and those +formed by early B-type stars, respectively. +Table A.7 lists runaway candidates identified in this work. +Figure A.1 shows the UVES spectra, Fig. A.2 the GIRAFFE +spectra, and Fig. A.3 the GOSSS spectra. +16 + +Berlanas et al.: Massive stars in the Carina Nebula +Table A.1. Stars in the field of the Carina Nebula analyzed in this paper sorted by G′ +3. +Name +RA +dec +GOS/GBS/GWS ID +GES ID +Gaia ID +ALS ID +G′ +3 +GBP3 − GRP3 +ϖc(mas) +group +η Car +10:45:03.546 −59:41:03.95 +287.60−00.63 01 +— +5 350 358 584 482 202 880 1868 +4.1133 +1.3306 +— +O-025 +HD 93 420 +10:45:50.661 −59:29:19.33 +287.59−00.41 01 +— +5 350 343 775 447 416 704 — +5.9483 +3.0309 0.658±0.071 +foreground +QZ Car Aa,Ac +10:44:22.910 −59:59:35.95 +287.67−00.94 01 +— +5 350 346 970 905 044 480 1839 Aa,Ac 6.2727 +0.3369 0.771±0.436 +O-028 +WR 24 +10:43:52.258 −60:07:04.02 +287.67−01.08 01 +— +5 254 268 071 479 968 512 1817 +6.3929 +0.1271 0.408±0.062 +O-028 +HD 93 281 +10:44:57.341 −59:56:06.38 +287.70−00.86 03 +— +5 350 302 097 082 780 416 — +6.7871 +2.3637 0.429±0.049 +O-028 +HDE 303 310 +10:44:47.146 −59:24:48.16 +287.44−00.41 01 +— +5 350 389 095 946 525 568 — +6.9176 +3.3193 0.470±0.069 +O-027 +HD 93 129 Aa +10:43:57.462 −59:32:51.27 +287.41−00.57 01 +— +5 350 363 910 256 783 488 1820 A +7.2004 +0.4867 0.407±0.044 +O-002 +HD 93 403 +10:45:44.122 −59:24:28.15 +287.54−00.34 01 +— +5 350 391 535 488 119 552 1881 +7.2139 +0.4003 0.196±0.261 +Car OB1 +HD 93 250 A,B +10:44:45.027 −59:33:54.67 +287.51−00.54 01 +— +5 350 383 460 949 215 232 1859 A,B +7.2967 +0.3561 0.413±0.050 +O-029 +HD 93 205 +10:44:33.740 −59:44:15.46 +287.57−00.71 02 +— +5 350 357 313 186 767 104 1849 +7.6825 +0.1716 0.437±0.059 +O-003 +HD 93 160 A,B +10:44:07.267 −59:34:30.61 +287.44−00.59 01 +— +5 350 362 982 528 878 976 1831 A,B +7.7571 +0.3651 0.384±0.113 +O-002 +HD 93 162 +10:44:10.389 −59:43:11.09 +287.51−00.71 01 +— +5 350 357 519 345 171 200 1833 +7.8260 +0.9840 0.442±0.048 +O-003 +HD 93 130 +10:44:00.371 −59:52:27.50 +287.57−00.86 01 +— +5 350 350 303 799 853 056 1825 +7.9828 +0.4657 0.373±0.049 +O-028 +HD 93 222 A,B +10:44:36.250 −60:05:28.88 +287.74−01.02 01 +— +5 254 222 888 417 319 424 1852 A,B +8.0353 +0.2111 0.415±0.052 +O-028 +HDE 303 308 A,B 10:45:05.919 −59:40:05.92 +287.59−00.61 01 +— +5 350 358 683 250 920 704 1869 A,B +8.0909 +0.2861 0.447±0.048 +O-025 +HD 93 249 A +10:44:43.875 −59:21:25.15 +287.41−00.36 01 +u10444388−5921252 5 350 395 383 778 733 568 1857 +8.3192 +0.2393 0.403±0.039 +O-027 +HD 93 028 +10:43:15.340 −60:12:04.21 +287.64−01.19 01 +— +5 254 262 161 604 257 408 1803 +8.3596 +−0.0956 0.411±0.059 +O-028 +HD 93 146 A +10:44:00.158 −60:05:09.86 +287.67−01.05 01 +— +5 254 269 617 668 289 664 1826 +8.3766 +0.1140 0.344±0.076 +O-028 +HD 93 204 +10:44:32.336 −59:44:31.00 +287.57−00.71 01 +u10443234−5944310 5 350 357 205 782 177 664 1847 +8.3869 +0.2200 0.431±0.049 +O-003 +HD 93 190 +10:44:19.615 −59:16:58.81 +287.33−00.32 01 +— +5 350 396 620 729 366 656 1838 +8.3965 +0.6452 0.419±0.039 +O-027 +HDE 305 523 +10:44:29.479 −59:57:18.11 +287.66−00.90 01 +— +5 350 347 520 660 979 840 1845 +8.4190 +0.3661 0.417±0.046 +O-028 +CPD −59 2600 +10:44:41.795 −59:46:56.42 +287.60−00.74 01 +u10444177−5946563 5 350 356 419 833 915 904 1856 +8.4820 +0.4480 0.398±0.048 +O-028 +HD 93 161 B +10:44:09.080 −59:34:35.30 +287.44−00.59 03 +u10440921−5934353 5 350 362 982 543 828 352 1832 B +8.4865 +0.4579 0.403±0.074 +O-002 +HD 93 161 A +10:44:08.840 −59:34:34.49 +287.44−00.59 02 +u10440884−5934349 5 350 362 982 543 827 456 1832 A +8.4979 +0.5124 0.348±0.059 +O-002 +HD 93 129 Ab +10:43:57.463 −59:32:51.23 +287.41−00.57 03 +— +— +1820 B +8.6∗ +0.5∗ +— +O-002 +HDE 305 520 +10:44:05.860 −59:59:41.54 +287.64−00.96 01 +— +5 350 347 108 343 919 104 1830 +8.6130 +0.4070 0.424±0.041 +O-028 +HD 93 128 +10:43:54.372 −59:32:57.37 +287.40−00.58 01 +— +5 350 363 807 162 637 696 1819 +8.6455 +0.4752 0.434±0.032 +O-002 +HD 93 027 +10:43:17.953 −60:08:03.29 +287.61−01.13 01 +u10431795−6008033 5 254 268 518 156 437 888 1804 +8.7047 +0.0084 0.370±0.049 +O-028 +V572 Car +10:44:47.307 −59:43:53.23 +287.59−00.69 01 +u10444730−5943532 5 350 358 069 101 053 184 1861 +8.7108 +0.2978 0.379±0.043 +O-025 +HD 93 129 B +10:43:57.638 −59:32:53.50 +287.41−00.57 02 +— +5 350 363 910 256 783 744 19 309 +8.7126 +0.4321 0.416±0.034 +O-002 +HD 92 877 A +10:42:07.635 −59:54:24.47 +287.38−01.00 01 +— +5 254 287 175 496 161 024 18 203 A +8.7442 +−0.0488 0.493±0.045 +O-028 +HDE 305 438 +10:42:43.771 −59:54:16.47 +287.44−00.96 01 +— +5 254 275 630 622 585 344 1791 +8.7894 +−0.0587 0.416±0.051 +O-028 +CPD −59 2554 +10:44:00.433 −60:05:59.96 +287.67−01.06 01 +u10440043−6005599 5 254 269 514 589 043 200 15 960 +8.8206 +0.0306 0.390±0.045 +O-028 +HD 93 342 +10:45:17.570 −59:23:37.49 +287.49−00.36 01 +— +5 350 392 012 197 117 568 1875 +8.8300 +1.0955 0.288±0.029 background +HDE 305 536 +10:44:11.078 −60:03:21.58 +287.67−01.01 01 +— +5 254 269 961 265 754 368 1834 +8.9207 +0.1134 0.445±0.041 +O-028 +HDE 303 311 +10:44:37.462 −59:32:55.44 +287.48−00.54 01 +— +5 350 383 529 668 697 472 1851 +8.9295 +0.2375 0.447±0.029 +O-029 +HD 93 056 +10:43:27.401 −60:05:54.77 +287.61−01.09 01 +u10432740−6005548 5 254 269 067 912 338 048 1806 +8.9843 +0.0127 0.406±0.038 +O-028 +HD 93 501 +10:46:22.033 −60:01:18.93 +287.90−00.85 01 +— +5 350 303 024 796 002 304 15 965 +9.0400 +0.2339 0.543±0.032 +foreground +HDE 305 437 +10:42:45.176 −59:52:19.59 +287.43−00.93 01 +— +5 350 352 567 217 507 840 1792 +9.0531 +0.0047 0.414±0.046 +O-028 +CPD −59 2641 +10:45:16.517 −59:43:36.98 +287.64−00.65 01 +— +5 350 311 339 852 481 792 1874 +9.0761 +0.6220 0.457±0.029 +O-025 +HD 93 620 +10:47:09.194 −59:47:29.83 +287.88−00.60 01 +— +5 350 329 791 034 238 464 1905 +9.0978 +0.2233 0.418±0.040 +O-030 +CPD −59 2635 +10:45:12.715 −59:44:46.17 +287.64−00.68 02 +u10451271−5944460 5 350 310 927 535 609 728 1872 +9.1320 +0.5190 0.530±0.036 +O-025 +CPD −59 2592 +10:44:36.788 −59:54:24.77 +287.65−00.85 01 +— +5 350 349 169 928 548 352 1853 +9.1367 +1.2724 0.220±0.027 background +HDE 305 524 +10:44:45.238 −59:54:41.55 +287.67−00.85 01 +u10444523−5954416 5 350 349 101 209 092 224 1860 +9.1951 +0.4566 0.420±0.029 +O-028 +CPD −58 2620 +10:43:59.917 −59:32:25.36 +287.41−00.56 01 +— +5 350 363 944 616 553 216 1823 +9.2309 +0.3401 0.376±0.040 +O-002 +CPD −59 2551 +10:43:57.488 −60:05:28.15 +287.67−01.05 02 +— +5 254 269 613 320 423 040 1821 +9.2862 +0.0483 0.418±0.034 +O-028 +HDE 305 439 A +10:42:10.330 −59:58:00.95 +287.41−01.05 01 +g10421033−5958009 5 254 285 251 350 707 072 1780 +9.3724 +0.7778 0.229±0.024 background +HDE 303 299 +10:43:01.547 −59:20:23.77 +287.21−00.45 01 +— +5 350 401 534 171 661 696 1797 +9.3789 +0.3566 0.462±0.027 +Car OB1 +HD 93 249 B +10:44:43.754 −59:21:17.30 +287.41−00.36 02 +u10444375−5921173 5 350 395 379 453 207 296 15 853 +9.3847 +0.2613 0.432±0.025 +O-027 +HDE 305 535 +10:42:54.642 −59:58:19.71 +287.49−01.01 01 +u10425464−5958197 5 254 274 840 348 522 240 — +9.3990 +0.0683 0.465±0.031 +O-028 +HDE 305 452 +10:42:02.304 −60:08:38.62 +287.48−01.21 01 +u10420230−6008386 5 254 266 250 413 019 392 1776 +9.4296 +0.1269 0.435±0.027 +Car OB1 +HD 93 343 +10:45:12.217 −59:45:00.42 +287.64−00.68 01 +— +5 350 310 824 456 389 504 16 717 +9.4518 +0.5035 0.460±0.028 +O-025 +CPD −58 2611 +10:43:46.695 −59:32:54.82 +287.39−00.59 01 +— +5 350 363 875 896 996 480 1814 +9.4805 +0.5629 0.441±0.024 +O-002 +V573 Car +10:45:08.226 −59:40:49.48 +287.60−00.62 01 +u10450823−5940495 5 350 358 481 418 098 944 1871 +9.4925 +0.2370 0.459±0.035 +O-025 +∗ Estimated from Ma´ız Apell´aniz et al. (2017) and the Gaia DR3 results for Aa. The photometry listed for Aa is actually the combined one for Aa,Ab, G′ +3 for Aa should be ∼7.6. +17 + +Berlanas et al.: Massive stars in the Carina Nebula +Table A.1. (Continued). +Name +RA +dec +GOS/GBS/GWS ID +GES ID +Gaia ID +ALS ID +G′ +3 +GBP3 − GRP3 +ϖc(mas) +group +CPD −59 2636 A,B 10:45:12.870 −59:44:19.24 +287.64−00.67 01 +u10451288−5944192 5 350 357 862 947 802 880 15 194 A,B +9.5201 +0.6791 +— +O-025 +HDE 303 316 A +10:43:11.178 −59:44:21.02 +287.41−00.79 01 +— +5 350 355 251 602 278 528 1800 +9.5280 +0.5305 0.439±0.025 +O-028 +HDE 305 518 +10:43:44.006 −59:48:17.96 +287.51−00.81 01 +— +5 350 353 567 958 779 392 1810 +9.5862 +0.6167 0.503±0.033 +O-028 +HDE 305 534 +10:44:47.517 −59:57:58.88 +287.70−00.89 01 +— +5 350 301 994 003 543 936 1862 +9.6264 +0.2318 0.446±0.028 +O-028 +CPD −59 2624 +10:45:05.828 −59:43:07.57 +287.62−00.66 01 +g10450584−5943077 5 350 357 966 022 158 720 15 197 +9.6423 +0.4992 0.445±0.149 +O-025 +HDE 305 522 +10:44:14.961 −60:00:05.71 +287.66−00.96 01 +g10441496−6000057 5 350 347 142 704 520 576 1836 +9.6518 +0.1651 0.414±0.030 +O-028 +HDE 305 543 +10:43:10.072 −60:02:11.78 +287.55−01.05 01 +— +5 254 273 672 117 339 520 1799 +9.6785 +0.0534 0.410±0.028 +O-028 +HDE 303 300 +10:45:20.453 −59:17:06.16 +287.44−00.26 01 +g10452045−5917062 5 350 394 490 425 712 128 1877 +9.7012 +0.5080 0.378±0.031 +Car OB1 +HD 93 097 +10:43:46.989 −60:05:49.24 +287.65−01.07 01 +u10434698−6005493 5 254 270 304 862 985 728 1815 +9.7668 +0.0101 0.408±0.032 +O-028 +HDE 305 525 +10:46:05.704 −59:50:49.45 +287.79−00.71 01 +— +5 350 306 426 409 875 200 1886 +9.7840 +1.1869 0.035±0.225 +O-030 +HDE 305 521 +10:43:49.397 −59:57:22.67 +287.59−00.94 01 +— +5 350 348 242 215 309 440 1816 +9.8040 +0.1888 0.453±0.043 +O-028 +CPD −59 2574 +10:44:26.474 −59:41:02.86 +287.53−00.67 01 +— +5 350 359 099 893 220 992 15 198 +9.8124 +0.1943 0.462±0.025 +Car OB1 +HDE 305 516 +10:43:15.767 −59:51:05.57 +287.48−00.88 01 +u10431575−5951055 5 350 352 812 060 632 192 1802 +9.8265 +0.1084 0.410±0.024 +O-028 +CPD −59 2626 A,B 10:45:05.794 −59:45:19.60 +287.63−00.69 01 +g10450579−5945196 5 350 357 725 503 681 664 1870 A,B +9.8675 +0.8015 0.415±0.073 +O-025 +HDE 303 312 +10:43:30.842 −59:29:23.80 +287.33−00.55 01 +g10433085−5929239 5 350 376 004 884 719 232 1807 +9.8680 +0.5499 0.475±0.022 +Car OB1 +CPD −58 2605 +10:43:33.353 −59:35:11.19 +287.38−00.63 01 +g10433335−5935111 5 350 363 497 939 741 056 1808 +9.8854 +0.8669 0.323±0.108 +O-002 +ALS 15 196 +10:43:55.354 −59:32:48.61 +287.41−00.58 01 +— +5 350 363 910 241 888 768 15 196 +9.9065 +0.3814 0.452±0.021 +O-002 +HD 93 146 B +10:43:59.454 −60:05:13.33 +287.67−01.05 03 +— +5 254 269 617 668 284 416 15 959 +9.9135 +0.0998 0.415±0.026 +O-028 +CPD −59 2644 +10:45:20.573 −59:42:51.25 +287.64−00.64 01 +g10452057−5942513 5 350 311 374 212 228 736 1878 +9.9453 +0.3738 0.444±0.023 +O-025 +HDE 305 532 +10:45:34.066 −59:57:26.66 +287.78−00.84 01 +— +5 350 301 306 809 073 280 1880 +9.9812 +0.7215 0.417±0.021 +O-030 +CPD −58 2656 +10:44:42.342 −59:23:03.80 +287.42−00.39 01 +g10444234−5923038 5 350 389 267 745 250 944 1855 +9.9900 +0.3154 0.475±0.067 +O-027 +CPD −58 2623 +10:44:00.624 −59:25:49.25 +287.36−00.47 01 +g10440062−5925493 5 350 388 374 391 885 312 1822 +9.9967 +0.2815 0.461±0.023 +Car OB1 +CPD −59 2610 +10:44:54.714 −59:56:01.91 +287.70−00.86 01 +— +5 350 302 097 082 779 136 1865 +10.0030 +0.3709 0.432±0.027 +O-028 +CPD −59 2627 +10:45:06.721 −59:41:56.58 +287.61−00.64 01 +g10450673−5941565 5 350 358 343 979 094 912 15 200 +10.0272 +0.4217 0.406±0.020 +O-025 +CPD −58 2649 A +10:44:30.366 −59:37:26.44 +287.51−00.61 01 +g10443037−5937267 5 350 359 752 713 348 480 1844 A +10.0378 +0.5842 0.411±0.055 +Car OB1 +CPD −58 2627 +10:44:02.445 −59:29:36.77 +287.39−00.52 01 +g10440248−5929368 5 350 387 549 760 231 808 1827 +10.0786 +0.4051 0.420±0.031 +Car OB1 +CPD −59 2595 +10:44:38.647 −59:48:14.12 +287.61−00.76 01 +— +5 350 356 007 516 638 720 15 201 +10.1274 +0.1342 0.484±0.026 +O-028 +CPD −59 2673 +10:46:22.462 −59:53:20.45 +287.84−00.73 01 +g10462246−5953205 5 350 306 082 812 792 576 1892 +10.1319 +0.9613 0.379±0.023 +O-030 +ALS 15 210 +10:44:13.199 −59:43:10.33 +287.52−00.71 01 +g10441320−5943103 5 350 357 519 345 176 192 15 210 +10.1517 +1.5978 0.413±0.020 +O-003 +HDE 303 313 +10:42:50.183 −59:25:31.11 +287.23−00.53 01 +g10425018−5925311 5 350 377 447 993 712 896 1795 +10.2378 +0.2513 0.444±0.020 +Car OB1 +CPD −59 2593 +10:44:36.871 −60:01:11.71 +287.70−00.95 01 +g10443687−6001116 5 350 299 966 778 959 616 15 957 +10.2430 +0.2354 0.412±0.025 +O-028 +HDE 305 528 +10:45:16.722 −59:54:45.78 +287.73−00.82 01 +— +5 350 302 371 960 706 432 18 776 +10.2610 +0.2184 0.415±0.023 +O-028 +HDE 305 439 B +10:42:10.269 −59:57:57.74 +287.41−01.05 02 +— +5 254 285 457 509 138 944 16 053 +10.2910 +0.7888 0.213±0.020 background +CPD −59 2537 +10:43:45.076 −59:53:25.39 +287.55−00.89 01 +— +5 350 348 860 690 746 240 1813 +10.3537 +0.2046 0.365±0.060 +O-028 +ALS 15 205 +10:44:56.288 −59:33:03.54 +287.52−00.52 01 +g10445629−5933035 5 350 383 667 107 695 744 15 205 +10.4065 +0.4306 0.463±0.021 +O-029 +ALS 15 961 +10:44:00.131 −60:06:07.49 +287.68−01.06 01 +— +5 254 269 514 589 038 080 15 961 +10.4125 +0.0506 0.388±0.025 +O-028 +CPD −59 2469 +10:41:54.279 −59:55:45.01 +287.36−01.03 01 +g10415472−5955589 5 254 286 385 222 091 136 — +10.4319 +−0.0078 0.383±0.025 +O-028 +HDE 305 533 +10:45:13.383 −59:57:53.83 +287.75−00.87 01 +g10451338−5957538 5 350 301 061 972 444 416 1873 +10.4737 +0.4609 0.400±0.024 +O-028 +ALS 15 860 +10:44:36.359 −59:24:20.34 +287.42−00.41 01 +g10443636−5924203 5 350 389 336 464 678 784 15 860 +10.5021 +1.8303 0.306±0.019 background +CPD −58 2625 +10:44:00.927 −59:35:45.80 +287.44−00.61 01 +g10440093−5935459 5 350 362 638 946 360 960 15 206 +10.5087 +0.7150 0.401±0.026 +O-002 +HD 93 249 C +10:44:44.482 −59:21:32.80 +287.41−00.36 03 +g10444448−5921327 5 350 395 383 778 731 648 15 854 +10.5584 +0.3242 0.497±0.103 +O-027 +HDE 305 515 A +10:43:03.963 −59:51:39.08 +287.46−00.90 01 +— +5 350 352 468 463 172 480 18 774 A +10.5707 +0.0388 0.451±0.026 +O-028 +ALS 15 207 +10:43:48.707 −59:33:24.10 +287.40−00.59 01 +— +5 350 363 841 537 246 592 15 207 +10.5901 +0.6416 0.443±0.020 +O-002 +ALS 15 865 +10:45:04.762 −59:40:53.56 +287.60−00.63 04 +— +5 350 358 584 502 466 560 15 865 +10.5923 +0.2986 0.384±0.045 +O-025 +CPD −58 2634 +10:44:16.776 −59:20:09.63 +287.35−00.37 01 +g10441678−5920096 5 350 390 367 256 883 328 — +10.6184 +0.2344 0.538±0.021 +foreground +HD 93 128 B +10:43:53.629 −59:33:00.69 +287.40−00.58 02 +g10435366−5933006 5 350 363 875 897 024 256 15 862 +10.6209 +0.5218 0.418±0.018 +O-002 +CPD −59 2629 +10:45:08.225 −59:46:06.96 +287.64−00.70 01 +g10450823−5946070 5 350 357 725 503 668 096 15 218 +10.6244 +0.8850 0.440±0.021 +O-025 +CPD −58 2657 +10:44:42.763 −59:21:51.06 +287.41−00.37 02 +g10444276−5921511 5 350 395 383 780 746 496 15 857 +10.6261 +0.5021 1.273±0.737 +O-027 +CPD −59 2591 +10:44:36.688 −59:47:29.63 +287.60−00.75 01 +g10443669−5947296 5 350 356 007 521 809 664 15 217 +10.6371 +0.8688 0.422±0.024 +O-028 +CPD −58 2644 +10:44:29.116 −59:20:04.88 +287.37−00.35 01 +g10442912−5920049 5 350 396 208 412 448 000 1842 +10.6508 +0.2679 0.436±0.173 +O-027 +CPD −59 2598 +10:44:40.315 −59:41:48.96 +287.56−00.66 01 +— +5 350 358 859 375 074 816 1854 +10.6518 +0.5085 0.467±0.027 +O-025 +CPD −59 2570 +10:44:15.868 −60:09:04.44 +287.73−01.09 01 +— +5 254 222 269 942 017 152 1837 +10.6557 +0.0812 0.425±0.023 +O-028 +CPD −59 2622 +10:45:03.158 −59:40:12.52 +287.59−00.62 01 +— +5 350 358 687 576 533 120 15 214 +10.6851 +0.2917 0.440±0.025 +O-025 +CPD −59 2535 +10:43:40.817 −60:10:03.15 +287.67−01.14 01 +— +5 254 267 521 724 038 144 — +10.6883 +0.2638 0.321±0.022 background +18 + +Berlanas et al.: Massive stars in the Carina Nebula +Table A.1. (Continued). +Name +RA +dec +GOS/GBS/GWS ID +GES ID +Gaia ID +ALS ID +G′ +3 +GBP3 − GRP3 +ϖc(mas) +group +CPD −59 2632 +10:45:11.197 −59:41:11.29 +287.61−00.62 01 +g10451120−5941113 5 350 358 481 418 096 896 15 209 +10.7331 +0.2975 0.416±0.024 +O-025 +CPD −59 2495 +10:42:36.164 −59:59:26.20 +287.47−01.04 01 +g10423616−5959262 5 254 273 500 318 621 184 15 956 +10.7438 +0.0837 0.426±0.026 +O-028 +ALS 15 204 +10:43:41.237 −59:35:48.18 +287.40−00.63 02 +— +5 350 363 326 141 056 768 15 204 +10.7551 +1.0114 0.434±0.024 +O-002 +CPD −58 2655 +10:44:42.119 −59:22:30.53 +287.41−00.38 02 +g10444212−5922305 5 350 389 508 263 432 320 — +10.7725 +0.3943 0.434±0.022 +O-027 +CPD −59 2614 +10:44:57.331 −60:00:46.79 +287.74−00.93 01 +g10445734−6000467 5 350 300 207 297 141 120 1866 +10.7763 +0.4319 0.471±0.085 +O-028 +ALS 15 219 +10:43:59.863 −59:35:24.40 +287.43−00.61 01 +g10435986−5935244 5 350 362 638 946 370 688 15 219 +10.7772 +0.8030 0.426±0.026 +O-002 +CPD −59 2581 +10:44:30.480 −59:41:40.41 +287.54−00.67 01 +g10443049−5941406 5 350 358 893 734 785 664 15 215 +10.7796 +0.4012 0.462±0.021 +Car OB1 +[ARV2008] 206 +10:45:22.276 −59:50:47.07 +287.71−00.75 01 +g10452228−5950471 5 350 308 865 923 993 600 — +10.7818 +1.4876 0.395±0.022 +Car OB1 +CPD −59 2583 +10:44:32.901 −59:40:26.12 +287.53−00.65 01 +g10443290−5940261 5 350 359 237 337 379 456 15 211 +10.8035 +0.3430 0.410±0.037 +Car OB1 +CPD −58 2640 +10:44:24.622 −59:30:35.88 +287.44−00.51 01 +g10442462−5930359 5 350 386 931 282 820 736 1840 +10.8063 +0.3732 0.432±0.022 +O-029 +CPD −59 2606 +10:44:54.078 −59:41:29.36 +287.58−00.65 01 +g10445408−5941294 5 350 358 240 899 842 816 15 216 +10.8226 +0.4345 0.438±0.024 +O-025 +Tyc 8626-02506-1 +10:44:30.218 −59:26:12.97 +287.42−00.44 01 +g10443022−5926130 5 350 388 821 068 542 848 — +10.8724 +0.9641 0.420±0.020 +O-027 +CPD −59 2497 +10:42:39.575 −59:51:38.53 +287.41−00.93 01 +g10423957−5951386 5 350 364 322 572 800 768 — +10.8980 +0.1132 0.433±0.025 +O-028 +ALS 15 229 +10:43:55.220 −59:33:14.72 +287.41−00.58 02 +g10435522−5933147 5 350 363 807 177 545 216 15 229 +10.9253 +0.5420 0.388±0.022 +O-002 +CPD −59 2605 +10:44:50.412 −59:55:45.03 +287.69−00.86 01 +g10445041−5955450 5 350 302 165 802 254 592 1864 +10.9255 +0.4419 0.444±0.026 +O-028 +CPD −59 2640 +10:45:16.560 −59:39:57.10 +287.61−00.60 01 +g10451656−5939571 5 350 381 914 760 913 024 15 223 +10.9383 +0.2694 0.435±0.028 +O-025 +ALS 15 228 +10:45:12.652 −59:42:48.79 +287.63−00.65 01 +g10451265−5942488 5 350 358 275 259 605 888 15 228 +10.9387 +0.7716 0.432±0.024 +O-025 +Gaia DR3 5350358378338864768 10:45:05.139 −59:40:57.25 +287.60−00.63 02 +g10450520−5940574 5 350 358 378 338 864 768 — +10.9440 +0.3175 0.455±0.031 +O-025 +ALS 15 855 +10:44:40.582 −59:21:13.76 +287.40−00.36 01 +g10444058−5921138 5 350 395 383 778 729 472 15 855 +10.9449 +0.3293 0.368±0.025 +O-027 +CPD −59 2504 +10:42:46.164 −60:00:57.62 +287.50−01.06 01 +g10424616−6000576 5 254 273 191 085 477 888 — +10.9542 +0.2761 0.381±0.049 +O-028 +CPD −59 2579 +10:44:30.080 −59:52:14.14 +287.62−00.83 01 +g10443008−5952141 5 350 349 754 044 153 216 15 222 +10.9558 +0.3769 0.502±0.151 +O-028 +CPD −59 2543 +10:43:48.871 −60:09:00.92 +287.68−01.11 01 +g10434887−6009009 5 254 267 865 321 470 208 18 775 +10.9704 +−0.0353 0.378±0.032 +O-028 +ALS 15 227 +10:44:05.830 −59:35:11.70 +287.44−00.60 01 +g10440583−5935117 5 350 362 948 184 049 280 15 227 +10.9838 +0.4994 0.469±0.035 +O-002 +CPD −58 2647 +10:44:30.750 −59:21:26.31 +287.38−00.37 01 +g10443075−5921263 5 350 389 576 983 239 424 15 861 +11.0133 +0.4067 0.441±0.025 +O-027 +ALS 15 224 +10:43:57.564 −59:33:38.53 +287.42−00.59 02 +g10435756−5933385 5 350 363 807 162 568 704 15 224 +11.0165 +0.4218 0.450±0.026 +O-002 +CPD −59 2510 +10:42:57.179 −60:07:41.49 +287.57−01.15 01 +g10425717−6007414 5 254 271 473 093 882 752 — +11.0294 +0.0816 0.398±0.027 +O-028 +CPD −59 2619 +10:45:02.156 −59:42:01.03 +287.60−00.64 02 +g10450216−5942010 5 350 358 550 137 500 672 15 220 +11.0762 +0.4514 0.446±0.029 +O-025 +CPD −59 2596 +10:44:40.978 −59:40:10.39 +287.55−00.64 01 +g10444098−5940104 5 350 359 065 533 563 136 19 743 +11.0934 +0.7025 0.452±0.027 +O-025 +ALS 15 234 +10:43:43.891 −59:33:46.15 +287.39−00.60 01 +— +5 350 363 463 580 089 856 15 234 +11.0958 +0.5879 0.459±0.025 +O-002 +2MASS J10424532−6012063 +10:42:45.322 −60:12:06.33 +287.59−01.22 02 +g10424533−6012063 5 254 265 013 462 466 944 — +11.1095 +0.0455 0.423±0.031 +Car OB1 +2MASS J10424476−6005020 +10:42:44.766 −60:05:02.10 +287.53−01.12 01 +g10424477−6005021 5 254 271 782 331 569 664 — +11.1174 +1.1337 0.253±0.022 background +CPD −58 2653 A +10:44:40.676 −59:22:28.58 +287.41−00.38 01 +g10444065−5922285 5 350 389 503 936 214 912 15 859 A 11.1249 +0.4786 0.413±0.083 +O-027 +CPD −59 2571 +10:44:22.515 −59:39:25.81 +287.51−00.65 01 +g10442251−5939258 5 350 359 306 051 691 136 15 230 +11.1263 +0.2942 0.439±0.027 +Car OB1 +ALS 15 863 +10:43:53.636 −59:33:28.45 +287.41−00.59 01 +— +5 350 363 807 177 527 680 15 863 +11.1408 +0.6491 0.360±0.028 +O-002 +2MASS J10415981−5955075 +10:41:59.816 −59:55:07.50 +287.37−01.02 01 +g10415981−5955075 5 254 286 419 581 867 520 — +11.1440 +0.0249 0.417±0.039 +O-028 +2MASS J10461906−5957543 +10:46:19.062 −59:57:54.32 +287.87−00.80 01 +g10461906−5957543 5 350 304 021 228 445 824 — +11.1478 +0.8827 0.428±0.028 +O-030 +CPD −59 2661 +10:45:53.709 −59:57:03.85 +287.81−00.82 01 +g10455371−5957038 5 350 304 227 386 859 776 1883 +11.1558 +0.6849 0.432±0.027 +O-030 +ALS 15 232 +10:44:25.184 −59:28:16.01 +287.43−00.48 01 +g10442518−5928160 5 350 387 274 880 260 736 15 232 +11.1629 +0.4284 0.441±0.027 +O-029 +ALS 15 233 +10:44:08.280 −59:29:59.38 +287.41−00.52 01 +g10440828−5929594 5 350 387 549 758 083 840 15 233 +11.1969 +0.4927 0.460±0.029 +Car OB1 +2MASS J10431897−5946569 +10:43:18.977 −59:46:56.92 +287.45−00.82 01 +g10431898−5946569 5 350 354 628 805 549 824 — +11.1976 +0.1876 0.459±0.035 +O-028 +ALS 19 739 +10:43:59.524 −59:32:31.61 +287.41−00.57 09 +g10435952−5932316 5 350 363 944 616 548 608 19 739 +11.2540 +0.6952 0.427±0.030 +O-002 +ALS 15 235 +10:44:25.491 −59:33:09.25 +287.46−00.55 01 +g10442549−5933093 5 350 386 415 886 679 424 15 235 +11.2560 +0.3918 0.453±0.028 +O-029 +CPD −59 2569 B +10:44:15.162 −60:07:51.11 +287.72−01.07 02 +g10441513−6007509 5 254 269 239 711 125 376 15 955 +11.2643 +0.3218 0.417±0.031 +O-028 +2MASS J10441791−5925204 +10:44:17.916 −59:25:20.42 +287.39−00.44 01 +g10441792−5925204 5 350 389 645 702 254 080 — +11.2698 +0.5511 +negative background +CPD −58 2667 +10:44:53.761 −59:37:48.32 +287.55−00.59 01 +g10445376−5937483 5 350 359 546 570 052 096 15 236 +11.2707 +0.7409 0.452±0.028 +O-025 +ALS 15 856 +10:44:41.780 −59:21:32.97 +287.40−00.36 02 +g10444178−5921330 5 350 395 383 778 723 200 15 856 +11.3168 +0.4167 0.426±0.031 +O-027 +2MASS J10460477−5949217 +10:46:04.779 −59:49:21.79 +287.77−00.69 01 +g10460478−5949218 5 350 309 518 786 341 888 — +11.3292 +1.1054 0.453±0.029 +O-030 +ALS 15 237 +10:45:06.160 −59:31:23.06 +287.53−00.48 01 +g10450616−5931231 5 350 385 213 295 984 640 15 237 +11.3499 +0.5550 0.422±0.028 +O-029 +CPD −59 2617 +10:44:58.788 −59:49:21.05 +287.65−00.76 01 +g10445887−5949210 5 350 355 698 279 011 200 — +11.3593 +0.2541 0.440±0.033 +O-028 +CPD −58 2648 +10:44:29.422 −59:38:38.10 +287.51−00.63 01 +— +5 350 359 267 370 892 544 15 240 +11.3633 +0.5307 0.390±0.029 +Car OB1 +2MASS J10460606−5956339 +10:46:06.068 −59:56:33.94 +287.83−00.80 01 +g10460607−5956339 5 350 304 296 106 353 152 — +11.3664 +0.5043 0.437±0.031 +O-030 +2MASS J10450531−5919347 +10:45:05.320 −59:19:34.80 +287.43−00.31 01 +g10450531−5919349 5 350 395 693 016 478 592 — +11.3828 +0.2700 0.414±0.033 +O-027 +CPD −59 2585 +10:44:31.742 −60:05:44.82 +287.73−01.02 01 +g10443174−6005449 5 254 222 785 298 659 840 16 054 +11.3910 +0.1757 0.468±0.034 +O-028 +[HSB2012] 1443 +10:43:53.841 −59:32:47.36 +287.40−00.58 04 +— +5 350 363 978 961 349 120 — +11.4019 +0.4616 0.431±0.038 +O-002 +19 + +Berlanas et al.: Massive stars in the Carina Nebula +Table A.1. (Continued). +Name +RA +dec +GOS/GBS/GWS ID +GES ID +Gaia ID +ALS ID +G′ +3 +GBP3 − GRP3 +ϖc(mas) +group +CPD −59 2625 +10:45:05.877 −59:44:18.88 +287.63−00.68 01 +g10450588−5944189 5 350 357 755 541 882 880 15 238 +11.4258 +0.3799 0.407±0.033 +O-025 +ALS 15 245 +10:45:14.985 −59:43:23.28 +287.64−00.65 02 +g10451499−5943233 5 350 358 275 259 591 296 15 245 +11.4535 +0.6717 0.432±0.028 +O-025 +CPD −58 2677 +10:45:22.137 −59:37:38.53 +287.60−00.56 01 +g10452214−5937385 5 350 382 086 559 684 480 15 244 +11.4716 +0.5113 0.453±0.035 +O-025 +CPD −59 2541 +10:43:48.824 −60:00:36.71 +287.61−00.99 01 +g10434882−6000366 5 254 271 056 429 426 688 15 962 +11.4791 +0.4154 0.317±0.079 +O-028 +ALS 15 246 +10:45:09.744 −59:42:57.21 +287.62−00.65 01 +g10450974−5942572 5 350 357 897 302 465 536 15 246 +11.4902 +0.5053 0.409±0.033 +O-025 +CPD −59 2616 +10:45:00.234 −59:43:34.53 +287.61−00.67 01 +g10450023−5943345 5 350 357 931 646 616 448 19 745 +11.4969 +0.5213 0.454±0.030 +O-025 +ALS 15 242 +10:44:37.185 −59:40:01.49 +287.54−00.64 01 +g10443719−5940015 5 350 359 065 533 555 712 15 242 +11.5250 +0.6599 0.439±0.028 +O-025 +CPD −58 2662 +10:44:46.522 −59:21:53.86 +287.42−00.36 01 +g10444652−5921538 5 350 395 177 620 297 472 — +11.5257 +0.4114 0.427±0.072 +O-027 +ALS 19 738 +10:43:58.464 −59:33:01.56 +287.41−00.58 04 +— +5 350 363 905 936 139 904 19 738 +11.5773 +0.6335 0.412±0.031 +O-002 +2MASS J10444803−5954297 10:44:48.038 −59:54:29.71 +287.67−00.84 01 +g10444804−5954297 5 350 349 479 166 230 144 — +11.5782 +1.6920 0.222±0.031 background +ALS 15 247 +10:45:16.703 −59:43:14.09 +287.64−00.65 03 +g10451670−5943141 5 350 311 335 535 366 528 15 247 +11.5786 +0.6600 0.486±0.028 +O-025 +2MASS J10432014−5917582 10:43:20.149 −59:17:58.25 +287.22−00.39 01 +g10432015−5917582 5 350 403 007 313 388 672 — +11.5824 +0.3389 0.428±0.028 +Car OB1 +2MASS J10441242−5934091 10:44:12.430 −59:34:09.13 +287.45−00.58 01 +g10441243−5934091 5 350 363 085 623 078 272 — +11.5851 +0.5778 +negative background +2MASS J10435413−5918244 10:43:54.138 −59:18:24.43 +287.29−00.36 01 +g10435416−5918244 5 350 402 461 886 825 088 — +11.5885 +0.2229 0.453±0.043 +Car OB1 +ALS 19 741 +10:44:14.747 −59:42:51.79 +287.52−00.70 01 +— +5 350 357 553 704 923 520 19 741 +11.6226 +0.9663 0.433±0.026 +O-003 +ALS 15 243 +10:44:13.804 −59:42:57.12 +287.52−00.71 02 +— +5 350 357 549 383 373 440 15 243 +11.6330 +1.1806 0.393±0.030 +O-003 +QZ Car B +10:44:21.973 −59:59:35.20 +287.66−00.94 01 +g10442198−5959351 5 350 346 966 582 819 584 1839 B +11.6511 +0.2851 0.476±0.028 +O-028 +ALS 15 249 +10:45:06.362 −59:42:35.68 +287.61−00.65 01 +g10450636−5942357 5 350 357 961 696 418 048 15 249 +11.6562 +0.7131 0.458±0.028 +O-025 +CPD −59 2539 +10:43:46.794 −60:08:26.42 +287.67−01.11 01 +g10434679−6008264 5 254 267 934 040 957 056 — +11.6569 +0.3507 0.317±0.024 background +ALS 15 248 +10:45:22.586 −59:42:36.75 +287.64−00.63 01 +g10452258−5942368 5 350 311 374 212 232 960 15 248 +11.6647 +0.5182 0.439±0.027 +O-025 +2MASS J10440327−5919498 10:44:03.273 −59:19:49.81 +287.32−00.38 01 +g10440328−5919498 5 350 390 534 728 362 496 — +11.6707 +0.3067 0.424±0.027 +O-027 +[HSB2012] 3192 +10:44:57.890 −59:41:02.91 +287.59−00.63 01 +g10445796−5941031 5 350 358 550 137 930 624 — +11.6865 +0.3848 0.444±0.028 +O-025 +CPD −59 2618 +10:44:59.908 −59:43:14.90 +287.61−00.67 02 +g10445991−5943149 5 350 357 931 662 159 104 19 744 +11.6907 +0.5313 0.454±0.027 +O-025 +ALS 15 225 +10:44:00.988 −59:52:38.80 +287.57−00.86 02 +g10440099−5952388 5 350 350 303 799 849 088 15 225 +11.7138 +0.3903 0.455±0.031 +O-028 +V662 Car +10:45:36.318 −59:48:23.37 +287.71−00.71 01 +g10453632−5948234 5 350 310 480 858 998 656 16 081 +11.7142 +1.6326 0.429±0.025 +Car OB1 +HDE 303 316 B +10:43:11.925 −59:44:30.42 +287.42−00.79 01 +— +5 350 354 873 645 153 664 15 231 +11.7547 +0.7124 0.445±0.025 +O-028 +ALS 15 958 +10:44:14.439 −60:01:27.05 +287.66−00.98 01 +g10441444−6001270 5 350 347 005 264 666 368 15 958 +11.7724 +0.2725 0.421±0.026 +O-028 +ALS 15 864 +10:43:57.956 −59:33:53.67 +287.42−00.59 01 +g10435796−5933537 5 350 363 807 177 529 728 15 864 +11.7808 +0.5819 0.242±0.155 +O-002 +CPD −59 2638 A +10:45:13.402 −60:00:58.87 +287.77−00.91 01 +g10451341−6000589 5 350 300 894 465 285 376 — +11.7950 +0.6051 0.415±0.033 +O-028 +ALS 15 203 A +10:43:41.259 −59:35:52.55 +287.40−00.63 01 +g10434124−5935530 5 350 363 326 141 053 312 15 203 A 11.7978 +1.1068 0.452±0.086 +O-002 +ALS 19 733 +10:43:50.904 −59:33:50.56 +287.41−00.59 02 +g10435090−5933506 5 350 363 841 537 240 064 19 733 +11.8077 +0.7121 0.439±0.027 +O-002 +2MASS J10440973−6000432 10:44:09.732 −60:00:43.24 +287.65−00.97 02 +g10440974−6000432 5 350 347 000 942 101 120 — +11.8184 +0.6340 0.369±0.024 +O-028 +ALS 19 735 +10:43:56.035 −59:34:41.04 +287.42−00.60 01 +g10435603−5934410 5 350 363 768 497 176 320 19 735 +11.8248 +0.7094 0.426±0.028 +O-002 +2MASS J10441879−5951490 10:44:18.792 −59:51:49.02 +287.60−00.83 01 +g10441879−5951490 5 350 350 200 720 724 992 — +11.8308 +0.5924 0.453±0.025 +O-028 +2MASS J10443223−5933592 10:44:32.231 −59:33:59.27 +287.48−00.56 01 +g10443223−5933593 5 350 386 347 167 198 976 — +11.8355 +0.6170 0.439±0.022 +O-029 +HD 93 129 C +10:43:56.824 −59:32:51.34 +287.41−00.57 04 +g10435650−5932498 5 350 363 910 242 514 816 — +11.8378 +0.5435 0.381±0.051 +O-002 +2MASS J10443139−5944080 10:44:31.392 −59:44:08.02 +287.56−00.71 01 +g10443139−5944080 5 350 357 278 827 025 792 — +11.8542 +0.2898 0.464±0.029 +O-003 +ALS 16 082 +10:45:44.610 −59:50:41.13 +287.75−00.73 01 +g10454461−5950411 5 350 308 140 079 159 168 16 082 +11.8561 +1.5379 0.312±0.127 +O-030 +ALS 15 203 B +10:43:41.181 −59:35:53.64 +287.40−00.63 03 +— +5 350 363 326 125 475 712 15 203 B +11.8838 +1.0998 0.499±0.102 +O-002 +2MASS J10435902−5933196 10:43:59.024 −59:33:19.66 +287.42−00.58 02 +g10435902−5933197 5 350 363 910 256 772 608 — +11.8921 +0.5094 0.441±0.026 +O-002 +2MASS J10454701−6000271 10:45:47.026 −60:00:27.19 +287.83−00.87 01 +g10454701−6000272 5 350 298 141 394 249 088 — +11.9041 +0.4997 0.486±0.033 +O-030 +2MASS J10432556−5919175 10:43:25.561 −59:19:17.50 +287.24−00.41 01 +g10432557−5919175 5 350 402 835 514 697 728 — +11.9224 +0.3310 0.424±0.029 +Car OB1 +2MASS J10451588−5929563 10:45:15.889 −59:29:56.37 +287.53−00.45 01 +g10451589−5929564 5 350 384 629 180 484 224 — +11.9227 +0.5950 0.426±0.020 +O-029 +2MASS J10422722−6000052 10:42:27.224 −60:00:05.30 +287.46−01.06 01 +g10422722−6000052 5 254 273 427 251 373 184 — +11.9452 +0.1805 0.370±0.027 +O-028 +ALS 19 740 +10:44:05.088 −59:33:41.25 +287.43−00.58 01 +g10440509−5933413 5 350 363 150 021 919 360 19 740 +11.9791 +0.9392 +— +O-002 +ALS 19 746 +10:45:05.184 −59:41:42.36 +287.60−00.64 03 +g10450523−5941426 5 350 358 343 963 943 424 19 746 +12.0054 +0.4348 0.432±0.030 +O-025 +2MASS J10452314−5940033 10:45:23.143 −59:40:03.38 +287.63−00.60 01 +g10452314−5940034 5 350 381 910 438 554 624 — +12.0061 +0.4046 0.408±0.026 +O-025 +2MASS J10440516−5921165 10:44:05.160 −59:21:16.57 +287.33−00.40 01 +g10440517−5921165 5 350 390 504 695 776 768 — +12.0104 +0.5822 0.427±0.024 +O-027 +ALS 19 742 +10:44:28.972 −59:42:34.31 +287.54−00.69 01 +g10442897−5942343 5 350 357 416 266 008 960 19 742 +12.0148 +0.3911 0.476±0.030 +O-003 +2MASS J10433865−5934444 10:43:38.658 −59:34:44.45 +287.39−00.62 01 +g10433866−5934444 5 350 363 631 058 143 488 — +12.0225 +0.7373 0.427±0.024 +O-002 +2MASS J10441729−5917154 10:44:17.298 −59:17:15.41 +287.32−00.32 01 +g10441729−5917154 5 350 396 620 729 353 088 — +12.0396 +0.3942 0.439±0.026 +O-027 +2MASS J10445053−5957227 10:44:50.533 −59:57:22.73 +287.70−00.88 01 +g10445053−5957227 5 350 301 959 643 812 480 — +12.0650 +0.4132 0.546±0.036 +O-028 +2MASS J10425900−6000240 10:42:59.006 −60:00:24.02 +287.52−01.04 01 +g10425898−6000242 5 254 273 946 995 268 608 — +12.0792 +0.1634 0.465±0.043 +O-028 +2MASS J10435207−5932401 10:43:52.078 −59:32:40.14 +287.40−00.58 05 +g10435208−5932401 5 350 363 875 897 031 296 — +12.0884 +0.4237 0.426±0.028 +O-002 +20 + +Berlanas et al.: Massive stars in the Carina Nebula +Table A.1. (Continued). +Name +RA +dec +GOS/GBS/GWS ID +GES ID +Gaia ID +ALS ID +G′ +3 +GBP3 − GRP3 +ϖc(mas) +group +2MASS J10435478−6006207 +10:43:54.787 −60:06:20.77 +287.67−01.07 01 +g10435479−6006208 5 254 269 475 881 471 104 — +12.0899 +0.1552 0.453±0.033 +O-028 +ALS 19 748 +10:45:19.427 −59:39:37.44 +287.62−00.59 01 +g10451943−5939374 5 350 381 914 760 925 056 19 748 +12.0908 +0.4645 0.423±0.028 +O-025 +ALS 19 732 +10:43:48.812 −59:33:35.17 +287.40−00.59 02 +— +5 350 363 841 537 241 088 19 732 +12.0957 +0.5980 0.435±0.023 +O-002 +ALS 19 747 +10:45:09.651 −59:40:08.46 +287.60−00.61 01 +g10450968−5940088 5 350 358 515 777 872 512 19 747 +12.1571 +0.4232 0.484±0.033 +O-025 +2MASS J10435198−6010368 +10:43:51.984 −60:10:36.81 +287.70−01.13 01 +g10435198−6010368 5 254 267 448 656 913 152 — +12.1638 +0.1415 0.391±0.030 +O-028 +2MASS J10442909−5948207 +10:44:29.098 −59:48:20.75 +287.59−00.77 01 +g10442910−5948207 5 350 356 656 026 369 280 — +12.1802 +1.4850 0.582±0.134 +O-028 +ALS 17 185 +10:43:43.556 −59:34:03.47 +287.39−00.61 01 +g10434356−5934035 5 350 363 459 259 473 664 17 185 +12.1839 +0.8092 0.424±0.023 +O-002 +2MASS J10451925−5929522 +10:45:19.255 −59:29:52.24 +287.54−00.45 01 +g10451925−5929522 5 350 384 629 180 493 952 — +12.2402 +0.9340 0.425±0.021 +O-029 +[HSB2012] 3545 +10:45:09.867 −59:42:13.79 +287.62−00.64 02 +g10450990−5942139 5 350 358 343 979 096 704 — +12.2560 +0.6066 0.422±0.027 +O-025 +2MASS J10440371−5948141 +10:44:03.714 −59:48:14.12 +287.54−00.79 01 +g10440371−5948141 5 350 351 055 389 015 040 — +12.2654 +0.8280 0.448±0.024 +O-028 +2MASS J10434797−6001201 +10:43:47.975 −60:01:20.10 +287.62−01.00 01 +g10434797−6001201 5 254 270 953 350 213 248 — +12.3016 +0.2349 0.423±0.031 +O-028 +2MASS J10443829−6005449 +10:44:38.295 −60:05:44.93 +287.74−01.02 04 +g10443829−6005450 5 254 222 682 258 888 576 — +12.3027 +0.4803 0.332±0.038 +O-028 +2MASS J10454824−5922041 +10:45:48.246 −59:22:04.20 +287.53−00.31 01 +g10454824−5922042 5 350 391 707 286 868 608 — +12.3031 +0.3918 0.445±0.027 +Car OB1 +2MASS J10441829−5942296 +10:44:18.297 −59:42:29.61 +287.52−00.70 02 +g10441830−5942296 5 350 357 652 462 592 896 — +12.3089 +0.3907 0.450±0.031 +O-003 +2MASS J10445080−5918005 +10:44:50.809 −59:18:00.57 +287.39−00.30 01 +g10445081−5918004 5 350 395 967 894 386 176 — +12.3290 +0.4284 0.412±0.026 +O-027 +[HSB2012] 3416 +10:45:05.700 −59:41:23.83 +287.60−00.63 03 +g10450576−5941240 5 350 358 378 338 847 744 — +12.3333 +0.4009 0.441±0.036 +O-025 +2MASS J10451894−5942184 +10:45:18.943 −59:42:18.40 +287.64−00.63 03 +g10451894−5942184 5 350 311 374 212 231 936 — +12.3552 +0.6042 0.398±0.033 +O-025 +2MASS J10442894−5943473 +10:44:28.948 −59:43:47.31 +287.55−00.70 02 +g10442895−5943473 5 350 357 377 584 700 928 — +12.3794 +0.3535 0.452±0.031 +O-003 +2MASS J10413434−5958474 +10:41:34.347 −59:58:47.42 +287.35−01.10 01 +g10413434−5958474 5 254 285 968 555 876 224 — +12.4053 +0.5480 0.212±0.021 background +ALS 16 078 +10:43:21.816 −59:24:22.79 +287.28−00.48 01 +g10432183−5924227 5 350 400 022 343 131 008 16 078 +12.4227 +0.4214 0.397±0.025 +Car OB1 +2MASS J10444278−5921383 +10:44:42.783 −59:21:38.31 +287.41−00.36 04 +g10444278−5921383 5 350 395 379 457 415 296 — +12.4254 +0.6128 0.426±0.027 +O-027 +2MASS J10433443−5943264 +10:43:34.435 −59:43:26.49 +287.45−00.75 01 +g10433443−5943265 5 350 355 144 197 874 048 — +12.4277 +1.3156 0.419±0.046 +Car OB1 +2MASS J10450836−5938475 +10:45:08.367 −59:38:47.52 +287.59−00.59 01 +g10450837−5938475 5 350 382 189 638 824 192 — +12.4502 +0.6893 0.407±0.035 +O-025 +2MASS J10453807−5944095 +10:45:38.077 −59:44:09.55 +287.69−00.64 01 +g10453808−5944095 5 350 311 202 413 541 760 — +12.4689 +1.5293 0.443±0.029 +O-025 +2MASS J10430420−5948591 +10:43:04.207 −59:48:59.13 +287.44−00.86 01 +g10430420−5948591 5 350 353 185 692 794 880 — +12.4933 +0.4511 0.503±0.039 +O-028 +2MASS J10444235−5922029 +10:44:42.353 −59:22:02.97 +287.41−00.37 01 +g10444236−5922029 5 350 389 508 263 449 600 — +12.4998 +0.5674 0.399±0.031 +O-027 +2MASS J10453134−5941133 +10:45:31.342 −59:41:13.31 +287.65−00.60 01 +g10453134−5941133 5 350 334 910 634 790 144 — +12.5033 +0.6492 0.428±0.025 +O-025 +2MASS J10444726−5928154 +10:44:47.270 −59:28:15.49 +287.47−00.46 01 +g10444727−5928155 5 350 385 591 253 129 984 — +12.5055 +0.4944 0.452±0.026 +O-029 +Gaia DR3 5350302097055370496 10:44:53.834 −59:56:13.54 +287.70−00.86 02 +g10445392−5956134 5 350 302 097 055 370 496 — +12.5385 +0.4596 0.402±0.030 +O-028 +2MASS J10434812−5950443 +10:43:48.127 −59:50:44.34 +287.53−00.85 01 +g10434813−5950443 5 350 353 357 491 488 896 — +12.5419 +0.4722 0.413±0.028 +O-028 +2MASS J10443591−5923356 +10:44:35.919 −59:23:35.63 +287.41−00.40 01 +g10443592−5923356 5 350 389 366 497 928 448 — +12.5614 +0.4567 0.431±0.028 +O-027 +2MASS J10440236−5952046 +10:44:02.368 −59:52:04.68 +287.57−00.85 01 +g10440237−5952047 5 350 350 338 159 613 312 — +12.5664 +0.5889 0.421±0.029 +O-028 +2MASS J10460277−5950192 +10:46:02.773 −59:50:19.28 +287.78−00.71 01 +g10460277−5950193 5 350 309 484 426 592 768 — +12.5815 +1.1325 0.407±0.026 +O-030 +[HSB2012] 3314 +10:45:01.789 −59:42:01.31 +287.60−00.65 01 +g10450180−5942014 5 350 358 550 122 199 424 — +12.5821 +0.3860 0.432±0.038 +O-025 +2MASS J10440744−5916399 +10:44:07.449 −59:16:39.98 +287.30−00.33 01 +g10440746−5916399 5 350 408 333 079 066 112 — +12.6005 +1.5270 0.232±0.025 background +2MASS J10443822−5943056 +10:44:38.222 −59:43:05.61 +287.57−00.68 01 +g10443822−5943056 5 350 357 343 224 979 200 — +12.6092 +0.5159 0.409±0.029 +O-025 +2MASS J10440576−5927078 +10:44:05.763 −59:27:07.88 +287.38−00.48 01 +g10440576−5927079 5 350 388 060 827 209 344 — +12.6094 +0.7435 0.331±0.056 +Car OB1 +2MASS J10443510−5923281 +10:44:35.106 −59:23:28.16 +287.41−00.40 02 +g10443511−5923282 5 350 389 370 824 434 944 — +12.6150 +0.4994 0.444±0.031 +O-027 +2MASS J10444550−5952537 +10:44:45.505 −59:52:53.77 +287.66−00.82 01 +g10444551−5952538 5 350 349 509 200 925 824 — +12.6357 +1.0904 0.341±0.028 +O-028 +2MASS J10454536−5958530 +10:45:45.362 −59:58:53.09 +287.81−00.85 01 +— +5 350 298 278 833 689 728 — +12.6513 +1.0703 0.449±0.034 +O-030 +2MASS J10451297−5946059 +10:45:12.973 −59:46:05.96 +287.65−00.69 01 +g10451297−5946060 5 350 310 790 096 642 816 — +12.6680 +0.8258 0.428±0.027 +O-025 +Gaia DR3 5350388683629610752 10:44:47.350 −59:26:59.53 +287.46−00.44 01 +g10444735−5926595 5 350 388 683 629 610 752 — +12.6745 +0.6993 0.421±0.023 +O-029 +2MASS J10450792−5939011 +10:45:07.928 −59:39:01.19 +287.59−00.60 01 +g10450793−5939012 5 350 382 189 638 820 352 — +12.6798 +0.5717 0.446±0.031 +O-025 +[HSB2012] 3211 +10:44:58.410 −59:39:43.50 +287.58−00.61 01 +g10445846−5939437 5 350 358 756 296 011 008 — +12.7158 +0.6691 0.423±0.029 +O-025 +[HSB2012] 1498 +10:43:55.834 −59:32:52.26 +287.41−00.58 05 +g10435580−5932520 5 350 363 910 242 737 920 — +12.7240 +0.4671 0.395±0.037 +O-002 +2MASS J10435501−5936242 +10:43:55.016 −59:36:24.23 +287.43−00.63 01 +g10435502−5936242 5 350 362 600 261 090 816 — +12.7507 +0.7025 0.415±0.027 +O-002 +2MASS J10444208−5926353 +10:44:42.086 −59:26:35.39 +287.44−00.44 01 +g10444209−5926354 5 350 388 649 269 869 952 — +12.7537 +0.5922 0.408±0.026 +Car OB1 +2MASS J10443769−5929316 +10:44:37.691 −59:29:31.61 +287.46−00.48 01 +g10443769−5929316 5 350 387 034 362 093 696 — +12.7839 +0.4714 0.423±0.026 +O-029 +[ESK2003] 148 +10:45:36.610 −59:44:11.07 +287.68−00.64 01 +g10453661−5944111 5 350 311 095 013 575 296 — +12.7958 +2.0482 0.444±0.093 +O-025 +2MASS J10471498−5953374 +10:47:14.989 −59:53:37.44 +287.94−00.69 01 +g10471499−5953374 5 350 293 850 745 988 096 — +12.8177 +0.8072 0.326±0.027 background +[HSB2012] 3482 +10:45:07.885 −59:41:34.01 +287.61−00.63 01 +g10450790−5941341 5 350 358 378 338 850 304 — +12.8236 +0.7083 0.434±0.031 +O-025 +2MASS J10464886−5950409 +10:46:48.869 −59:50:40.96 +287.87−00.67 01 +g10464887−5950410 5 350 305 945 373 896 192 — +12.8277 +0.8870 0.430±0.025 +O-030 +2MASS J10453254−5942359 +10:45:32.546 −59:42:35.91 +287.66−00.62 01 +g10453255−5942359 5 350 311 442 931 718 400 — +12.8331 +0.9362 0.408±0.030 +O-025 +2MASS J10440105−6006377 +10:44:01.051 −60:06:37.72 +287.68−01.07 01 +g10440104−6006378 5 254 269 514 589 025 024 — +12.8586 +0.4092 0.451±0.029 +O-028 +21 + +Berlanas et al.: Massive stars in the Carina Nebula +Table A.1. (Continued). +Name +RA +dec +GOS/GBS/GWS ID +GES ID +Gaia ID +ALS ID +G′ +3 +GBP3 − GRP3 +ϖc(mas) +group +2MASS J10425293−6003478 10:42:52.934 −60:03:47.83 +287.53−01.09 01 +g10425293−6003478 5 254 272 297 727 705 728 — +12.8649 +0.8109 0.239±0.025 background +2MASS J10440866−5933488 10:44:08.669 −59:33:48.81 +287.44−00.58 01 +g10440867−5933488 5 350 363 085 623 074 432 — +12.8705 +0.6870 0.398±0.028 +O-002 +2MASS J10444609−5946056 10:44:46.092 −59:46:05.69 +287.60−00.72 01 +g10444609−5946057 5 350 356 385 474 179 840 — +12.9082 +0.5019 0.416±0.035 +Car OB1 +2MASS J10454060−5937041 10:45:40.609 −59:37:04.20 +287.63−00.53 01 +g10454061−5937042 5 350 335 597 829 590 400 — +12.9289 +0.7568 0.455±0.028 +Car OB1 +2MASS J10470063−5957242 10:47:00.639 −59:57:24.23 +287.94−00.76 01 +g10470064−5957242 5 350 293 232 270 641 152 — +12.9382 +0.6593 0.396±0.026 +O-030 +2MASS J10440384−5934344 10:44:03.843 −59:34:34.44 +287.43−00.59 01 +g10440384−5934344 5 350 363 051 263 281 920 — +12.9416 +0.5507 0.442±0.029 +O-002 +2MASS J10420949−6002265 10:42:09.498 −60:02:26.60 +287.44−01.12 01 +g10420950−6002267 5 254 284 598 515 540 608 — +12.9549 +1.1432 0.237±0.028 background +2MASS J10445602−5938530 10:44:56.022 −59:38:53.04 +287.57−00.60 01 +g10445602−5938530 5 350 358 751 974 862 848 — +12.9838 +1.1511 0.379±0.085 +O-025 +2MASS J10443089−5914461 10:44:30.898 −59:14:46.11 +287.33−00.27 01 +g10443090−5914461 5 350 396 964 328 794 624 — +12.9867 +2.1690 0.218±0.131 +Car OB1 +2MASS J10440683−5936116 10:44:06.836 −59:36:11.64 +287.45−00.61 01 +g10440684−5936116 5 350 362 570 226 891 648 — +12.9975 +0.7657 0.391±0.032 +Car OB1 +2MASS J10431388−5954584 10:43:13.880 −59:54:58.45 +287.50−00.94 01 +g10431388−5954585 5 350 351 472 030 662 400 — +13.0061 +1.0351 0.212±0.027 background +2MASS J10434798−5933590 10:43:47.984 −59:33:59.06 +287.40−00.60 01 +g10434798−5933591 5 350 363 463 580 099 968 — +13.0216 +0.7676 0.442±0.028 +O-002 +2MASS J10434580−5934359 10:43:45.802 −59:34:35.94 +287.40−00.61 01 +g10434580−5934359 5 350 363 463 564 711 168 — +13.0216 +0.6789 0.426±0.030 +O-002 +[HSB2012] 3150 +10:44:56.737 −59:40:02.36 +287.58−00.62 01 +g10445680−5940026 5 350 358 653 216 770 048 — +13.0397 +0.6459 0.429±0.025 +O-025 +2MASS J10452875−5930037 10:45:28.755 −59:30:03.79 +287.56−00.44 01 +g10452876−5930038 5 350 384 938 418 159 872 — +13.0641 +1.5256 0.437±0.023 +O-029 +2MASS J10445837−5932062 10:44:58.378 −59:32:06.29 +287.52−00.50 01 +g10445838−5932063 5 350 383 770 186 937 856 — +13.0713 +0.7362 0.433±0.021 +O-029 +2MASS J10460608−5957394 10:46:06.086 −59:57:39.42 +287.84−00.81 01 +g10460609−5957394 5 350 304 055 588 173 568 — +13.0983 +1.1034 0.415±0.027 +O-030 +2MASS J10442945−5933437 10:44:29.458 −59:33:43.75 +287.48−00.55 01 +g10442946−5933438 5 350 386 347 167 197 824 — +13.1063 +0.6184 0.439±0.024 +O-029 +2MASS J10460257−5957372 10:46:02.572 −59:57:37.27 +287.83−00.82 01 +g10460257−5957373 5 350 304 158 675 916 672 — +13.1141 +1.1942 0.428±0.030 +O-030 +2MASS J10452056−5942212 10:45:20.568 −59:42:21.29 +287.64−00.63 02 +g10452057−5942213 5 350 311 374 212 233 472 — +13.1151 +0.6445 0.443±0.022 +O-025 +2MASS J10453834−5942078 10:45:38.341 −59:42:07.87 +287.67−00.61 01 +g10453834−5942079 5 350 334 704 476 358 912 — +13.1477 +1.1618 0.420±0.104 +O-025 +2MASS J10443766−5923073 10:44:37.662 −59:23:07.31 +287.41−00.39 01 +— +5 350 389 473 903 666 176 — +13.1500 +0.5337 0.423±0.057 +O-027 +2MASS J10444710−5939201 10:44:47.107 −59:39:20.19 +287.55−00.62 01 +g10444711−5939202 5 350 359 409 131 005 056 — +13.1919 +1.0657 0.405±0.123 +O-025 +[HSB2012] 3526 +10:45:09.278 −59:41:28.23 +287.61−00.63 02 +g10450933−5941283 5 350 358 378 323 816 704 — +13.1951 +0.6212 0.445±0.027 +O-025 +2MASS J10452415−5942313 10:45:24.155 −59:42:31.38 +287.65−00.63 01 +g10452416−5942314 5 350 311 477 291 450 880 — +13.2065 +0.7001 0.445±0.023 +O-025 +2MASS J10452190−5945249 10:45:21.904 −59:45:24.94 +287.66−00.68 01 +g10452190−5945249 5 350 310 858 816 135 936 — +13.2175 +0.9117 0.440±0.022 +O-025 +2MASS J10433596−5933179 10:43:35.965 −59:33:17.92 +287.37−00.60 01 +g10433597−5933179 5 350 363 738 457 983 232 — +13.2571 +0.9318 0.419±0.022 +O-002 +2MASS J10463643−5948048 10:46:36.433 −59:48:04.89 +287.82−00.64 01 +g10463643−5948049 5 350 330 237 710 808 576 — +13.3165 +1.3039 0.404±0.022 +O-030 +2MASS J10453185−6000293 10:45:31.859 −60:00:29.36 +287.80−00.89 01 +g10453186−6000294 5 350 300 791 412 952 448 — +13.4142 +2.0645 0.404±0.026 +O-030 +2MASS J10435009−5947024 10:43:50.098 −59:47:02.48 +287.51−00.79 01 +g10435010−5947025 5 350 353 980 292 003 456 — +13.4565 +0.9572 0.415±0.022 +O-028 +2MASS J10462657−5956131 10:46:26.577 −59:56:13.11 +287.87−00.77 01 +g10462658−5956131 5 350 303 776 387 591 296 — +13.4573 +1.1028 0.473±0.028 +O-030 +2MASS J10460291−5950259 10:46:02.912 −59:50:25.91 +287.78−00.71 02 +g10460291−5950259 5 350 309 484 426 592 256 — +13.4955 +1.1179 0.419±0.024 +O-030 +2MASS J10453819−5942157 10:45:38.200 −59:42:15.74 +287.67−00.61 02 +g10453820−5942157 5 350 334 704 476 358 016 — +13.5218 +0.9537 0.428±0.022 +O-025 +[ARV2008] 217 +10:45:36.748 −59:47:02.00 +287.70−00.69 02 +g10453675−5947020 5 350 310 583 938 228 352 — +13.5888 +2.9145 0.396±0.083 +Car OB1 +2MASS J10434303−5945333 10:43:43.031 −59:45:33.37 +287.48−00.77 01 +g10434303−5945334 5 350 354 289 529 667 968 — +13.6020 +1.2323 0.420±0.022 +O-028 +2MASS J10460116−5949420 10:46:01.166 −59:49:42.03 +287.77−00.70 01 +g10460116−5949420 5 350 309 480 105 783 040 — +13.7741 +1.0953 0.454±0.024 +O-030 +2MASS J10454661−5948404 10:45:46.616 −59:48:40.43 +287.74−00.70 01 +g10454661−5948404 5 350 309 759 304 499 456 — +14.6338 +2.2879 0.438±0.029 +Car OB1 +2MASS J10431945−5944488 10:43:19.457 −59:44:48.81 +287.43−00.79 01 +g10431946−5944488 5 350 355 045 443 864 832 — +14.6466 +−0.3289 1.266±0.043 +foreground +2MASS J10454595−5949075 10:45:45.953 −59:49:07.59 +287.74−00.71 01 +— +5 350 309 656 225 280 000 — +14.8208 +1.8609 0.431±0.031 +Car OB1 +2MASS J10453024−5948206 10:45:30.249 −59:48:20.64 +287.70−00.71 03 +— +5 350 310 446 472 013 440 — +15.2266 +4.0558 0.334±0.153 +Car OB1 +2MASS J10452013−5950104 10:45:20.136 −59:50:10.43 +287.70−00.75 01 +— +5 350 308 900 311 308 928 — +15.4819 +2.0970 0.380±0.036 +Car OB1 +2MASS J10452648−5946188 10:45:26.489 −59:46:18.88 +287.68−00.68 01 +— +5 350 310 652 657 701 760 — +15.4955 +1.9808 +negative background +2MASS J10451717−5947013 10:45:17.180 −59:47:01.36 +287.67−00.70 01 +— +5 350 310 721 377 468 544 — +16.0174 +2.9979 0.339±0.049 +Car OB1 +2MASS J10453470−5947537 10:45:34.706 −59:47:53.76 +287.71−00.70 01 +— +5 350 310 476 539 330 944 — +17.1801 +4.7233 0.335±0.174 +Car OB1 +2MASS J10450879−5950537 10:45:08.798 −59:50:53.72 +287.68−00.77 01 +— +5 350 308 659 792 838 016 — +17.2377 +3.3508 0.423±0.117 +O-028 +2MASS J10452862−5947553 10:45:28.623 −59:47:55.31 +287.70−00.71 01 +— +5 350 310 510 899 068 544 — +17.4992 +4.8127 0.183±0.180 +Car OB1 +22 + +Berlanas et al.: Massive stars in the Carina Nebula +Table A.2. Spectral classifications for the stars in the field of the Carina Nebula analyzed in this paper. +Name +Gaia-ESO +GOSSS +LiLiMaRlin + STIS + literature +ST +LC +qual. +sec. +ST +LC +qual. +sec. +ref. +ST +LC +qual. +sec. +ref. +η Car +— +— +— +— +LBV +— +— +— +M22 +F: +I +— +— +W77 +HD 93 420 +— +— +— +— +— +— +— +— +— +M4 +Ib +— +— +M23b +QZ Car Aa,Ac +— +— +— +— +O9.7 +Ib +n +— +S14 +O9.7 +Ib +— +O9 II: +M23a +WR 24 +— +— +— +— +WN6 +— +ha +— +TW +WN6 +— +ha-w... — +H06 +HD 93 281 +— +— +— +— +— +— +— +— +— +M1.5 +Iab — +— +M23b +HDE 303 310 +— +— +— +— +— +— +— +— +— +M3 +Iab — +— +M23b +HD 93 129 Aa +— +— +— +— +O2 +I +f* +— +S14 +O2 +I +f* +OB? +M17 +HD 93 403 +— +— +— +— +O5 +I +fc +O7.5 V +M23a O5 +I +fc +O7.5 V +M23a +HD 93 250 A,B +— +— +— +— +O4 +IV +(fc) +— +M16 +O3.5 +V +((f+)) +— +W02 +HD 93 205 +— +— +— +— +O3.5 +V +((f)) +O8 V +S14 +O3.5 +V +((f)) +O8 V +M20 +HD 93 160 A,B +— +— +— +— +O7 +III +((f)) +— +S14 +O6 +III +(f) +— +W72 +HD 93 162 +— +— +— +— +O2.5 +I +f*/WN6 OB +S14 +WN6 +— +h +O4 +V01 +HD 93 130 +— +— +— +— +O6.5 +III +(f) +— +S14 +O6 +III +(f) +— +W72 +HD 93 222 A,B +— +— +— +— +O7 +V +((f)) +— +M16 +O7 +III +((f)) +— +W73b +HDE 303 308 A,B — +— +— +— +O4.5 +V +((fc)) +— +S14 +O4 +V +((f+)) +— +W02 +HD 93 249 A +O9 +III +— +— +O9 +III +— +— +S14 +O9 +III +— +— +W73a +HD 93 028 +— +— +— +— +O9 +IV +— +— +S14 +O9 +V +— +— +W72 +HD 93 146 A +— +— +— +— +O7 +V +((f)) +— +M16 +O6.5 +V +((f)) +— +W73b +HD 93 204 +O5.5 V +((f)) +— +O5.5 +V +((f)) +— +S14 +O5 +V +((f)) +— +W02 +HD 93 190 +— +— +— +— +O9.7: V: +(n)e +— +S14 +B0 +IV +pe +— +M55 +HDE 305 523 +— +— +— +— +O9 +II-III — +— +S14 +O9 +II +— +— +W73a +CPD −59 2600 +— +— +— +— +O6 +V +((f)) +— +S14 +O6 +V +((f)) +— +W73b +HD 93 161 B +O6.5 IV +((f)) +— +O6.5 +IV +((f)) +— +S14 +O5 +— +— +— +T73 +HD 93 161 A +O7 +V +((f)) +O9 IV +O7.5 +V +— +O9 V +S14 +O7 +V +((f)) +O9 IV +M23a +HD 93 129 Ab +— +— +— +— +— +— +— +— +— +O2 +I +f* +— +M17 +HDE 305 520 +— +— +— +— +— +— +— +— +— +B0.7 +Iab — +— +M23b +HD 93 128 +— +— +— +— +O3.5 +V +((fc))z +— +S14 +O3.5 +V +((f+)) +— +W02 +HD 93 027 +O9.5 IV +— +— +O9.5 +IV +— +— +S14 +O9.5 +V +— +— +W73b +V572 Car +O6.5 V +z +B0 V + B0.5: V O7.5 +V +(n) +B0 V(n) +M16 +O6.5 +V +z +B0 V +B0.2 V M23a +HD 93 129 B +— +— +— +— +O3.5 +V +((fc))z +— +M22 +O3.5 +V +((f+)) +— +W02 +HD 92 877 A +— +— +— +— +— +— +— +— +— +B2 +III +— +— +M23b +HDE 305 438 +— +— +— +— +O8 +V +z +— +S14 +O8 +— +— +— +T74 +CPD −59 2554 +O9.2 V +— +B1: V +O9.5 +IV +— +— +S14 +O9.2 +V +— +B1: V +M23a +HD 93 342 +— +— +— +— +— +— +— +— +— +B1.5 +Ib +— +— +M23b +HDE 305 536 +— +— +— +— +O9.5 +V +— +— +S14 +O9.5 +V +— +— +M01 +HDE 303 311 +— +— +— +— +O6 +V +((f))z +— +S14 +O5 +V +z +— +W09 +HD 93 056 +B1: +V: +n +— +— +— +— +— +— +O9 +V +— +B2 V +A16 +HD 93 501 +— +— +— +— +— +— +— +— +— +B1.5: +III: (n)e +— +M23b +HDE 305 437 +— +— +— +— +B0 +V +— +— +TW +B0 +V +— +— +A16 +CPD −59 2641 +— +— +— +— +O6 +V +((fc)) +— +S14 +O5.5-6 V +((fc)) +B2 V-III +R09 +HD 93 620 +— +— +— +— +— +— +— +— +— +B2 +III +— +— +M23b +CPD −59 2635 +O8 +V +z +O9.2 V +O8 +V +(n) +O9.5 V +S14 +O8 +V +z +O9.2 V +M23a +CPD −59 2592 +— +— +— +— +— +— +— +— +— +B2.5 +Ia +— +— +M23b +HDE 305 524 +O6.5 V +n((f))z — +O6.5 +V +n((f))z +— +S14 +O6 +III +n +— +V93 +CPD −58 2620 +— +— +— +— +O7 +V +((f))z +— +S14 +O6.5 +V +((f)) +— +W73b +CPD −59 2551 +— +— +— +— +O9 +V +— +— +S14 +O9 +V +— +— +V93 +HDE 305 439 A +B0 +Ia +— +— +— +— +— +— +— +B0 +Ia +— +— +V93 +HDE 303 299 +— +— +— +— +— +— +— +— +— +B2.5: +III: (n)e +— +M23b +HD 93 249 B +B0: +V +(n) +— +B0.2 +V +(n) +— +TW +B0 +V +— +— +A16 +HDE 305 535 +B4 +III +(n) +— +— +— +— +— +— +B2.5 +V +— +— +A16 +HDE 305 452 +B2 +III +— +— +— +— +— +— +— +B8/9 +— +— +— +L76 +HD 93 343 +— +— +— +— +O8 +V +— +— +M16 +O7.5 +V +z +O7.5: V(n) +M22 +CPD −58 2611 +— +— +— +— +O6 +V +((f))z +— +S14 +O6 +V +((f)) +— +W82 +V573 Car +— +— +— +— +O9.5 +V +(n) +B0.5 V(n) S14 +O9.5 +IV +— +B0.5 V +M23a +23 + +Berlanas et al.: Massive stars in the Carina Nebula +Table A.2. (Continued). +Name +Gaia-ESO +GOSSS +LiLiMaRlin + STIS + literature +ST +LC +qual. +sec. +ST +LC +qual. +sec. +ref. +ST +LC +qual. +sec. +ref. +CPD −59 2636 A,B O7.5 +V +— +O8 V + O8 V O8 +V +— +O8 V +S14 +O7 +V +— +O8 V + O9 V A02 +HDE 303 316 A +— +— +— +— +O7 +V +((f))z +— +S14 +O6 +V +— +— +F81 +HDE 305 518 +— +— +— +— +O9.7 III +— +— +S14 +O9.5 +V +— +— +L82 +HDE 305 534 +— +— +— +— +— +— +— +— +— +B0 +V +— +B1: V +M23b +CPD −59 2624 +O9.7 +V +— +— +O9.7 IV +— +— +S14 +O9.5 +V +— +— +A16 +HDE 305 522 +B0.2 +V +— +B1: V +— +— +— +— +— +B0 +V +— +B1 V +A16 +HDE 305 543 +— +— +— +— +— +— +— +— +— +B0.2 +V +(n) +B1: V(n) +M23b +HDE 303 300 +B0.5 +V +— +B +— +— +— +— +— +B1 +V +— +— +A16 +HD 93 097 +B0.2 +V: +n +— +— +— +— +— +— +B0.5 +IV +— +— +A16 +HDE 305 525 +— +— +— +— +O5.5 V +((f))z +O7.5 V + B M16 +O4 +V +— +— +V93 +HDE 305 521 +— +— +— +— +— +— +— +— +— +B0.7 +V +(n) +— +M23b +CPD −59 2574 +— +— +— +— +— +— +— +— +— +B1.5 +V +— +— +M23b +HDE 305 516 +B0.5 +V +(n) +B2: V +— +— +— +— +— +B0.5 +V +— +— +G11b +CPD −59 2626 A,B O7.5 +V +(n) +— +O7.5 V +(n) +— +M16 +O7 +V +n +— +L82 +HDE 303 312 +O9.5 +III +— +B0.5: V +O9.7 IV +— +— +S14 +O9 +V +— +— +F81 +CPD −58 2605 +B0.5 +II +— +— +— +— +— +— +— +B0 +III-IV: — +— +M88 +ALS 15 196 +— +— +— +— +O8.5 V +— +— +S14 +O8 +V +— +— +M88 +HD 93 146 B +— +— +— +— +O9.7 IV +— +— +S14 +O9.5 +V +— +— +L81 +CPD −59 2644 +O9 +V +— +— +O9 +V +— +— +S14 +O8.5 +V +— +— +M93 +HDE 305 532 +— +— +— +— +O6.5 V +((f))z +— +S14 +O6 +V +((f)) +— +W82 +CPD −58 2656 +B0.2 +V +— +— +— +— +— +— +— +B1 +V +— +— +A16 +CPD −58 2623 +B0.2 +V +— +— +— +— +— +— +— +B0 +V +— +— +K93 +CPD −59 2610 +— +— +— +— +O8.5 V +— +— +S14 +O8.5 +V +((f)) +— +M01 +CPD −59 2627 +O9.5 +IV +— +— +O9.5 V +— +— +S14 +O9.5 +V +— +— +A16 +CPD −58 2649 A +O9.5: — +— +B0: +O9.7 III: — +B0: V: +M23a O7 +V +— +O8 V +A16 +CPD −58 2627 +O9.5 +V +(n) +— +O9.5 V +(n) +— +S14 +O9 +III +— +— +F81 +CPD −59 2595 +— +— +— +— +B2.5 +V +— +— +TW +B2 +V +— +— +A16 +CPD −59 2673 +O5.5 +V +(n)((f))z — +O5.5 V +(n)((f))z — +S14 +O5 +V +n +— +W82 +ALS 15 210 +O3.5 +I +f* Nwk +— +O3.5 I +f* Nwk +— +S14 +O3/4 +I +f +— +M93 +HDE 303 313 +B2 +V +— +B2 V +— +— +— +— +— +B2 +V +— +B2 V +A16 +CPD −59 2593 +B2.5: +IV +nnn +— +— +— +— +— +— +B2 +V +— +— +A16 +HDE 305 528 +— +— +— +— +— +— +— +— +— +B2 +V +— +— +A16 +HDE 305 439 B +— +— +— +— +— +— +— +— +— +B0.7 +Ib +— +— +M23b +CPD −59 2537 +— +— +— +— +— +— +— +— +— +B1.5 +V +— +— +M23b +ALS 15 205 +B0.2 +V +(n) +— +— +— +— +— +— +B2 +V +— +— +A16 +ALS 15 961 +— +— +— +— +B1 +V +— +— +TW +B0 +V +— +— +A16 +CPD −59 2469 +B9 +III +— +— +— +— +— +— +— +B2/5: — +— +— +L76 +HDE 305 533 +B0.5 +V +(n) +— +— +— +— +— +— +B0 +V +— +— +A16 +ALS 15 860 +B1 +Iab — +— +B1 +Iab — +— +TW +O9.5 +I/II +— +— +F80 +CPD −58 2625 +O9.5 +V +— +— +O9.2 V +— +— +S14 +O9 +V +— +— +A16 +HD 93 249 C +B1.5 +V +(n) +— +— +— +— +— +— +B2 +V +n +— +M88 +HDE 305 515 A +— +— +— +— +— +— +— +— +— +B1.5 +V +sn: +— +G11b +ALS 15 207 +— +— +— +— +O9 +V +— +— +S14 +O9 +V +— +— +M88 +ALS 15 865 +— +— +— +— +— +— +— +— +— +B1.5 +V: +b +— +W11 +CPD −58 2634 +B1.5 +V +— +— +— +— +— +— +— +B3/6 +— +— +— +L76 +HD 93 128 B +B0.2: +V +— +B1: V +B0.2 +V +— +— +M22 +— +— +— +— +— +CPD −59 2629 +O8.5 +V +p +— +O8.5 V +p +— +S14 +O9 +V +— +— +A16 +CPD −58 2657 +B0.7 +V +(n) +— +— +— +— +— +— +B2.5 +V +n +— +G11b +CPD −59 2591 +O8.5 +V +— +B0/1 +O8 +V +z +B0.5: V: +M16 +O8 +V +z +B0.5: V: +M23a +CPD −58 2644 +B1.5 +V +— +— +— +— +— +— +— +B0 +V +— +— +F80 +CPD −59 2598 +— +— +— +— +B1.5 +V +— +— +M22 +B0 +V +— +— +A16 +CPD −59 2570 +— +— +— +— +— +— +— +— +— +B0.5 +V +— +— +A16 +CPD −59 2622 +— +— +— +— +— +— +— +— +— +B0.5 +V +— +— +W11 +CPD −59 2535 +— +— +— +— +— +— +— +— +— +B2 +V +— +— +A16 +24 + +Berlanas et al.: Massive stars in the Carina Nebula +Table A.2. (Continued). +Name +Gaia-ESO +GOSSS +LiLiMaRlin + STIS + literature +ST +LC qual. +sec. +ST +LC qual. +sec. +ref. +ST +LC qual. +sec. +ref. +CPD −59 2632 +B0.5 +V +n +— +— +— +— +— +— +B1 +V +— +— +A16 +CPD −59 2495 +B1.5 +V +— +— +— +— +— +— +— +B0.5 +V +— +— +A16 +ALS 15 204 +— +— +— +— +O7.5 +V +z +O9: V M16 — +— +— +— +— +CPD −58 2655 +B1: +V +— +— +— +— +— +— +— +B1 +V +— +— +F80 +CPD −59 2614 +B1 +V +— +— +— +— +— +— +— +B1 +III +— +— +A16 +ALS 15 219 +B0.5: V +— +B1: V +B0.7 +V +— +B1 V +M22 B1 +V +— +— +M93 +CPD −59 2581 +B0.5 +V +(n) +— +— +— +— +— +— +B1 +V +— +— +M93 +[ARV2008] 206 +O6 +V +((f)) +— +O6 +V +((f)) +— +S14 +O5: +V +— +— +A16 +CPD −59 2583 +B1.5: V +— +B2: V +— +— +— +— +— +B1 +V +— +— +A16 +CPD −58 2640 +B1.5 +V +— +B2: V +— +— +— +— +— +B1 +V +— +B2 V A16 +CPD −59 2606 +B0.7 +V +n +— +— +— +— +— +— +B1 +V +— +— +W11 +Tyc 8626-02506-1 +O9 +V +(n) +— +O9 +V +(n) +— +S14 +O9.5 V +— +— +A16 +CPD −59 2497 +B2 +V +— +— +— +— +— +— +— +B7 +— +— +— +L76 +ALS 15 229 +B0 +V +— +— +B0 +V +— +— +S14 +B0 +V +— +— +M88 +CPD −59 2605 +B0 +V +— +— +— +— +— +— +— +B1 +V +— +— +G11b +CPD −59 2640 +B1.5 +V +(n) +— +B1.5 +V +(n) +— +M22 — +— +— +— +— +ALS 15 228 +B1 +V +— +— +— +— +— +— +— +B2: +V +— +— +W11 +Gaia DR3 5350358378338864768 B1.5: V +(n) +— +— +— +— +— +— +B1.5 +V: +— +— +L82 +ALS 15 855 +B1 +V +(n) +— +— +— +— +— +— +B1 +V +n +— +G11b +CPD −59 2504 +B7 +II: +nn +— +— +— +— +— +— +B2/5 +— +— +— +L76 +CPD −59 2579 +B1.5 +V +— +— +— +— +— +— +— +— +— +— +— +— +CPD −59 2543 +B2 +V +(n) +— +— +— +— +— +— +B2 +V +— +— +A16 +ALS 15 227 +B0.7 +V +— +— +— +— +— +— +— +B1 +V +— +— +A16 +CPD −58 2647 +B1.5 +V +— +— +— +— +— +— +— +B0 +V: +— +— +M88 +ALS 15 224 +B0.5 +V +— +— +— +— +— +— +— +B1 +V +— +— +G11a +CPD −59 2510 +B2 +V +— +— +— +— +— +— +— +B3/4 +— +— +— +L76 +CPD −59 2619 +B0.7 +V +— +— +— +— +— +— +— +B1 +V +— +— +A16 +CPD −59 2596 +B0 +V +— +— +B0 +V +— +— +TW +B0 +V +— +— +M93 +ALS 15 234 +— +— +— +— +B1.5 +V +— +— +M22 O9 +V +— +— +M88 +2MASS J10424532−6012063 +B2 +V +— +— +— +— +— +— +— +— +— +— +— +— +2MASS J10424476−6005020 +B0.2 +V +— +B0.2 V B0 +IV: +— +— +TW +O: +— +— +— +P11 +CPD −58 2653 A +B1.5 +V +— +— +— +— +— +— +— +B2 +V +— +— +A16 +CPD −59 2571 +B1.5 +V +n +— +— +— +— +— +— +B3 +V +— +— +A16 +ALS 15 863 +— +— +— +— +B0.2 +V +— +— +M22 O9: +V +— +— +M88 +2MASS J10415981−5955075 +B2 +V +— +— +— +— +— +— +— +— +— +— +— +— +2MASS J10461906−5957543 +B0.7 +V +— +— +— +— +— +— +— +B1 +III +— +— +A16 +CPD −59 2661 +B0: +— +— +B +B0 +V +(n)e +B2: V +TW +O9.5 V +— +— +W84 +ALS 15 232 +B1 +V +— +— +— +— +— +— +— +— +— +— +— +— +ALS 15 233 +B0.7 +V +— +— +B0.7 +V +— +— +TW +— +— +— +— +— +2MASS J10431897−5946569 +B2: +V +(n) +B +— +— +— +— +— +— +— +— +— +— +ALS 19 739 +B1: +V +— +— +— +— +— +— +— +B1 +V +— +— +M93 +ALS 15 235 +B1.5 +V +— +— +— +— +— +— +— +B1.5 +V +— +— +A16 +CPD −59 2569 B +B1.5 +V +— +— +— +— +— +— +— +— +— +— +— +— +2MASS J10441791−5925204 +B2.5 +V +— +— +— +— +— +— +— +— +— +— +— +— +CPD −58 2667 +B2 +IV +— +— +— +— +— +— +— +B2 +III +— +— +A16 +ALS 15 856 +B1.5 +V +— +— +— +— +— +— +— +— +— +— +— +— +2MASS J10460477−5949217 +O9.7 +V +(n) +— +— +— +— +— +— +— +— +— +— +— +ALS 15 237 +B1.5: V +— +B2: V +— +— +— +— +— +— +— +— +— +— +CPD −59 2617 +B2 +V +(n)e +— +— +— +— +— +— +— +— +— +— +— +CPD −58 2648 +— +— +— +— +— +— +— +— +— +B1 +III +— +— +A16 +2MASS J10460606−5956339 +B2 +V +(n) +— +— +— +— +— +— +— +— +— +— +— +2MASS J10450531−5919347 +B2 +V +(n) +— +— +— +— +— +— +— +— +— +— +— +CPD −59 2585 +B2 +V +— +— +— +— +— +— +— +— +— +— +— +— +[HSB2012] 1443 +— +— +— +— +B1.5: V +(n) +— +M22 — +— +— +— +— +25 + +Berlanas et al.: Massive stars in the Carina Nebula +Table A.2. (Continued). +Name +Gaia-ESO +GOSSS +LiLiMaRlin + STIS + literature +ST +LC +qual. +sec. +ST +LC +qual. +sec. +ref. +ST +LC qual. +sec. +ref. +CPD −59 2625 +B1.5: V +— +B2: V — +— +— +— +— +B2 +V +— +— +A16 +ALS 15 245 +B0.5 +V +— +— +— +— +— +— +— +— +— +— +— +— +CPD −58 2677 +B1 +V +— +— +— +— +— +— +— +B2 +III +— +— +A16 +CPD −59 2541 +B1.5 +V +(n) +— +— +— +— +— +— +— +— +— +— +— +ALS 15 246 +B0.7 +V +— +— +— +— +— +— +— +B1 +V +— +— +A16 +CPD −59 2616 +B2 +V +(n) +— +B2 +V +(n) +— +M22 +B2 +V +— +— +A16 +ALS 15 242 +B0.7 +V +— +— +— +— +— +— +— +B1 +V +— +— +A16 +CPD −58 2662 +B2 +V +— +— +— +— +— +— +— +B2 +V +— +— +G11b +ALS 19 738 +— +— +— +— +— +— +— +— +— +B1 +V +— +— +M93 +2MASS J10444803−5954297 O7: +V +e +— +O8: +V +e +— +M23a — +— +— +— +— +ALS 15 247 +B1 +V +— +— +— +— +— +— +— +— +— +— +— +— +2MASS J10432014−5917582 B2 +V +— +— +— +— +— +— +— +— +— +— +— +— +2MASS J10441242−5934091 B2 +V +n +— +— +— +— +— +— +— +— +— +— +— +2MASS J10435413−5918244 B2 +V +n +— +— +— +— +— +— +— +— +— +— +— +ALS 19 741 +— +— +— +— +— +— +— +— +— +B1 +V +— +— +A16 +ALS 15 243 +— +— +— +— +B0.2 +V +(n) +— +M22 +B0 +V +— +— +M93 +QZ Car B +B1.5 +V +— +— +— +— +— +— +— +— +— +— +— +— +ALS 15 249 +B1 +V +— +— +— +— +— +— +— +B1: +— +— +— +M93 +CPD −59 2539 +B2 +V +— +— +— +— +— +— +— +— +— +— +— +— +ALS 15 248 +B2: +IV +nnn +— +— +— +— +— +— +— +— +— +— +— +2MASS J10440327−5919498 B1.5 +V +— +— +— +— +— +— +— +— +— +— +— +— +[HSB2012] 3192 +B2: +V +ne +— +— +— +— +— +— +— +— +— +— +— +CPD −59 2618 +B1: +V +(n)p He r +— +— +— +— +— +— +B2 +V +— +— +A16 +ALS 15 225 +B1.5 +V +p He rich — +B1.5 +V +p He rich — +TW +— +— +— +— +— +V662 Car +O +— +— +O+B +O5 +V +(n)z +B0: V M16 +O5.5 V +z +O9.5 V N06 +HDE 303 316 B +— +— +— +— +B1 +V +— +— +TW +— +— +— +— +— +ALS 15 958 +B1.5 +V +— +— +— +— +— +— +— +— +— +— +— +— +ALS 15 864 +B1 +V +— +— +— +— +— +— +— +O9 +V +— +— +M88 +CPD −59 2638 A +B2.5 +V +— +— +— +— +— +— +— +— +— +— +— +— +ALS 15 203 A +B +— +— +B +B0.5 +V +— +B1: V TW +O7 +V +— +— +V93 +ALS 19 733 +B1 +V +(n) +— +B1.5: V +— +— +TW +B1 +V +— +— +M93 +2MASS J10440973−6000432 B1.5 +V +e +— +— +— +— +— +— +— +— +— +— +— +ALS 19 735 +B1 +V +— +— +— +— +— +— +— +B2 +V +— +— +M93 +2MASS J10441879−5951490 B1.5 +V +(n) +— +— +— +— +— +— +— +— +— +— +— +2MASS J10443223−5933592 B2: +V +n +— +— +— +— +— +— +— +— +— +— +— +HD 93 129 C +B0.5 +V +(n) +— +— +— +— +— +— +— +— +— +— +— +2MASS J10443139−5944080 B2 +V +— +— +— +— +— +— +— +— +— +— +— +— +ALS 16 082 +B0.7 +V +(n) +— +— +— +— +— +— +— +— +— +— +— +ALS 15 203 B +— +— +— +— +B0 +V +— +— +TW +— +— +— +— +— +2MASS J10435902−5933196 B2 +IV +p He rich — +— +— +— +— +— +— +— +— +— +— +2MASS J10454701−6000271 B2 +V +(n) +— +— +— +— +— +— +— +— +— +— +— +2MASS J10432556−5919175 B2 +V +— +— +— +— +— +— +— +— +— +— +— +— +2MASS J10451588−5929563 B2 +IV +n +— +— +— +— +— +— +B +— +— +— +D17 +2MASS J10422722−6000052 B2 +V +(n) +— +— +— +— +— +— +— +— +— +— +— +ALS 19 740 +B1.5 +V +— +— +— +— +— +— +— +B1.5 +V +— +— +M93 +ALS 19 746 +B1.5 +V +— +— +— +— +— +— +— +B1 +V +— +— +M93 +2MASS J10452314−5940033 B2 +V +(n) +— +— +— +— +— +— +— +— +— +— +— +2MASS J10440516−5921165 B1.5 +V +— +— +— +— +— +— +— +— +— +— +— +— +ALS 19 742 +B1.5 +V +— +— +— +— +— +— +— +B2 +V +— +— +M93 +2MASS J10433865−5934444 B1: +IV +nnn +— +— +— +— +— +— +B +— +— +— +D17 +2MASS J10441729−5917154 B2.5 +V +— +— +— +— +— +— +— +— +— +— +— +— +2MASS J10445053−5957227 B2 +IV +— +— +— +— +— +— +— +— +— +— +— +— +2MASS J10425900−6000240 B2 +V +— +— +— +— +— +— +— +— +— +— +— +— +2MASS J10435207−5932401 B2 +V +(n) +— +— +— +— +— +— +— +— +— +— +— +26 + +Berlanas et al.: Massive stars in the Carina Nebula +Table A.2. (Continued). +Name +Gaia-ESO +GOSSS +LiLiMaRlin + STIS + literature +ST +LC +qual. +sec. +ST +LC qual. sec. +ref. +ST +LC qual. sec. +ref. +2MASS J10435478−6006207 +B2 +V +— +— +— +— +— +— +— +— +— +— +— +— +ALS 19 748 +B1.5 +V +— +— +— +— +— +— +— +B1.5 V +— +— +M93 +ALS 19 732 +— +— +— +— +B2: +V +e +— +TW +B2 +V +— +— +A16 +ALS 19 747 +B2 +V +n +— +— +— +— +— +— +B2 +V +— +— +M93 +2MASS J10435198−6010368 +B2.5 +V +n +— +— +— +— +— +— +— +— +— +— +— +2MASS J10442909−5948207 +B0 +V +— +— +B0 +V +— +— +TW +— +— +— +— +— +ALS 17 185 +B1 +V +— +— +— +— +— +— +— +B2 +III +— +— +A16 +2MASS J10451925−5929522 +B1 +V +— +— +— +— +— +— +— +B +I +— +— +D17 +[HSB2012] 3545 +B2.5: +IV +nn +— +— +— +— +— +— +B +— +— +— +D17 +2MASS J10440371−5948141 +B1 +V +— +— +— +— +— +— +— +— +— +— +— +— +2MASS J10434797−6001201 +B2.5 +V +(n) +— +— +— +— +— +— +— +— +— +— +— +2MASS J10443829−6005449 +B2 +V +(n) +— +— +— +— +— +— +— +— +— +— +— +2MASS J10454824−5922041 +B2.5 +V +n +— +— +— +— +— +— +— +— +— +— +— +2MASS J10441829−5942296 +B2: +V +p He rich — +— +— +— +— +— +— +— +— +— +— +2MASS J10445080−5918005 +B2 +V +— +— +— +— +— +— +— +— +— +— +— +— +[HSB2012] 3416 +B2.5: +V +nn +— +— +— +— +— +— +— +— +— +— +— +2MASS J10451894−5942184 +B2 +V +— +— +— +— +— +— +— +B2 +V +— +— +V15 +2MASS J10442894−5943473 +B2 +V +n +— +— +— +— +— +— +— +— +— +— +— +2MASS J10413434−5958474 +B1 +V +— +— +— +— +— +— +— +— +— +— +— +— +ALS 16 078 +B2 +V +n +— +— +— +— +— +— +— +— +— +— +— +2MASS J10444278−5921383 +B2 +V +— +— +— +— +— +— +— +— +— +— +— +— +2MASS J10433443−5943264 +B0 +V +— +— +— +— +— +— +— +— +— +— +— +— +2MASS J10450836−5938475 +B2 +V +(n) +— +— +— +— +— +— +B +— +— +— +D17 +2MASS J10453807−5944095 +O8 +V +z +— +— +— +— +— +— +— +— +— +— +— +2MASS J10430420−5948591 +B2 +V +(n) +— +— +— +— +— +— +— +— +— +— +— +2MASS J10444235−5922029 +B2 +V +(n) +— +— +— +— +— +— +B2 +V +— +— +A16 +2MASS J10453134−5941133 +B2 +V +— +— +— +— +— +— +— +B +— +— +— +D17 +2MASS J10444726−5928154 +B2 +V +(n) +— +— +— +— +— +— +— +— +— +— +— +Gaia DR3 5350302097055370496 B2 +V +(n) +— +— +— +— +— +— +— +— +— +— +— +2MASS J10434812−5950443 +B2 +V +— +— +— +— +— +— +— +— +— +— +— +— +2MASS J10443591−5923356 +B2.5 +V +n +— +— +— +— +— +— +B2 +V +— +— +A16 +2MASS J10440236−5952046 +B2 +V +— +— +— +— +— +— +— +— +— +— +— +— +2MASS J10460277−5950192 +B2: +V +nnn +— +B2: +V +nnn +— +TW +— +— +— +— +— +[HSB2012] 3314 +B2.5: +V +nn +— +— +— +— +— +— +— +— +— +— +— +2MASS J10440744−5916399 +O9.7: — +— +B0.5: +— +— +— +— +— +— +— +— +— +— +2MASS J10443822−5943056 +B2 +V +nnn +— +— +— +— +— +— +B2 +V +(e) +— +V15 +2MASS J10440576−5927078 +B2 +V +— +— +— +— +— +— +— +— +— +— +— +— +2MASS J10443510−5923281 +B2 +V +— +— +— +— +— +— +— +O9 +V: +— +— +G11b +2MASS J10444550−5952537 +B1: +V +— +B1.5: V — +— +— +— +— +— +— +— +— +— +2MASS J10454536−5958530 +— +— +— +— +B0.5 +V +— +— +TW +B1 +III +— +— +A16 +2MASS J10451297−5946059 +B2 +V +n +— +— +— +— +— +— +— +— +— +— +— +Gaia DR3 5350388683629610752 B2 +V +n +— +— +— +— +— +— +— +— +— +— +— +2MASS J10450792−5939011 +B2 +V +n +— +— +— +— +— +— +B +— +— +— +D17 +[HSB2012] 3211 +B2 +V +— +— +— +— +— +— +— +B1 +V +— +— +V15 +[HSB2012] 1498 +B2 +V +(n) +— +— +— +— +— +— +— +— +— +— +— +2MASS J10435501−5936242 +B2 +V +(n) +— +— +— +— +— +— +— +— +— +— +— +2MASS J10444208−5926353 +B2: +V +nnn +— +— +— +— +— +— +— +— +— +— +— +2MASS J10443769−5929316 +B2.5 +V +n +— +— +— +— +— +— +— +— +— +— +— +[ESK2003] 148 +O9.2 +V +(n) +— +O9.2 V +(n) +— +M23a O9 +V +— +— +A16 +2MASS J10471498−5953374 +B6: +III +e +— +— +— +— +— +— +— +— +— +— +— +[HSB2012] 3482 +B2: +V +— +B3: V +— +— +— +— +— +B3 +— +— +— +V15 +2MASS J10464886−5950409 +B1.5 +V +(n) +— +— +— +— +— +— +— +— +— +— +— +2MASS J10453254−5942359 +B2: +V +nnn +— +— +— +— +— +— +B +— +— +— +D17 +2MASS J10440105−6006377 +B2 +V +— +— +— +— +— +— +— +— +— +— +— +— +27 + +Berlanas et al.: Massive stars in the Carina Nebula +Table A.2. (Continued). +Name +Gaia-ESO +GOSSS +LiLiMaRlin + STIS + literature +ST +LC qual. sec. +ST +LC qual. sec. +ref. +ST +LC qual. sec. +ref. +2MASS J10425293−6003478 B0.7 +V +— +— +— +— +— +— +— +— +— +— +— +— +2MASS J10440866−5933488 B2: +V +nnn +— +— +— +— +— +— +— +— +— +— +— +2MASS J10444609−5946056 B2 +V +— +— +— +— +— +— +— +— +— +— +— +— +2MASS J10454060−5937041 B2: +V +nn +— +— +— +— +— +— +B +— +— +— +D17 +2MASS J10470063−5957242 B2 +V +— +— +— +— +— +— +— +— +— +— +— +— +2MASS J10440384−5934344 B2 +V +— +— +— +— +— +— +— +B +I +— +— +D17 +2MASS J10420949−6002265 B0.7 +V +— +— +— +— +— +— +— +— +— +— +— +— +2MASS J10445602−5938530 B2 +V +n +— +— +— +— +— +— +— +— +— +— +— +2MASS J10443089−5914461 O7.5 +II +(f) +— +O7.5 II +(f) +— +M23a O8 +V +— +— +A16 +2MASS J10440683−5936116 B2 +V +nnn +— +— +— +— +— +— +B2 +V +— +— +V15 +2MASS J10431388−5954584 B2 +V +— +— +— +— +— +— +— +— +— +— +— +— +2MASS J10434798−5933590 B2 +V +n +— +— +— +— +— +— +B +— +— +— +D17 +2MASS J10434580−5934359 B2: +V +nnn +— +— +— +— +— +— +— +— +— +— +— +[HSB2012] 3150 +B2 +V +— +— +— +— +— +— +— +B +— +— +— +D17 +2MASS J10452875−5930037 B1.5 +V +p +— +— +— +— +— +— +B +I +— +— +D17 +2MASS J10445837−5932062 B2.5 +V +n +— +— +— +— +— +— +B +— +— +— +D17 +2MASS J10460608−5957394 B2 +V +— +— +— +— +— +— +— +— +— +— +— +— +2MASS J10442945−5933437 B2: +V +nnn +— +— +— +— +— +— +B +— +— +— +D17 +2MASS J10460257−5957372 B2: +V +ne +— +— +— +— +— +— +— +— +— +— +— +2MASS J10452056−5942212 B2: +V +nne +— +— +— +— +— +— +— +— +— +— +— +2MASS J10453834−5942078 B +— +e +— +— +— +— +— +— +— +— +— +— +— +2MASS J10443766−5923073 — +— +— +— +— +— +— +— +— +B1 +V: +— +— +A16 +2MASS J10444710−5939201 B2: +V +nn +— +— +— +— +— +— +B +— +— +— +D17 +[HSB2012] 3526 +B2 +V +(n) +— +— +— +— +— +— +— +— +— +— +— +2MASS J10452415−5942313 B2.5: V +nn +— +— +— +— +— +— +B +— +— +— +D17 +2MASS J10452190−5945249 B2 +V +— +— +— +— +— +— +— +B2 +V +— +— +V15 +2MASS J10433596−5933179 B2 +V +— +— +— +— +— +— +— +— +— +— +— +— +2MASS J10463643−5948048 B1.5 +V +— +— +— +— +— +— +— +— +— +— +— +— +2MASS J10453185−6000293 O8.5 +V +— +— +O7.5 V +— +— +M23a — +— +— +— +— +2MASS J10435009−5947024 B2.5 +V +(n) +— +— +— +— +— +— +— +— +— +— +— +2MASS J10462657−5956131 B1.5: V +— +— +— +— +— +— +— +— +— +— +— +— +2MASS J10460291−5950259 B2 +V +— +— +B2 +V +— +— +TW +— +— +— +— +— +2MASS J10453819−5942157 B +— +e +— +— +— +— +— +— +B +— +— +— +D17 +[ARV2008] 217 +O3: +III: (n) +— +O3: +III: — +— +M23a — +— +— +— +— +2MASS J10434303−5945333 B2: +V +e +— +— +— +— +— +— +— +— +— +— +— +2MASS J10460116−5949420 B2: +V +— +— +— +— +— +— +— +— +— +— +— +— +2MASS J10454661−5948404 B0.7: V +e +— +— +— +— +— +— +— +— +— +— +— +2MASS J10431945−5944488 sdO +— +— +sec +— +— +— +— +— +— +— +— +— +— +2MASS J10454595−5949075 — +— +— +— +— +— +— +— +— +O9-B1 — +— +— +P21 +2MASS J10453024−5948206 — +— +— +— +— +— +— +— +— +O5-6 +— +— +— +P21 +2MASS J10452013−5950104 — +— +— +— +— +— +— +— +— +O8-9 +— +— +— +P21 +2MASS J10452648−5946188 — +— +— +— +— +— +— +— +— +B1-2 +— +— +— +P21 +2MASS J10451717−5947013 — +— +— +— +— +— +— +— +— +O9-B1 — +— +— +P21 +2MASS J10453470−5947537 — +— +— +— +— +— +— +— +— +O9-B1 — +— +— +P21 +2MASS J10450879−5950537 — +— +— +— +— +— +— +— +— +O6-7 +— +— +— +P21 +2MASS J10452862−5947553 — +— +— +— +— +— +— +— +— +O6-7 +— +— +— +P21 +A02: Albacete Colombo et al. (2002), A16: Alexander et al. (2016), D17: Damiani et al. (2017), F80: Feinstein et al. (1980), +F81: Forte & Orsatti (1981), G11a: Gvaramadze et al. (2011), G11b: Gagn´e et al. (2011), H06: Hamann et al. (2006), K93: +Kilkenny (1993), L76: Loden et al. (1976), L81: Levato & Malaroda (1981), L82: Levato & Malaroda (1982), M01: Massey +et al. (2001), M16: Ma´ız Apell´aniz et al. (2016), M17: Ma´ız Apell´aniz et al. (2017), M20: Ma´ız Apell´aniz et al. (2020), M22: +Ma´ız Apell´aniz et al. (2022a), M23a: GOSSS IV, M23b: Villafranca III, M55: Morgan et al. (1955), M88: Morrell et al. +(1988), M93: Massey & Johnson (1993), N06: Niemel¨a et al. (2006), P11: Povich et al. (2011), P21: Preibisch et al. (2021), +R09: Rauw et al. (2009), S14: Sota et al. (2014), T73: Thackeray et al. (1973), T74: Thackeray & Andrews (1974), TW: +This work, V01: van der Hucht (2001), V15: Vaidya et al. (2015), V93: Vijapurkar & Drilling (1993), W02: Walborn et al. +(2002b), W09: Walborn (2009), W11: Wolk et al. (2011), W72: Walborn (1972), W73a: Walborn (1973b), W73b: Walborn +(1973a), W77: Walborn & Liller (1977), W82: Walborn (1982b), W84: Walsh (1984). +28 + +Berlanas et al.: Massive stars in the Carina Nebula +Table A.3. Stars of the census whose distance is not compatible with Car OB1 and are located in the foreground and in the +background. See Table A.2 for reference acronyms. +Name +Gaia-ESO +GOSSS +LiLiMaRlin + STIS + literature group +ST +LC qual. +sec. +ST +LC qual. sec. +ref. +ST +LC qual. sec. +ref. +HD 93 420 +— +— +— +— +— +— +— +— +— +M4 +Ib +— +— +H72 +foreground +HD 93 342 +— +— +— +— +— +— +— +— +— +B1 +Ia +— +— +A16 +background +HD 93 501 +— +— +— +— +— +— +— +— +— +B0 +V +— +— +A16 +foreground +CPD −59 2592 +— +— +— +— +— +— +— +— +— +B1 +Ib +— +— +A16 +background +HDE 305 439 A +B0 +Ia +— +— +— +— +— +— +— +B0 +Ia +— +— +V93 +background +HDE 305 439 B +— +— +— +— +— +— +— +— +— +B0.7 +Ib +— +— +M23b +background +ALS 15 860 +B1 +Iab — +— +B1 +Iab — +— +TW +O9.5 +I/II — +— +F80 +background +CPD −58 2634 +B1.5 +V +— +— +— +— +— +— +— +B3/6 +— +— +— +L76 +foreground +CPD −59 2535 +— +— +— +— +— +— +— +— +— +B2 +V +— +— +A16 +background +2MASS J10424476−6005020 B0.2 +V +— +B0.2 V B0 +IV: +— +— +TW +O: +— +— +— +P11 +background +2MASS J10441791−5925204 B2.5 +V +— +— +— +— +— +— +— +— +— +— +— +— +background +2MASS J10444803−5954297 O7: +V +e +— +O8: V +e +— +M22b — +— +— +— +— +background +2MASS J10441242−5934091 B2 +V +n +— +— +— +— +— +— +— +— +— +— +— +background +CPD −59 2539 +B2 +V +— +— +— +— +— +— +— +— +— +— +— +— +background +2MASS J10413434−5958474 B1 +V +— +— +— +— +— +— +— +— +— +— +— +— +background +2MASS J10440744−5916399 O9.7: — +— +B0.5: +— +— +— +— +— +— +— +— +— +— +background +2MASS J10471498−5953374 B6: +III +e +— +— +— +— +— +— +— +— +— +— +— +background +2MASS J10425293−6003478 B0.7 +V +— +— +— +— +— +— +— +— +— +— +— +— +background +2MASS J10420949−6002265 B0.7 +V +— +— +— +— +— +— +— +— +— +— +— +— +background +2MASS J10431388−5954584 B2 +V +— +— +— +— +— +— +— +— +— +— +— +— +background +2MASS J10431945−5944488 sdO +— +— +sec +— +— +— +— +— +— +— +— +— +— +foreground +2MASS J10452648−5946188 — +— +— +— +— +— +— +— +— +B1-2 — +— +— +P21 +background +Table A.4. Stars identified in the Car OB1 region as of WR or non-O supergiant (II to Ia) spectral class. See Table A.2 for reference +acronyms. +Name +Gaia-ESO +GOSSS +LiLiMaRlin + STIS + literature +group +ST +LC qual. sec. +ST +LC qual. sec. +ref. +ST +LC +qual. +sec. +ref. +η Car +— +— +— +— +LBV +— +— +— +M22a F: +I +— +— +W77 +O-025 +WR 24 +— +— +— +— +WN6 — +ha +— +TW +WN6 — +ha-w... — +H06 +O-028 +HD 93 281 +— +— +— +— +— +— +— +— +— +M1.5 +Iab: +— +— +M23b O-028 +HDE 303 310 +— +— +— +— +— +— +— +— +— +M3 +Iab +— +— +M23b O-027 +CPD −58 2605 B0.5 II +— +— +— +— +— +— +— +B0 +III-IV: — +— +M88 +O-002 +CPD −59 2504 B7 +II: +nn +— +— +— +— +— +— +B2/5 +— +— +— +L76 +O-028 +Table A.5. Spectroscopic binary systems identified in the census containing at least one O-type star. See Table A.2 for reference +acronyms. +Name +Gaia-ESO +GOSSS +LiLiMaRlin + STIS + literature +group +Notes +ST +LC qual. +sec. +ST +LC +qual. +sec. +ref. +ST +LC +qual. +sec. +ref. +QZ Car Aa,Ac +— +— +— +— +O9.7 Ib +n +— +S14 +O9.7 +Ib +— +O9 II: +M22b O-028 +new SB2 +HD 93 129 Aa +— +— +— +— +O2 +I +f* +— +S14 +O2 +I +f* +OB? +M17 +O-002 +HD 93 403 +— +— +— +— +O5 +I +fc +O7.5 V +M22b O5 +I +fc +O7.5 V +M22b Car OB1 new SB2 +HD 93 205 +— +— +— +— +O3.5 V +((f)) +O8 V +S14 +O3.5 +V +((f)) +O8 V +M20 +O-003 +HD 93 162 +— +— +— +— +O2.5 I +f*/WN6 OB +S14 +WN6 +— +h +O4 +V01 +O-003 +HD 93 161 A +O7 +V +((f)) +O9 IV +O7.5 V +— +O9 V +S14 +O7 +V +((f)) +O9 IV +M22b O-002 +V572 Car +O6.5 +V +z +B0 V + B0.5: V O7.5 V +(n) +B0 V(n) +M16 +O6.5 +V +z +B0 V +B0.2 V M22b O-025 +new SB3 +CPD −59 2554 +O9.2 +V +— +B1: V +O9.5 IV +— +— +S14 +O9.2 +V +— +B1: V +M22b O-028 +new SB2 +CPD −59 2641 +— +— +— +— +O6 +V +((fc)) +— +S14 +O5.5-6 V +((fc)) B2 V-III +R09 +O-025 +CPD −59 2635 +O8 +V +z +O9.2 V +O8 +V +(n) +O9.5 V +S14 +O8 +V +z +O9.2 V +M22b O-025 +HD 93 343 +— +— +— +— +O8 +V +— +— +M16 +O7.5 +V +z +O7.5: V(n) +M22a +O-025 +V573 Car +— +— +— +— +O9.5 V +(n) +B0.5 V(n) +S14 +O9.5 +IV +— +B0.5 V +M22b O-025 +CPD −59 2636 A,B +O7.5 +V +— +O8 V + O8 V +O8 +V +— +O8 V +S14 +O7 +V +— +O8 V + O9 V +A02 +O-025 +HDE 305 525 +— +— +— +— +O5.5 V +((f))z +O7.5 V + B M16 +O4 +V +— +— +V93 +O-030 +HDE 303 312 +O9.5 +III +— +B0.5: V +O9.7 IV +— +— +S14 +O9 +V +— +— +F81 +Car OB1 +new SB2 +CPD −58 2649 A +O9.5: — +— +B0: +O9.7 III: — +B0: V: +M22b O7 +V +— +O8 V +A16 +Car OB1 +CPD −59 2591 +O8.5 +V +— +B0/1 +O8 +V +z +B0.5: V: +M16 +O8 +V +z +B0.5: V: +M22b O-028 +ALS 15 204 +— +— +— +— +O7.5 V +z +O9: V +M16 +— +— +— +— +— +O-002 +V662 Car +O +— +— +O+B +O5 +V +(n)z +B0: V +M16 +O5.5 +V +z +O9.5 V +N06 +Car OB1 +2MASS J10440744−5916399 O9.7: — +— +B0.5: +— +— +— +— +— +— +— +— +— +— +back. +new SB2 +29 + +Berlanas et al.: Massive stars in the Carina Nebula +Table A.6. Spectroscopic binary systems identified in the census formed by early B-type stars. See Table A.2 for reference acronyms. +Name +Gaia-ESO +GOSSS +LiLiMaRlin + STIS + literature +group +Notes +ST +LC qual. +sec. +ST +LC qual. +sec. +ref. +ST +LC qual. +sec. +ref. +HDE 305 534 +— +— +— +— +— +— +— +— +— +B0 +V +— +B0 V +A16 +O-028 +HDE 305 522 +B0.2 +V +— +B1: V +— +— +— +— +— +B0 +V +— +B1 V +A16 +O-028 +HDE 305 543 +— +— +— +— +— +— +— +— +— +B0.2 +V +(n) +B1: V(n) M23b O-028 +new SB2 +HDE 303 300 +B0.5 +V +— +B +— +— +— +— +— +B1 +V +— +— +A16 +Car OB1 new SB2 +HDE 305 516 +B0.5 +V +(n) +B2: V +— +— +— +— +— +B0.5 +V +— +— +G11b +O-028 +new SB2 +HDE 303 313 +B2 +V +— +B2 V +— +— +— +— +— +B2 +V +— +B2 V +A16 +Car OB1 +HD 93 128 B +B0.2: V +— +B1: V +B0.2 V +— +— +M22a — +— +— +— +— +O-002 +new SB2 +ALS 15 219 +B0.5: V +— +B1: V +B0.7 V +— +B1 V +M22a B1 +V +— +— +M93 +O-002 +CPD −59 2583 +B1.5: V +— +B2: V +— +— +— +— +— +B1 +V +— +— +A16 +Car OB1 new SB2 +CPD −58 2640 +B1.5 +V +— +B2: V +— +— +— +— +— +B1 +V +— +B2 V +A16 +O-029 +2MASS J10424476−6005020 B0.2 +V +— +B0.2 V +B0 +IV: +— +— +TW +O: +— +— +— +P11 +back. +new SB2 +CPD −59 2661 +B0: +— +— +B +B0 +V +(n)e +B2: V TW +O9.5 V +— +— +W84 +O-030 +new SB2 +2MASS J10431897−5946569 B2: +V +(n) +B +— +— +— +— +— +— +— +— +— +— +O-028 +new SB2 +ALS 15 237 +B1.5: V +— +B2: V +— +— +— +— +— +— +— +— +— +— +O-029 +new SB2 +CPD −59 2625 +B1.5: V +— +B2: V +— +— +— +— +— +B2 +V +— +— +A16 +O-025 +new SB2 +ALS 15 203 A +B +— +— +B +B0.5 V +— +B1: V TW +O7 +V +— +— +V93 +O-002 +new SB2 +2MASS J10444550−5952537 B1: +V +— +B1.5: V — +— +— +— +— +— +— +— +— +— +O-028 +new SB2 +[HSB2012] 3482 +B2: +V +— +B3: V +— +— +— +— +— +B3 +— +— +— +V15 +O-025 +new SB2 +Table A.7. Proper motions from Gaia EDR3 for the identified runaway candidates. +Name +Gaia Source +µα∗ +µδ +RUWE +group +n-qualifer +[mas a−1] +[mas a−1] +2MASS J10440866−5933488 5350363085623074432 -7.181 ± 0.015 3.419 ± 0.014 +0.972 +O-002 +nnn +CPD −58 2657 +5350395383780746496 -6.622 ± 0.329 0.385 ± 0.300 +12.985 +O-027 +(n) +HDE 303 310 +5350389095946525568 -7.432 ± 0.031 2.851 ± 0.026 +0.704 +O-027 +CPD −59 2541 +5254271056429426688 -5.647 ± 0.032 2.545 ± 0.031 +1.582 +O-028 +(n) +HD 93 281 +5350302097082780416 -8.028 ± 0.022 2.029 ± 0.020 +1.008 +O-028 +QZ Car Aa,Ac +5350346970905044480 -5.750 ± 0.123 2.455 ± 0.108 +3.471 +O-028 +HDE 305 523 +5350347520660979840 -5.804 ± 0.025 2.165 ± 0.023 +1.083 +O-028 +2MASS J10451588−5929563 5350384629180484224 -6.893 ± 0.011 4.030 ± 0.010 +0.797 +O-029 +n +30 + +Berlanas et al.: Massive stars in the Carina Nebula +Hγ +Hβ +He I 4388 +He I 4438 +He I 4471 +He I 4713 +He I 4922 +He I 5016 +He I 5048 +He II 4200 +He II 4542 +He II 4686 +C II 4267 +C III 4326 +C III 4647 / 50 / 51 +N II 4530 +N II 4602 / 07 +N II 4614 / 21 +N II 4780 / 88 +N II 4803 +N II 5001 / 04 / 07 +N III 4379 +N III 4511 / 15 +N III 4634 / 41 / 42 +O II 4254 +O II 4276 / 85 +O II 4317 / 20 +O II 4349 / 67 +O II 4415 / 17 +O II 4448 / 52 +O II 4591 / 96 +O II 4662 / 76 +O II 4699 / 705 +O II 4907 +O II 4943 +Si III 4553 +Si III 4568 / 75 +Si III 4813 / 20 / 29 +Si IV 4631 +Mg II 4481 +DIB is 4428 +DIB is 4502 +DIB is 4727 +DIB is 4762 / 65 / 80 +DIB is 4880 / 87 +DIB is 4964 +CH is 4300 +CH+ is 4233 +HD 93 249 A O9 III +HD 93 204 O5.5 V((f)) +HD 93 161 B O6.5 IV((f)) +HD 93 161 A O7 V((f)) + O9 IV +HD 93 027 O9.5 IV +V572 Car O6.5 Vz + B0 V + B0.5: V +CPD −59 2554 O9.2 V + B1: V +HD 93 056 B1: V:n +4200 +4300 +4400 +4500 +4600 +4700 +4800 +4900 +5000 +Wavelength (Å) +0 +1 +2 +3 +4 +5 +Fig. A.1. UVES spectra shown at their original resolution. +31 + +Berlanas et al.: Massive stars in the Carina Nebula +Hγ +Hβ +He I 4388 +He I 4438 +He I 4471 +He I 4713 +He I 4922 +He I 5016 +He I 5048 +He II 4200 +He II 4542 +He II 4686 +C II 4267 +C III 4326 +C III 4647 / 50 / 51 +N II 4530 +N II 4602 / 07 +N II 4614 / 21 +N II 4780 / 88 +N II 4803 +N II 5001 / 04 / 07 +N III 4379 +N III 4511 / 15 +N III 4634 / 41 / 42 +O II 4254 +O II 4276 / 85 +O II 4317 / 20 +O II 4349 / 67 +O II 4415 / 17 +O II 4448 / 52 +O II 4591 / 96 +O II 4662 / 76 +O II 4699 / 705 +O II 4907 +O II 4943 +Si III 4553 +Si III 4568 / 75 +Si III 4813 / 20 / 29 +Si IV 4631 +Mg II 4481 +DIB is 4428 +DIB is 4502 +DIB is 4727 +DIB is 4762 / 65 / 80 +DIB is 4880 / 87 +DIB is 4964 +CH is 4300 +CH+ is 4233 +CPD −59 2635 O8 Vz + O9.2 V +HDE 305 524 O6.5 Vn((f))z +HD 93 249 B B0: V(n) +HDE 305 535 B4 III(n) +HDE 305 452 B2 III +CPD −59 2636 AB O7.5 V + O8 V + O8 V +HD 93 097 B0.2 V:n +HDE 305 516 B0.5 V(n) + B2: V +4200 +4300 +4400 +4500 +4600 +4700 +4800 +4900 +5000 +Wavelength (Å) +0 +1 +2 +3 +4 +5 +Fig. A.1. (Continued). +32 + +Berlanas et al.: Massive stars in the Carina Nebula +Hδ +Hγ +He I 4121 +He I 4144 +He I 4169 +He I 4388 +He I 4438 +He I 4471 +He I 4713 +He II 4200 +He II 4542 +He II 4686 +C II 4267 +C III 4068 / 69 / 70 +C III 4187 +C III 4326 +C III 4647 / 50 / 51 +N II 4530 +N II 4602 / 07 +N II 4614 / 21 +N III 4097 +N III 4379 +N III 4511 / 15 +N III 4634 / 41 / 42 +N IV 4058 +O II 4070 / 72 / 76 +O II 4133 +O II 4154 +O II 4186 / 90 +O II 4254 +O II 4276 / 85 +O II 4317 / 20 +O II 4349 / 67 +O II 4415 / 17 +O II 4448 / 52 +O II 4591 / 96 +O II 4662 / 76 +O II 4699 / 705 +Si III 4553 +Si III 4568 / 75 +Si IV 4089 +Si IV 4116 +Si IV 4631 +Mg II 4481 +DIB is 4428 +DIB is 4502 +DIB is 4727 +Ca I is 4227 +CH is 4300 +HDE 305 439 A B0 Ia +CPD −59 2624 O9.7 V +HDE 305 522 B0.2 V + B1: V +HDE 303 300 B0.5 V + B +CPD −59 2626 AB O7.5 V(n) +HDE 303 312 O9.5 III + B0.5: V +CPD −58 2605 B0.5 II +CPD −59 2644 O9 V +4100 +4200 +4300 +4400 +4500 +4600 +4700 +Wavelength (Å) +0 +1 +2 +3 +4 +5 +Fig. A.2. GIRAFFE spectra shown at a resolution R = 2500. +33 + +Berlanas et al.: Massive stars in the Carina Nebula +Hδ +Hγ +He I 4121 +He I 4144 +He I 4169 +He I 4388 +He I 4438 +He I 4471 +He I 4713 +He II 4200 +He II 4542 +He II 4686 +C II 4267 +C III 4068 / 69 / 70 +C III 4187 +C III 4326 +C III 4647 / 50 / 51 +N II 4530 +N II 4602 / 07 +N II 4614 / 21 +N III 4097 +N III 4379 +N III 4511 / 15 +N III 4634 / 41 / 42 +N IV 4058 +O II 4070 / 72 / 76 +O II 4133 +O II 4154 +O II 4186 / 90 +O II 4254 +O II 4276 / 85 +O II 4317 / 20 +O II 4349 / 67 +O II 4415 / 17 +O II 4448 / 52 +O II 4591 / 96 +O II 4662 / 76 +O II 4699 / 705 +Si III 4553 +Si III 4568 / 75 +Si IV 4089 +Si IV 4116 +Si IV 4631 +Mg II 4481 +DIB is 4428 +DIB is 4502 +DIB is 4727 +Ca I is 4227 +CH is 4300 +CPD −58 2656 B0.2 V +CPD −58 2623 B0.2 V +CPD −59 2627 O9.5 IV +CPD −58 2649 A O9.5: + B0: +CPD −58 2627 O9.5 V(n) +CPD −59 2673 O5.5 V(n)((f))z +ALS 15 210 O3.5 If* Nwk +HDE 303 313 B2 V + B2 V +4100 +4200 +4300 +4400 +4500 +4600 +4700 +Wavelength (Å) +0 +1 +2 +3 +4 +5 +Fig. A.2. (Continued). +34 + +Berlanas et al.: Massive stars in the Carina Nebula +Hδ +Hγ +He I 4121 +He I 4144 +He I 4169 +He I 4388 +He I 4438 +He I 4471 +He I 4713 +He II 4200 +He II 4542 +He II 4686 +C II 4267 +C III 4068 / 69 / 70 +C III 4187 +C III 4326 +C III 4647 / 50 / 51 +N II 4530 +N II 4602 / 07 +N II 4614 / 21 +N III 4097 +N III 4379 +N III 4511 / 15 +N III 4634 / 41 / 42 +N IV 4058 +O II 4070 / 72 / 76 +O II 4133 +O II 4154 +O II 4186 / 90 +O II 4254 +O II 4276 / 85 +O II 4317 / 20 +O II 4349 / 67 +O II 4415 / 17 +O II 4448 / 52 +O II 4591 / 96 +O II 4662 / 76 +O II 4699 / 705 +Si III 4553 +Si III 4568 / 75 +Si IV 4089 +Si IV 4116 +Si IV 4631 +Mg II 4481 +DIB is 4428 +DIB is 4502 +DIB is 4727 +Ca I is 4227 +CH is 4300 +CPD −59 2593 B2.5: IVnnn +ALS 15 205 B0.2 V(n) +CPD −59 2469 B9 III +HDE 305 533 B0.5 V(n) +ALS 15 860 B1 Iab +CPD −58 2625 O9.5 V +HD 93 249 C B1.5 V(n) +CPD −58 2634 B1.5 V +4100 +4200 +4300 +4400 +4500 +4600 +4700 +Wavelength (Å) +0 +1 +2 +3 +4 +5 +Fig. A.2. (Continued). +35 + +Berlanas et al.: Massive stars in the Carina Nebula +Hδ +Hγ +He I 4121 +He I 4144 +He I 4169 +He I 4388 +He I 4438 +He I 4471 +He I 4713 +He II 4200 +He II 4542 +He II 4686 +C II 4267 +C III 4068 / 69 / 70 +C III 4187 +C III 4326 +C III 4647 / 50 / 51 +N II 4530 +N II 4602 / 07 +N II 4614 / 21 +N III 4097 +N III 4379 +N III 4511 / 15 +N III 4634 / 41 / 42 +N IV 4058 +O II 4070 / 72 / 76 +O II 4133 +O II 4154 +O II 4186 / 90 +O II 4254 +O II 4276 / 85 +O II 4317 / 20 +O II 4349 / 67 +O II 4415 / 17 +O II 4448 / 52 +O II 4591 / 96 +O II 4662 / 76 +O II 4699 / 705 +Si III 4553 +Si III 4568 / 75 +Si IV 4089 +Si IV 4116 +Si IV 4631 +Mg II 4481 +DIB is 4428 +DIB is 4502 +DIB is 4727 +Ca I is 4227 +CH is 4300 +HD 93 128 B B0.2: V + B1: V +CPD −59 2629 O8.5 Vp +CPD −58 2657 B0.7 V(n) +CPD −59 2591 O8.5 V + B0/1 +CPD −58 2644 B1.5 V +CPD −59 2632 B0.5 Vn +CPD −59 2495 B1.5 V +CPD −58 2655 B1: V +4100 +4200 +4300 +4400 +4500 +4600 +4700 +Wavelength (Å) +0 +1 +2 +3 +4 +5 +Fig. A.2. (Continued). +36 + +Berlanas et al.: Massive stars in the Carina Nebula +Hδ +Hγ +He I 4121 +He I 4144 +He I 4169 +He I 4388 +He I 4438 +He I 4471 +He I 4713 +He II 4200 +He II 4542 +He II 4686 +C II 4267 +C III 4068 / 69 / 70 +C III 4187 +C III 4326 +C III 4647 / 50 / 51 +N II 4530 +N II 4602 / 07 +N II 4614 / 21 +N III 4097 +N III 4379 +N III 4511 / 15 +N III 4634 / 41 / 42 +N IV 4058 +O II 4070 / 72 / 76 +O II 4133 +O II 4154 +O II 4186 / 90 +O II 4254 +O II 4276 / 85 +O II 4317 / 20 +O II 4349 / 67 +O II 4415 / 17 +O II 4448 / 52 +O II 4591 / 96 +O II 4662 / 76 +O II 4699 / 705 +Si III 4553 +Si III 4568 / 75 +Si IV 4089 +Si IV 4116 +Si IV 4631 +Mg II 4481 +DIB is 4428 +DIB is 4502 +DIB is 4727 +Ca I is 4227 +CH is 4300 +CPD −59 2614 B1 V +ALS 15 219 B0.5: V + B1: V +CPD −59 2581 B0.5 V(n) +[ARV2008] 206 O6 V((f)) +CPD −59 2583 B1.5: V + B2: V +CPD −58 2640 B1.5 V + B2: V +CPD −59 2606 B0.7 Vn +Tyc 8626−02506−1 O9 V(n) +4100 +4200 +4300 +4400 +4500 +4600 +4700 +Wavelength (Å) +0 +1 +2 +3 +4 +5 +Fig. A.2. (Continued). +37 + +Berlanas et al.: Massive stars in the Carina Nebula +Hδ +Hγ +He I 4121 +He I 4144 +He I 4169 +He I 4388 +He I 4438 +He I 4471 +He I 4713 +He II 4200 +He II 4542 +He II 4686 +C II 4267 +C III 4068 / 69 / 70 +C III 4187 +C III 4326 +C III 4647 / 50 / 51 +N II 4530 +N II 4602 / 07 +N II 4614 / 21 +N III 4097 +N III 4379 +N III 4511 / 15 +N III 4634 / 41 / 42 +N IV 4058 +O II 4070 / 72 / 76 +O II 4133 +O II 4154 +O II 4186 / 90 +O II 4254 +O II 4276 / 85 +O II 4317 / 20 +O II 4349 / 67 +O II 4415 / 17 +O II 4448 / 52 +O II 4591 / 96 +O II 4662 / 76 +O II 4699 / 705 +Si III 4553 +Si III 4568 / 75 +Si IV 4089 +Si IV 4116 +Si IV 4631 +Mg II 4481 +DIB is 4428 +DIB is 4502 +DIB is 4727 +Ca I is 4227 +CH is 4300 +CPD −59 2497 B2 V +ALS 15 229 B0 V +CPD −59 2605 B0 V +CPD −59 2640 B1.5 V(n) +ALS 15 228 B1 V +Gaia DR3 5350358378338864768 B1.5: V(n) +ALS 15 855 B1 V(n) +CPD −59 2504 B7 II:nn +4100 +4200 +4300 +4400 +4500 +4600 +4700 +Wavelength (Å) +0 +1 +2 +3 +4 +5 +Fig. A.2. (Continued). +38 + +Berlanas et al.: Massive stars in the Carina Nebula +Hδ +Hγ +He I 4121 +He I 4144 +He I 4169 +He I 4388 +He I 4438 +He I 4471 +He I 4713 +He II 4200 +He II 4542 +He II 4686 +C II 4267 +C III 4068 / 69 / 70 +C III 4187 +C III 4326 +C III 4647 / 50 / 51 +N II 4530 +N II 4602 / 07 +N II 4614 / 21 +N III 4097 +N III 4379 +N III 4511 / 15 +N III 4634 / 41 / 42 +N IV 4058 +O II 4070 / 72 / 76 +O II 4133 +O II 4154 +O II 4186 / 90 +O II 4254 +O II 4276 / 85 +O II 4317 / 20 +O II 4349 / 67 +O II 4415 / 17 +O II 4448 / 52 +O II 4591 / 96 +O II 4662 / 76 +O II 4699 / 705 +Si III 4553 +Si III 4568 / 75 +Si IV 4089 +Si IV 4116 +Si IV 4631 +Mg II 4481 +DIB is 4428 +DIB is 4502 +DIB is 4727 +Ca I is 4227 +CH is 4300 +CPD −59 2579 B1.5 V +CPD −59 2543 B2 V(n) +ALS 15 227 B0.7 V +CPD −58 2647 B1.5 V +ALS 15 224 B0.5 V +CPD −59 2510 B2 V +CPD −59 2619 B0.7 V +CPD −59 2596 B0 V +4100 +4200 +4300 +4400 +4500 +4600 +4700 +Wavelength (Å) +0 +1 +2 +3 +4 +5 +Fig. A.2. (Continued). +39 + +Berlanas et al.: Massive stars in the Carina Nebula +Hδ +Hγ +He I 4121 +He I 4144 +He I 4169 +He I 4388 +He I 4438 +He I 4471 +He I 4713 +He II 4200 +He II 4542 +He II 4686 +C II 4267 +C III 4068 / 69 / 70 +C III 4187 +C III 4326 +C III 4647 / 50 / 51 +N II 4530 +N II 4602 / 07 +N II 4614 / 21 +N III 4097 +N III 4379 +N III 4511 / 15 +N III 4634 / 41 / 42 +N IV 4058 +O II 4070 / 72 / 76 +O II 4133 +O II 4154 +O II 4186 / 90 +O II 4254 +O II 4276 / 85 +O II 4317 / 20 +O II 4349 / 67 +O II 4415 / 17 +O II 4448 / 52 +O II 4591 / 96 +O II 4662 / 76 +O II 4699 / 705 +Si III 4553 +Si III 4568 / 75 +Si IV 4089 +Si IV 4116 +Si IV 4631 +Mg II 4481 +DIB is 4428 +DIB is 4502 +DIB is 4727 +Ca I is 4227 +CH is 4300 +2MASS J10424532−6012063 B2 V +2MASS J10424476−6005020 B0.2 V + B0.2 V +CPD −58 2653 A B1.5 V +CPD −59 2571 B1.5 Vn +2MASS J10415981−5955075 B2 V +2MASS J10461906−5957543 B0.7 V +CPD −59 2661 B0: + B +ALS 15 232 B1 V +4100 +4200 +4300 +4400 +4500 +4600 +4700 +Wavelength (Å) +0 +1 +2 +3 +4 +5 +Fig. A.2. (Continued). +40 + +Berlanas et al.: Massive stars in the Carina Nebula +Hδ +Hγ +He I 4121 +He I 4144 +He I 4169 +He I 4388 +He I 4438 +He I 4471 +He I 4713 +He II 4200 +He II 4542 +He II 4686 +C II 4267 +C III 4068 / 69 / 70 +C III 4187 +C III 4326 +C III 4647 / 50 / 51 +N II 4530 +N II 4602 / 07 +N II 4614 / 21 +N III 4097 +N III 4379 +N III 4511 / 15 +N III 4634 / 41 / 42 +N IV 4058 +O II 4070 / 72 / 76 +O II 4133 +O II 4154 +O II 4186 / 90 +O II 4254 +O II 4276 / 85 +O II 4317 / 20 +O II 4349 / 67 +O II 4415 / 17 +O II 4448 / 52 +O II 4591 / 96 +O II 4662 / 76 +O II 4699 / 705 +Si III 4553 +Si III 4568 / 75 +Si IV 4089 +Si IV 4116 +Si IV 4631 +Mg II 4481 +DIB is 4428 +DIB is 4502 +DIB is 4727 +Ca I is 4227 +CH is 4300 +ALS 15 233 B0.7 V +2MASS J10431897−5946569 B2: V(n) + B +ALS 19 739 B1: V +ALS 15 235 B1.5 V +CPD −59 2569 B B1.5 V +2MASS J10441791−5925204 B2.5 V +CPD −58 2667 B2 IV +ALS 15 856 B1.5 V +4100 +4200 +4300 +4400 +4500 +4600 +4700 +Wavelength (Å) +0 +1 +2 +3 +4 +5 +Fig. A.2. (Continued). +41 + +Berlanas et al.: Massive stars in the Carina Nebula +Hδ +Hγ +He I 4121 +He I 4144 +He I 4169 +He I 4388 +He I 4438 +He I 4471 +He I 4713 +He II 4200 +He II 4542 +He II 4686 +C II 4267 +C III 4068 / 69 / 70 +C III 4187 +C III 4326 +C III 4647 / 50 / 51 +N II 4530 +N II 4602 / 07 +N II 4614 / 21 +N III 4097 +N III 4379 +N III 4511 / 15 +N III 4634 / 41 / 42 +N IV 4058 +O II 4070 / 72 / 76 +O II 4133 +O II 4154 +O II 4186 / 90 +O II 4254 +O II 4276 / 85 +O II 4317 / 20 +O II 4349 / 67 +O II 4415 / 17 +O II 4448 / 52 +O II 4591 / 96 +O II 4662 / 76 +O II 4699 / 705 +Si III 4553 +Si III 4568 / 75 +Si IV 4089 +Si IV 4116 +Si IV 4631 +Mg II 4481 +DIB is 4428 +DIB is 4502 +DIB is 4727 +Ca I is 4227 +CH is 4300 +2MASS J10460477−5949217 O9.7 V(n) +ALS 15 237 B1.5: V + B2: V +CPD −59 2617 B2 V(n)e +2MASS J10460606−5956339 B2 V(n) +2MASS J10450531−5919347 B2 V(n) +CPD −59 2585 B2 V +CPD −59 2625 B1.5: V + B2: V +ALS 15 245 B0.5 V +4100 +4200 +4300 +4400 +4500 +4600 +4700 +Wavelength (Å) +0 +1 +2 +3 +4 +5 +Fig. A.2. (Continued). +42 + +Berlanas et al.: Massive stars in the Carina Nebula +Hδ +Hγ +He I 4121 +He I 4144 +He I 4169 +He I 4388 +He I 4438 +He I 4471 +He I 4713 +He II 4200 +He II 4542 +He II 4686 +C II 4267 +C III 4068 / 69 / 70 +C III 4187 +C III 4326 +C III 4647 / 50 / 51 +N II 4530 +N II 4602 / 07 +N II 4614 / 21 +N III 4097 +N III 4379 +N III 4511 / 15 +N III 4634 / 41 / 42 +N IV 4058 +O II 4070 / 72 / 76 +O II 4133 +O II 4154 +O II 4186 / 90 +O II 4254 +O II 4276 / 85 +O II 4317 / 20 +O II 4349 / 67 +O II 4415 / 17 +O II 4448 / 52 +O II 4591 / 96 +O II 4662 / 76 +O II 4699 / 705 +Si III 4553 +Si III 4568 / 75 +Si IV 4089 +Si IV 4116 +Si IV 4631 +Mg II 4481 +DIB is 4428 +DIB is 4502 +DIB is 4727 +Ca I is 4227 +CH is 4300 +CPD −58 2677 B1 V +CPD −59 2541 B1.5 V(n) +ALS 15 246 B0.7 V +CPD −59 2616 B2 V(n) +ALS 15 242 B0.7 V +CPD −58 2662 B2 V +2MASS J10444803−5954297 O7: Ve +ALS 15 247 B1 V +4100 +4200 +4300 +4400 +4500 +4600 +4700 +Wavelength (Å) +0 +1 +2 +3 +4 +5 +Fig. A.2. (Continued). +43 + +Berlanas et al.: Massive stars in the Carina Nebula +Hδ +Hγ +He I 4121 +He I 4144 +He I 4169 +He I 4388 +He I 4438 +He I 4471 +He I 4713 +He II 4200 +He II 4542 +He II 4686 +C II 4267 +C III 4068 / 69 / 70 +C III 4187 +C III 4326 +C III 4647 / 50 / 51 +N II 4530 +N II 4602 / 07 +N II 4614 / 21 +N III 4097 +N III 4379 +N III 4511 / 15 +N III 4634 / 41 / 42 +N IV 4058 +O II 4070 / 72 / 76 +O II 4133 +O II 4154 +O II 4186 / 90 +O II 4254 +O II 4276 / 85 +O II 4317 / 20 +O II 4349 / 67 +O II 4415 / 17 +O II 4448 / 52 +O II 4591 / 96 +O II 4662 / 76 +O II 4699 / 705 +Si III 4553 +Si III 4568 / 75 +Si IV 4089 +Si IV 4116 +Si IV 4631 +Mg II 4481 +DIB is 4428 +DIB is 4502 +DIB is 4727 +Ca I is 4227 +CH is 4300 +2MASS J10432014−5917582 B2 V +2MASS J10441242−5934091 B2 Vn +2MASS J10435413−5918244 B2 Vn +QZ Car B B1.5 V +ALS 15 249 B1 V +CPD −59 2539 B2 V +ALS 15 248 B2: IVnnn +2MASS J10440327−5919498 B1.5 V +4100 +4200 +4300 +4400 +4500 +4600 +4700 +Wavelength (Å) +0 +1 +2 +3 +4 +5 +Fig. A.2. (Continued). +44 + +Berlanas et al.: Massive stars in the Carina Nebula +Hδ +Hγ +He I 4121 +He I 4144 +He I 4169 +He I 4388 +He I 4438 +He I 4471 +He I 4713 +He II 4200 +He II 4542 +He II 4686 +C II 4267 +C III 4068 / 69 / 70 +C III 4187 +C III 4326 +C III 4647 / 50 / 51 +N II 4530 +N II 4602 / 07 +N II 4614 / 21 +N III 4097 +N III 4379 +N III 4511 / 15 +N III 4634 / 41 / 42 +N IV 4058 +O II 4070 / 72 / 76 +O II 4133 +O II 4154 +O II 4186 / 90 +O II 4254 +O II 4276 / 85 +O II 4317 / 20 +O II 4349 / 67 +O II 4415 / 17 +O II 4448 / 52 +O II 4591 / 96 +O II 4662 / 76 +O II 4699 / 705 +Si III 4553 +Si III 4568 / 75 +Si IV 4089 +Si IV 4116 +Si IV 4631 +Mg II 4481 +DIB is 4428 +DIB is 4502 +DIB is 4727 +Ca I is 4227 +CH is 4300 +[HSB2012] 3192 B2: Vne +CPD −59 2618 B1: V(n)p He rich +ALS 15 225 B1.5 Vp He rich +V662 Car O + O+B +ALS 15 958 B1.5 V +ALS 15 864 B1 V +CPD −59 2638 A B2.5 V +ALS 15 203 A B + B +4100 +4200 +4300 +4400 +4500 +4600 +4700 +Wavelength (Å) +0 +1 +2 +3 +4 +5 +Fig. A.2. (Continued). +45 + +Berlanas et al.: Massive stars in the Carina Nebula +Hδ +Hγ +He I 4121 +He I 4144 +He I 4169 +He I 4388 +He I 4438 +He I 4471 +He I 4713 +He II 4200 +He II 4542 +He II 4686 +C II 4267 +C III 4068 / 69 / 70 +C III 4187 +C III 4326 +C III 4647 / 50 / 51 +N II 4530 +N II 4602 / 07 +N II 4614 / 21 +N III 4097 +N III 4379 +N III 4511 / 15 +N III 4634 / 41 / 42 +N IV 4058 +O II 4070 / 72 / 76 +O II 4133 +O II 4154 +O II 4186 / 90 +O II 4254 +O II 4276 / 85 +O II 4317 / 20 +O II 4349 / 67 +O II 4415 / 17 +O II 4448 / 52 +O II 4591 / 96 +O II 4662 / 76 +O II 4699 / 705 +Si III 4553 +Si III 4568 / 75 +Si IV 4089 +Si IV 4116 +Si IV 4631 +Mg II 4481 +DIB is 4428 +DIB is 4502 +DIB is 4727 +Ca I is 4227 +CH is 4300 +ALS 19 733 B1 V(n) +2MASS J10440973−6000432 B1.5 Ve +ALS 19 735 B1 V +2MASS J10441879−5951490 B1.5 V(n) +2MASS J10443223−5933592 B2: Vn +HD 93 129 C B0.5 V(n) +2MASS J10443139−5944080 B2 V +ALS 16 082 B0.7 V(n) +4100 +4200 +4300 +4400 +4500 +4600 +4700 +Wavelength (Å) +0 +1 +2 +3 +4 +5 +Fig. A.2. (Continued). +46 + +Berlanas et al.: Massive stars in the Carina Nebula +Hδ +Hγ +He I 4121 +He I 4144 +He I 4169 +He I 4388 +He I 4438 +He I 4471 +He I 4713 +He II 4200 +He II 4542 +He II 4686 +C II 4267 +C III 4068 / 69 / 70 +C III 4187 +C III 4326 +C III 4647 / 50 / 51 +N II 4530 +N II 4602 / 07 +N II 4614 / 21 +N III 4097 +N III 4379 +N III 4511 / 15 +N III 4634 / 41 / 42 +N IV 4058 +O II 4070 / 72 / 76 +O II 4133 +O II 4154 +O II 4186 / 90 +O II 4254 +O II 4276 / 85 +O II 4317 / 20 +O II 4349 / 67 +O II 4415 / 17 +O II 4448 / 52 +O II 4591 / 96 +O II 4662 / 76 +O II 4699 / 705 +Si III 4553 +Si III 4568 / 75 +Si IV 4089 +Si IV 4116 +Si IV 4631 +Mg II 4481 +DIB is 4428 +DIB is 4502 +DIB is 4727 +Ca I is 4227 +CH is 4300 +2MASS J10435902−5933196 B2 IVp He rich +2MASS J10454701−6000271 B2 V(n) +2MASS J10432556−5919175 B2 V +2MASS J10451588−5929563 B2 IVn +2MASS J10422722−6000052 B2 V(n) +ALS 19 740 B1.5 V +ALS 19 746 B1.5 V +2MASS J10452314−5940033 B2 V(n) +4100 +4200 +4300 +4400 +4500 +4600 +4700 +Wavelength (Å) +0 +1 +2 +3 +4 +5 +Fig. A.2. (Continued). +47 + +Berlanas et al.: Massive stars in the Carina Nebula +Hδ +Hγ +He I 4121 +He I 4144 +He I 4169 +He I 4388 +He I 4438 +He I 4471 +He I 4713 +He II 4200 +He II 4542 +He II 4686 +C II 4267 +C III 4068 / 69 / 70 +C III 4187 +C III 4326 +C III 4647 / 50 / 51 +N II 4530 +N II 4602 / 07 +N II 4614 / 21 +N III 4097 +N III 4379 +N III 4511 / 15 +N III 4634 / 41 / 42 +N IV 4058 +O II 4070 / 72 / 76 +O II 4133 +O II 4154 +O II 4186 / 90 +O II 4254 +O II 4276 / 85 +O II 4317 / 20 +O II 4349 / 67 +O II 4415 / 17 +O II 4448 / 52 +O II 4591 / 96 +O II 4662 / 76 +O II 4699 / 705 +Si III 4553 +Si III 4568 / 75 +Si IV 4089 +Si IV 4116 +Si IV 4631 +Mg II 4481 +DIB is 4428 +DIB is 4502 +DIB is 4727 +Ca I is 4227 +CH is 4300 +2MASS J10440516−5921165 B1.5 V +ALS 19 742 B1.5 V +2MASS J10433865−5934444 B1: IVnnn +2MASS J10441729−5917154 B2.5 V +2MASS J10445053−5957227 B2 IV +2MASS J10425900−6000240 B2 V +2MASS J10435207−5932401 B2 V(n) +2MASS J10435478−6006207 B2 V +4100 +4200 +4300 +4400 +4500 +4600 +4700 +Wavelength (Å) +0 +1 +2 +3 +4 +5 +Fig. A.2. (Continued). +48 + +Berlanas et al.: Massive stars in the Carina Nebula +Hδ +Hγ +He I 4121 +He I 4144 +He I 4169 +He I 4388 +He I 4438 +He I 4471 +He I 4713 +He II 4200 +He II 4542 +He II 4686 +C II 4267 +C III 4068 / 69 / 70 +C III 4187 +C III 4326 +C III 4647 / 50 / 51 +N II 4530 +N II 4602 / 07 +N II 4614 / 21 +N III 4097 +N III 4379 +N III 4511 / 15 +N III 4634 / 41 / 42 +N IV 4058 +O II 4070 / 72 / 76 +O II 4133 +O II 4154 +O II 4186 / 90 +O II 4254 +O II 4276 / 85 +O II 4317 / 20 +O II 4349 / 67 +O II 4415 / 17 +O II 4448 / 52 +O II 4591 / 96 +O II 4662 / 76 +O II 4699 / 705 +Si III 4553 +Si III 4568 / 75 +Si IV 4089 +Si IV 4116 +Si IV 4631 +Mg II 4481 +DIB is 4428 +DIB is 4502 +DIB is 4727 +Ca I is 4227 +CH is 4300 +ALS 19 748 B1.5 V +ALS 19 747 B2 Vn +2MASS J10435198−6010368 B2.5 Vn +2MASS J10442909−5948207 B0 V +ALS 17 185 B1 V +2MASS J10451925−5929522 B1 V +[HSB2012] 3545 B2.5: IVnn +2MASS J10440371−5948141 B1 V +4100 +4200 +4300 +4400 +4500 +4600 +4700 +Wavelength (Å) +0 +1 +2 +3 +4 +5 +Fig. A.2. (Continued). +49 + +Berlanas et al.: Massive stars in the Carina Nebula +Hδ +Hγ +He I 4121 +He I 4144 +He I 4169 +He I 4388 +He I 4438 +He I 4471 +He I 4713 +He II 4200 +He II 4542 +He II 4686 +C II 4267 +C III 4068 / 69 / 70 +C III 4187 +C III 4326 +C III 4647 / 50 / 51 +N II 4530 +N II 4602 / 07 +N II 4614 / 21 +N III 4097 +N III 4379 +N III 4511 / 15 +N III 4634 / 41 / 42 +N IV 4058 +O II 4070 / 72 / 76 +O II 4133 +O II 4154 +O II 4186 / 90 +O II 4254 +O II 4276 / 85 +O II 4317 / 20 +O II 4349 / 67 +O II 4415 / 17 +O II 4448 / 52 +O II 4591 / 96 +O II 4662 / 76 +O II 4699 / 705 +Si III 4553 +Si III 4568 / 75 +Si IV 4089 +Si IV 4116 +Si IV 4631 +Mg II 4481 +DIB is 4428 +DIB is 4502 +DIB is 4727 +Ca I is 4227 +CH is 4300 +2MASS J10434797−6001201 B2.5 V(n) +2MASS J10443829−6005449 B2 V(n) +2MASS J10454824−5922041 B2.5 Vn +2MASS J10441829−5942296 B2: Vp He rich +2MASS J10445080−5918005 B2 V +[HSB2012] 3416 B2.5: Vnn +2MASS J10451894−5942184 B2 V +2MASS J10442894−5943473 B2 Vn +4100 +4200 +4300 +4400 +4500 +4600 +4700 +Wavelength (Å) +0 +1 +2 +3 +4 +5 +Fig. A.2. (Continued). +50 + +Berlanas et al.: Massive stars in the Carina Nebula +Hδ +Hγ +He I 4121 +He I 4144 +He I 4169 +He I 4388 +He I 4438 +He I 4471 +He I 4713 +He II 4200 +He II 4542 +He II 4686 +C II 4267 +C III 4068 / 69 / 70 +C III 4187 +C III 4326 +C III 4647 / 50 / 51 +N II 4530 +N II 4602 / 07 +N II 4614 / 21 +N III 4097 +N III 4379 +N III 4511 / 15 +N III 4634 / 41 / 42 +N IV 4058 +O II 4070 / 72 / 76 +O II 4133 +O II 4154 +O II 4186 / 90 +O II 4254 +O II 4276 / 85 +O II 4317 / 20 +O II 4349 / 67 +O II 4415 / 17 +O II 4448 / 52 +O II 4591 / 96 +O II 4662 / 76 +O II 4699 / 705 +Si III 4553 +Si III 4568 / 75 +Si IV 4089 +Si IV 4116 +Si IV 4631 +Mg II 4481 +DIB is 4428 +DIB is 4502 +DIB is 4727 +Ca I is 4227 +CH is 4300 +2MASS J10413434−5958474 B1 V +ALS 16 078 B2 Vn +2MASS J10444278−5921383 B2 V +2MASS J10433443−5943264 B0 V +2MASS J10450836−5938475 B2 V(n) +2MASS J10453807−5944095 O8 Vz +2MASS J10430420−5948591 B2 V(n) +2MASS J10444235−5922029 B2 V(n) +4100 +4200 +4300 +4400 +4500 +4600 +4700 +Wavelength (Å) +0 +1 +2 +3 +4 +5 +Fig. A.2. (Continued). +51 + +Berlanas et al.: Massive stars in the Carina Nebula +Hδ +Hγ +He I 4121 +He I 4144 +He I 4169 +He I 4388 +He I 4438 +He I 4471 +He I 4713 +He II 4200 +He II 4542 +He II 4686 +C II 4267 +C III 4068 / 69 / 70 +C III 4187 +C III 4326 +C III 4647 / 50 / 51 +N II 4530 +N II 4602 / 07 +N II 4614 / 21 +N III 4097 +N III 4379 +N III 4511 / 15 +N III 4634 / 41 / 42 +N IV 4058 +O II 4070 / 72 / 76 +O II 4133 +O II 4154 +O II 4186 / 90 +O II 4254 +O II 4276 / 85 +O II 4317 / 20 +O II 4349 / 67 +O II 4415 / 17 +O II 4448 / 52 +O II 4591 / 96 +O II 4662 / 76 +O II 4699 / 705 +Si III 4553 +Si III 4568 / 75 +Si IV 4089 +Si IV 4116 +Si IV 4631 +Mg II 4481 +DIB is 4428 +DIB is 4502 +DIB is 4727 +Ca I is 4227 +CH is 4300 +2MASS J10453134−5941133 B2 V +2MASS J10444726−5928154 B2 V(n) +Gaia DR3 5350302097055370496 B2 V(n) +2MASS J10434812−5950443 B2 V +2MASS J10443591−5923356 B2.5 Vn +2MASS J10440236−5952046 B2 V +2MASS J10460277−5950192 B2: Vnnn +[HSB2012] 3314 B2.5: Vnn +4100 +4200 +4300 +4400 +4500 +4600 +4700 +Wavelength (Å) +0 +1 +2 +3 +4 +5 +Fig. A.2. (Continued). +52 + +Berlanas et al.: Massive stars in the Carina Nebula +Hδ +Hγ +He I 4121 +He I 4144 +He I 4169 +He I 4388 +He I 4438 +He I 4471 +He I 4713 +He II 4200 +He II 4542 +He II 4686 +C II 4267 +C III 4068 / 69 / 70 +C III 4187 +C III 4326 +C III 4647 / 50 / 51 +N II 4530 +N II 4602 / 07 +N II 4614 / 21 +N III 4097 +N III 4379 +N III 4511 / 15 +N III 4634 / 41 / 42 +N IV 4058 +O II 4070 / 72 / 76 +O II 4133 +O II 4154 +O II 4186 / 90 +O II 4254 +O II 4276 / 85 +O II 4317 / 20 +O II 4349 / 67 +O II 4415 / 17 +O II 4448 / 52 +O II 4591 / 96 +O II 4662 / 76 +O II 4699 / 705 +Si III 4553 +Si III 4568 / 75 +Si IV 4089 +Si IV 4116 +Si IV 4631 +Mg II 4481 +DIB is 4428 +DIB is 4502 +DIB is 4727 +Ca I is 4227 +CH is 4300 +2MASS J10440744−5916399 O9.7: + B0.5: +2MASS J10443822−5943056 B2 Vnnn +2MASS J10440576−5927078 B2 V +2MASS J10443510−5923281 B2 V +2MASS J10444550−5952537 B1: V + B1.5: V +2MASS J10451297−5946059 B2 Vn +Gaia DR3 5350388683629610752 B2 Vn +2MASS J10450792−5939011 B2 Vn +4100 +4200 +4300 +4400 +4500 +4600 +4700 +Wavelength (Å) +0 +1 +2 +3 +4 +5 +Fig. A.2. (Continued). +53 + +Berlanas et al.: Massive stars in the Carina Nebula +Hδ +Hγ +He I 4121 +He I 4144 +He I 4169 +He I 4388 +He I 4438 +He I 4471 +He I 4713 +He II 4200 +He II 4542 +He II 4686 +C II 4267 +C III 4068 / 69 / 70 +C III 4187 +C III 4326 +C III 4647 / 50 / 51 +N II 4530 +N II 4602 / 07 +N II 4614 / 21 +N III 4097 +N III 4379 +N III 4511 / 15 +N III 4634 / 41 / 42 +N IV 4058 +O II 4070 / 72 / 76 +O II 4133 +O II 4154 +O II 4186 / 90 +O II 4254 +O II 4276 / 85 +O II 4317 / 20 +O II 4349 / 67 +O II 4415 / 17 +O II 4448 / 52 +O II 4591 / 96 +O II 4662 / 76 +O II 4699 / 705 +Si III 4553 +Si III 4568 / 75 +Si IV 4089 +Si IV 4116 +Si IV 4631 +Mg II 4481 +DIB is 4428 +DIB is 4502 +DIB is 4727 +Ca I is 4227 +CH is 4300 +[HSB2012] 3211 B2 V +[HSB2012] 1498 B2 V(n) +2MASS J10435501−5936242 B2 V(n) +2MASS J10444208−5926353 B2: Vnnn +2MASS J10443769−5929316 B2.5 Vn +[ESK2003] 148 O9.2 V(n) +2MASS J10471498−5953374 B6: IIIe +[HSB2012] 3482 B2: V + B3: V +4100 +4200 +4300 +4400 +4500 +4600 +4700 +Wavelength (Å) +0 +1 +2 +3 +4 +5 +Fig. A.2. (Continued). +54 + +Berlanas et al.: Massive stars in the Carina Nebula +Hδ +Hγ +He I 4121 +He I 4144 +He I 4169 +He I 4388 +He I 4438 +He I 4471 +He I 4713 +He II 4200 +He II 4542 +He II 4686 +C II 4267 +C III 4068 / 69 / 70 +C III 4187 +C III 4326 +C III 4647 / 50 / 51 +N II 4530 +N II 4602 / 07 +N II 4614 / 21 +N III 4097 +N III 4379 +N III 4511 / 15 +N III 4634 / 41 / 42 +N IV 4058 +O II 4070 / 72 / 76 +O II 4133 +O II 4154 +O II 4186 / 90 +O II 4254 +O II 4276 / 85 +O II 4317 / 20 +O II 4349 / 67 +O II 4415 / 17 +O II 4448 / 52 +O II 4591 / 96 +O II 4662 / 76 +O II 4699 / 705 +Si III 4553 +Si III 4568 / 75 +Si IV 4089 +Si IV 4116 +Si IV 4631 +Mg II 4481 +DIB is 4428 +DIB is 4502 +DIB is 4727 +Ca I is 4227 +CH is 4300 +2MASS J10464886−5950409 B1.5 V(n) +2MASS J10453254−5942359 B2: Vnnn +2MASS J10440105−6006377 B2 V +2MASS J10425293−6003478 B0.7 V +2MASS J10440866−5933488 B2: Vnnn +2MASS J10444609−5946056 B2 V +2MASS J10454060−5937041 B2: Vnn +2MASS J10470063−5957242 B2 V +4100 +4200 +4300 +4400 +4500 +4600 +4700 +Wavelength (Å) +0 +1 +2 +3 +4 +5 +Fig. A.2. (Continued). +55 + +Berlanas et al.: Massive stars in the Carina Nebula +Hδ +Hγ +He I 4121 +He I 4144 +He I 4169 +He I 4388 +He I 4438 +He I 4471 +He I 4713 +He II 4200 +He II 4542 +He II 4686 +C II 4267 +C III 4068 / 69 / 70 +C III 4187 +C III 4326 +C III 4647 / 50 / 51 +N II 4530 +N II 4602 / 07 +N II 4614 / 21 +N III 4097 +N III 4379 +N III 4511 / 15 +N III 4634 / 41 / 42 +N IV 4058 +O II 4070 / 72 / 76 +O II 4133 +O II 4154 +O II 4186 / 90 +O II 4254 +O II 4276 / 85 +O II 4317 / 20 +O II 4349 / 67 +O II 4415 / 17 +O II 4448 / 52 +O II 4591 / 96 +O II 4662 / 76 +O II 4699 / 705 +Si III 4553 +Si III 4568 / 75 +Si IV 4089 +Si IV 4116 +Si IV 4631 +Mg II 4481 +DIB is 4428 +DIB is 4502 +DIB is 4727 +Ca I is 4227 +CH is 4300 +2MASS J10440384−5934344 B2 V +2MASS J10420949−6002265 B0.7 V +2MASS J10445602−5938530 B2 Vn +2MASS J10443089−5914461 O7.5 II(f) +2MASS J10440683−5936116 B2 Vnnn +2MASS J10431388−5954584 B2 V +2MASS J10434798−5933590 B2 Vn +2MASS J10434580−5934359 B2: Vnnn +4100 +4200 +4300 +4400 +4500 +4600 +4700 +Wavelength (Å) +0 +1 +2 +3 +4 +5 +Fig. A.2. (Continued). +56 + +Berlanas et al.: Massive stars in the Carina Nebula +Hδ +Hγ +He I 4121 +He I 4144 +He I 4169 +He I 4388 +He I 4438 +He I 4471 +He I 4713 +He II 4200 +He II 4542 +He II 4686 +C II 4267 +C III 4068 / 69 / 70 +C III 4187 +C III 4326 +C III 4647 / 50 / 51 +N II 4530 +N II 4602 / 07 +N II 4614 / 21 +N III 4097 +N III 4379 +N III 4511 / 15 +N III 4634 / 41 / 42 +N IV 4058 +O II 4070 / 72 / 76 +O II 4133 +O II 4154 +O II 4186 / 90 +O II 4254 +O II 4276 / 85 +O II 4317 / 20 +O II 4349 / 67 +O II 4415 / 17 +O II 4448 / 52 +O II 4591 / 96 +O II 4662 / 76 +O II 4699 / 705 +Si III 4553 +Si III 4568 / 75 +Si IV 4089 +Si IV 4116 +Si IV 4631 +Mg II 4481 +DIB is 4428 +DIB is 4502 +DIB is 4727 +Ca I is 4227 +CH is 4300 +[HSB2012] 3150 B2 V +2MASS J10452875−5930037 B1.5 Vp +2MASS J10445837−5932062 B2.5 Vn +2MASS J10460608−5957394 B2 V +2MASS J10442945−5933437 B2: Vnnn +2MASS J10460257−5957372 B2: Vne +2MASS J10452056−5942212 B2: Vnne +2MASS J10453834−5942078 Be +4100 +4200 +4300 +4400 +4500 +4600 +4700 +Wavelength (Å) +0 +1 +2 +3 +4 +5 +Fig. A.2. (Continued). +57 + +Berlanas et al.: Massive stars in the Carina Nebula +Hδ +Hγ +He I 4121 +He I 4144 +He I 4169 +He I 4388 +He I 4438 +He I 4471 +He I 4713 +He II 4200 +He II 4542 +He II 4686 +C II 4267 +C III 4068 / 69 / 70 +C III 4187 +C III 4326 +C III 4647 / 50 / 51 +N II 4530 +N II 4602 / 07 +N II 4614 / 21 +N III 4097 +N III 4379 +N III 4511 / 15 +N III 4634 / 41 / 42 +N IV 4058 +O II 4070 / 72 / 76 +O II 4133 +O II 4154 +O II 4186 / 90 +O II 4254 +O II 4276 / 85 +O II 4317 / 20 +O II 4349 / 67 +O II 4415 / 17 +O II 4448 / 52 +O II 4591 / 96 +O II 4662 / 76 +O II 4699 / 705 +Si III 4553 +Si III 4568 / 75 +Si IV 4089 +Si IV 4116 +Si IV 4631 +Mg II 4481 +DIB is 4428 +DIB is 4502 +DIB is 4727 +Ca I is 4227 +CH is 4300 +2MASS J10444710−5939201 B2: Vnn +[HSB2012] 3526 B2 V(n) +2MASS J10452415−5942313 B2.5: Vnn +2MASS J10452190−5945249 B2 V +2MASS J10433596−5933179 B2 V +2MASS J10463643−5948048 B1.5 V +2MASS J10453185−6000293 O8.5 V +2MASS J10435009−5947024 B2.5 V(n) +4100 +4200 +4300 +4400 +4500 +4600 +4700 +Wavelength (Å) +0 +1 +2 +3 +4 +5 +Fig. A.2. (Continued). +58 + +Berlanas et al.: Massive stars in the Carina Nebula +Hδ +Hγ +He I 4121 +He I 4144 +He I 4169 +He I 4388 +He I 4438 +He I 4471 +He I 4713 +He II 4200 +He II 4542 +He II 4686 +C II 4267 +C III 4068 / 69 / 70 +C III 4187 +C III 4326 +C III 4647 / 50 / 51 +N II 4530 +N II 4602 / 07 +N II 4614 / 21 +N III 4097 +N III 4379 +N III 4511 / 15 +N III 4634 / 41 / 42 +N IV 4058 +O II 4070 / 72 / 76 +O II 4133 +O II 4154 +O II 4186 / 90 +O II 4254 +O II 4276 / 85 +O II 4317 / 20 +O II 4349 / 67 +O II 4415 / 17 +O II 4448 / 52 +O II 4591 / 96 +O II 4662 / 76 +O II 4699 / 705 +Si III 4553 +Si III 4568 / 75 +Si IV 4089 +Si IV 4116 +Si IV 4631 +Mg II 4481 +DIB is 4428 +DIB is 4502 +DIB is 4727 +Ca I is 4227 +CH is 4300 +2MASS J10462657−5956131 B1.5: V +2MASS J10460291−5950259 B2 V +2MASS J10453819−5942157 Be +[ARV2008] 217 O3: III:(n) +2MASS J10434303−5945333 B2: Ve +2MASS J10460116−5949420 B2: V +2MASS J10454661−5948404 B0.7: Ve +2MASS J10431945−5944488 sdO + sec +4100 +4200 +4300 +4400 +4500 +4600 +4700 +Wavelength (Å) +0 +1 +2 +3 +4 +5 +Fig. A.2. (Continued). +59 + +Berlanas et al.: Massive stars in the Carina Nebula +Hε +Hδ +Hγ +Hβ +He I 4009 +He I 4121 +He I 4144 +He I 4169 +He I 4388 +He I 4438 +He I 4471 +He I 4713 +He I 4922 +He I 5016 +He I 5048 +He II 4200 +He II 4542 +He II 4686 +He I+II 4026 +C II 4267 +C III 4068 / 69 / 70 +C III 4187 +C III 4326 +C III 4647 / 50 / 51 +N II 3995 +N II 4530 +N II 4780 / 88 +N II 4803 +N II 5001 / 04 / 07 +N III 4097 +N III 4379 +N III 4511 / 15 +N III 4634 / 41 / 42 +N IV 4058 +N V 4604 / 20 +N V 4944 / 46 +O II 3983 +O II 4070 / 72 / 76 +O II 4133 +O II 4154 +O II 4186 / 90 +O II 4254 +O II 4276 / 85 +O II 4317 / 20 +O II 4349 / 67 +O II 4415 / 17 +O II 4448 / 52 +O II 4591 / 96 +O II 4662 / 76 +O II 4699 / 705 +O II 4907 +O II 4943 +Si III 4553 +Si III 4568 / 75 +Si III 4813 / 20 / 29 +Si IV 4089 +Si IV 4116 +Si IV 4631 +Mg II 4481 +DIB is 4428 +DIB is 4502 +DIB is 4727 +DIB is 4762 / 65 / 80 +DIB is 4880 / 87 +DIB is 4964 +Ca II is 3934 +Ca II is 3968 +WR 24 WN6ha +HDE 305 437 B0 V +HD 93 249 B B0.2 V(n) +CPD −59 2595 B2.5 V +ALS 15 961 B1 V +4000 +4250 +4500 +4750 +5000 +Wavelength (Å) +0 +1 +2 +3 +4 +5 +Fig. A.3. GOSSS spectra shown at a resolution R = 2500. +60 + +Berlanas et al.: Massive stars in the Carina Nebula +Hε +Hδ +Hγ +Hβ +He I 4009 +He I 4121 +He I 4144 +He I 4169 +He I 4388 +He I 4438 +He I 4471 +He I 4713 +He I 4922 +He I 5016 +He I 5048 +He II 4200 +He II 4542 +He II 4686 +He I+II 4026 +C II 4267 +C III 4068 / 69 / 70 +C III 4187 +C III 4326 +C III 4647 / 50 / 51 +N II 3995 +N II 4530 +N II 4602 / 07 +N II 4614 / 21 +N II 4780 / 88 +N II 4803 +N II 5001 / 04 / 07 +N III 4097 +N III 4379 +N III 4511 / 15 +N III 4634 / 41 / 42 +N IV 4058 +O II 3983 +O II 4070 / 72 / 76 +O II 4133 +O II 4154 +O II 4186 / 90 +O II 4254 +O II 4276 / 85 +O II 4317 / 20 +O II 4349 / 67 +O II 4415 / 17 +O II 4448 / 52 +O II 4591 / 96 +O II 4662 / 76 +O II 4699 / 705 +O II 4907 +O II 4943 +Si III 4553 +Si III 4568 / 75 +Si III 4813 / 20 / 29 +Si IV 4089 +Si IV 4116 +Si IV 4631 +Mg II 4481 +DIB is 4428 +DIB is 4502 +DIB is 4727 +DIB is 4762 / 65 / 80 +DIB is 4880 / 87 +DIB is 4964 +Ca II is 3934 +Ca II is 3968 +CPD −59 2661 B0 V(n)e + B2: V +ALS 15 860 B1 Iab +CPD −59 2596 B0 V +2MASS J10424476−6005020 B0 IV: +ALS 15 233 B0.7 V +ALS 15 225 B1.5 Vp He rich +ALS 15 203 A B0.5 V + B1: V +HDE 303 316 B B1 V +4000 +4250 +4500 +4750 +5000 +Wavelength (Å) +0 +1 +2 +3 +4 +5 +Fig. A.3. (Continued). +61 + +Berlanas et al.: Massive stars in the Carina Nebula +Hε +Hδ +Hγ +Hβ +He I 4009 +He I 4121 +He I 4144 +He I 4169 +He I 4388 +He I 4438 +He I 4471 +He I 4713 +He I 4922 +He I 5016 +He I 5048 +He II 4200 +He II 4542 +He II 4686 +He I+II 4026 +C II 4267 +C III 4068 / 69 / 70 +C III 4187 +C III 4326 +C III 4647 / 50 / 51 +N II 3995 +N II 4530 +N II 4602 / 07 +N II 4614 / 21 +N II 4780 / 88 +N II 4803 +N II 5001 / 04 / 07 +N III 4097 +N III 4379 +N III 4511 / 15 +N III 4634 / 41 / 42 +N IV 4058 +O II 3983 +O II 4070 / 72 / 76 +O II 4133 +O II 4154 +O II 4186 / 90 +O II 4254 +O II 4276 / 85 +O II 4317 / 20 +O II 4349 / 67 +O II 4415 / 17 +O II 4448 / 52 +O II 4591 / 96 +O II 4662 / 76 +O II 4699 / 705 +O II 4907 +O II 4943 +Si III 4553 +Si III 4568 / 75 +Si III 4813 / 20 / 29 +Si IV 4089 +Si IV 4116 +Si IV 4631 +Mg II 4481 +DIB is 4428 +DIB is 4502 +DIB is 4727 +DIB is 4762 / 65 / 80 +DIB is 4880 / 87 +DIB is 4964 +Ca II is 3934 +Ca II is 3968 +ALS 19 733 B1.5: V +ALS 15 203 B B0 V +ALS 19 732 B2: Ve +2MASS J10442909−5948207 B0 V +2MASS J10460277−5950192 B2: Vnnn +2MASS J10454536−5958530 B0.5 V +2MASS J10460291−5950259 B2 V +4000 +4250 +4500 +4750 +5000 +Wavelength (Å) +0 +1 +2 +3 +4 +5 +Fig. A.3. (Continued). +62 + diff --git a/UdE_T4oBgHgl3EQfxBzD/content/tmp_files/load_file.txt b/UdE_T4oBgHgl3EQfxBzD/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7c6f82aa024b836b3efa83c166d828baab82eaa7 --- /dev/null +++ b/UdE_T4oBgHgl3EQfxBzD/content/tmp_files/load_file.txt @@ -0,0 +1,7612 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf,len=7611 +page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' ges © ESO 2023 January 23, 2023 Gaia-ESO Survey: massive stars in the Carina Nebula I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' A new census of OB stars S.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Drew16, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Morbidelli17, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Vink18 1 Departamento de F´ısica Aplicada,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Universidad de Alicante,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' E-3690,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' San Vicente del Raspeig,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Alicante,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Spain 2 Astrophysics Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Keele University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Keele ST5 5BG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Staffordshire,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Spain 6 Royal Observatory of Belgium,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Ringlaan 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 1180 Brussels,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Belgium 7 School of Architecture,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Universidad Europea de Canarias,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Tenerife,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' All´ee du 6 Aoˆut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 19c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Bˆat B5c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 4000 Li`ege,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Belgium 10 Departamento de Astrof´ısica y F´ısica de la Atm´osfera,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Universidad Complutense de Madrid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' E-28 040 Madrid, Spain 11 Instituto de Astronom´ıa y F´ısica del Espacio, UBA-CONICET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' CC 67, Suc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 28, 1428 Buenos Aires, Argentina 12 Instituto de Astrof´ısica de Andaluc´ıa (IAA), CSIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Glorieta de la Astronom´ıa s/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' E-18 008 Granada,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Spain 13 Max Planck Institute for Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' K¨onigstuhl 17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 69117,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Heidelberg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Germany 14 Niels Bohr International Academy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Niels Bohr Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' University of Copenhagen Blegdamsvej 17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' DK-2100 Copenhagen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Denmark 15 Dipartimento di Fisica e Astronomia Galileo Galilei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Universit´a di Padova,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Vicolo Osservatorio 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' I-35122,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Padova,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Italy 16 Department of Physics & Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' University College London,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Gower Street,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' London,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' WC1E 6BT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' UK 17 INAF - Osservatorio Astrofisico di Arcetri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Largo E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Fermi 5, 50125 Florence, Italy 18 Armagh Observatory and Planetarium, College Hill, Armagh BT61 9DG, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Ireland Received month day, year;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' accepted month day, year ABSTRACT Context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The Carina Nebula is one of the major massive star-forming regions in the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Its relatively nearby distance (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='35 kpc) makes it an ideal laboratory for the study of massive star formation, structure and evolution, both for individual stars and stellar sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Thanks to the high-quality spectra provided by Gaia-ESO survey and the LiLiMaRlin library, as well as Gaia EDR3 astrometry, a detailed and homogeneous spectroscopic characterization of its massive stellar content can be carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Our main objective is to spectroscopically characterize all massive members of the Carina Nebula in the Gaia-ESO survey footprint to provide an updated census of massive stars in the region and an updated estimate of the binary fraction of O stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' We perform accurate spectral classification by using an interactive code that compares spectra with spectral libraries of OB standards, as well as line-based classic methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Membership is calculated using our own algorithm based on Gaia EDR3 astrometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' To check the correlation between the spectroscopic n-qualifier and the rotational velocity, we use the semi-automated tool for the line-broadening characterization of OB stars which is based on a combined Fourier Transform and Goodness-of-fit methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The Gaia-ESO survey sample of massive OB stars in the Carina Nebula consists of 234 stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The addition of brighter sources from the Galactic O-Star Spectroscopic Survey and additional sources from the literature allows us to create the most complete census of massive OB stars done so far in the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' It contains a total of 316 stars, being 18 of them in the background and four in the foreground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Of the 294 stellar systems in Car OB1, 74 are of O type, 214 are of non-supergiant B type and 6 are of WR or non-O supergiant (II to Ia) spectral class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' We identify 20 spectroscopic binary systems with an O-star primary, of which 6 are reported for the first time, and another 18 with a B-star primary, of which 13 are new detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The average observed double-lined binary fraction of O-type stars in the surveyed region is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='35, which represents a lower limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' We find a good correlation between the spectroscopic n-qualifier and the projected rotational velocity of the stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The fraction of candidate runaways among the stars with and without the n-qualifier is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='4% and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='4%, respectively, although non resolved double-lined binaries can be contaminating the fast rotators sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' stars: massive – stars: early-type – stars: rotation – binaries: spectroscopic – proper motions – open clusters and associa- tions: individual: Carina Nebula 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Introduction The Gaia-ESO Large Public Spectroscopic Survey (GES, Gilmore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Randich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2022) has obtained high qual- ity spectra of ∼105 stars in our Galaxy using FLAMES at the Very Large Telescope (VLT) with its high-resolution UVES and its intermediate-resolution GIRAFFE spectrographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' GES has systematically covered all the major components of the Milky Way, providing an homogeneous and unique overview of the kinematics, chemical composition, formation history, and evo- lution of young, mature and ancient Galactic populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Open 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='08310v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='SR] 19 Jan 2023 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula clusters are useful tools for this aim, where it is possible to study stellar populations of different ages in different evolution- ary stages (see Bragaglia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Numerous spectroscopic studies of massive stars have been carried out in Galactic young stellar clusters and OB associ- ations, the most extensive to date being the Galactic O-Star Spectroscopic Survey (GOSSS, Ma´ız Apell´aniz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2011)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' As some examples of such studies, Figer (2005) determined the upper mass limit of the Initial Mass Function (IMF) in the Arches Cluster, a result that was later challenged by studies in R136 (Crowther et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Other examples are the determina- tion of the chemical composition of stars in Orion (Sim´on-D´ıaz 2010);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' the membership, chemical and stellar parameter determi- nation studies in Cygnus OB2 (Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2018a,b, 2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' the characterization of very massive obscured clusters in the Milky Way like Westerlund 1 (Clark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Negueruela et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2010, 2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' and the analysis of the multiplicity of massive stars in clusters (De Becker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2004, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Mahy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Sana & Evans 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Mahy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Banyard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2022) and in the whole northern hemisphere (Ma´ız Apell´aniz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2019b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Trigueros P´aez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Mahy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Outside the Milky Way, the most thorough analysis is that of the many papers1 published by the VLT-FLAMES Tarantula Survey collaboration (VFTS, Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The Carina Nebula complex consists of several stellar groups, some bound and some not, immersed in the Car OB1 association (Ma´ız Apell´aniz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2020, 2022a, from now on Villafranca I and II, respectively, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' It represents a unique region to study Galactic massive stars with FLAMES since it contains a large number of O-type stars (Walborn 1972, 1973b, 1982b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Levato & Malaroda 1982;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Morrell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Sota et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Ma´ız Apell´aniz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Alexander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Mohr-Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' It is the most massive star-forming region within 3 kpc of the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The distance to its most famous member, η Car, was geometrically determined with excellent precision to be 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='35±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='05 kpc by Smith (2006b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The recent Gaia EDR3 (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2021) analysis in Villafranca I+II has not only con- firmed that value but has also found that there are little distance variations between at least Trumpler 14, Trumpler 16 W, and Trumpler 16 E, three of the stellar groups in the complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In a new installment of the series (Villafranca III, Molina Lera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' in preparation) the authors show that those distance variations are still small when including other stellar groups in Car OB1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Even though the Carina Nebula harbors hundreds of massive stars, there is no systematic spectroscopic analysis of its early- type members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Thanks to the high-quality spectra provided by GES and astrometry by Gaia EDR3, a detailed and homoge- neous spectroscopic study of its massive stellar content can be carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The analysis of the Carina massive stellar popula- tion will be highly relevant for problems like the initial mass function (IMF, Crowther et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2010), the chemical composition, rotation and internal mixing (Meynet & Maeder 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Ram´ırez- Agudelo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Sim´on-D´ıaz & Herrero 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Herrero 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Holgado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2022), or the stellar multiplicity of massive stars (see Langer 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Sana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Sota et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' de Mink et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In particular, although Sana & Evans (2011) and Sana (2017) quote fractions of binary systems in excess of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 for the O-type star population in the Milky Way, the former au- thors give a null fraction in a cluster like Trumpler 14, making clear the need for a systematic survey in the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In addition, binarity may be the origin of fast rotating and runaway stars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 1 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='roe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='uk/˜cje/tarantula/f2-pubs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Wavelength range and resolving power of the GES spectra obtained with different gratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Grating Wavelength Resolving Data release range (Å) power GIRAFFE HR03 4033 − 4201 24 800 iDR3-6 HR04 4188 − 4392 20 350 iDR3-6 HR05A 4340 − 4587 18 470 iDR5-6 HR06 4538 − 4759 20 350 iDR3-6 HR14A 6308 − 6701 17 740 iDR3-6 UVES 520 4140 − 6210 47 000 iDR3-6 580 4760 − 6840 47 000 iDR5-6 de Mink et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2013, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Mahy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Holgado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2022) by ejecting stars that have gained mass and angular momentum from the binary system after the explosion of the primary as su- pernova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The Carina region, containing a large number of mas- sive stars at a relatively nearby distance, is an ideal place to test the theories of massive star evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' As a first step this work focuses on the creation of the most complete to date census of massive stars and the identification of double-lined spectroscopic binaries (SB2) in Car OB1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' It is orga- nized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In Section 2 we describe how we have obtained our spectroscopy, compiled our spectral types, and used Gaia to determine the distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In Section 3 we present our census of massive stars in the central part of the Carina Nebula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' We dis- cuss the results in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 4, where we explore the completeness of the census, determine the binary fraction of OB stars and inves- tigate the correlation between the n spectroscopic qualifier, the projected rotational velocity and the runaway status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Finally, we summarize the conclusions in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Data and methods 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' GES strategy and spectroscopy GES spectroscopic data for hot stars were obtained using the FLAMES intermediate-resolution (R∼20 000) GIRAFFE and the high-resolution (R∼47 000) UVES spectrographs on the Very Large Telescope (VLT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' See Blomme et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2022) for further details on the analysis of GES hot stars and Table 1 for the wavelength range covered by each of the setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In the rest of this subsection we present the aspects that are more relevant to the Carina Nebula GES data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' A previous GES paper on the Carina Nebula (Damiani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2017) used a different data set and concentrated on stars of lower mass than the ones analyzed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The central part of the Carina Nebula can be divided into six stellar groups: Trumpler 14, Trumpler 15, Trumpler 16 W, Trumpler 16 E, Collinder 228, and Collinder 232 (Walborn 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Smith 2006a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Villafranca I+II+III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Of those stellar groups, only Trumpler 14 and Trumpler 15 and possibly Trumpler 16 E appear to be real bound clusters, with the rest being parts of the association defined by (apparent or real) struc- tures seen in the stellar distribution and nebulosity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' the sepa- ration of Collinder 228 from the other groups likely originates in the prominent V-shaped dust lane that crosses the H ii region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Other stellar groups farther away from the central region but likely members of the Car OB1 association include NGC 3293 (see Morel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2022), NGC 3324, Bochum 10, Bochum 11, Loden 153, IC 2581, Ruprecht 90, and ASCC 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Given the large 2 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula O-002 O-002 Tr 14 Tr 14 O-003 O-003 Tr 16W Tr 16W O-025 O-025 Tr 16E Tr 16E O-027 O-027 Tr 15 Tr 15 O-028 O-028 Coll 228 Coll 228 O-029 O-029 Coll 232 Coll 232 O-030 O-030 Bochum 11 Bochum 11 Carina_Gendler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='jpg 15’ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='14´� x 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='24’ N E Powered by Aladin Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Negative image of the Great Carina Nebula by Robert Gendler and Stephane Guisard showing the location of the whole census of massive stars in the GES surveyed area presented in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Yellow and cyan colors indicate O and B-type stars, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Green, red, purple and pink colors have been used to represent the sdO, LBV, WR and RSG stars, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Small filled-circles refer to the GES sample while rhombuses and squares refer to stars from GOSSS/LiLiMarlin and other works (Smith 2006a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Alexander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Preibisch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2021) not present in GES, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Red circles indicate the observing GES pointings while the blue ones indicate the Villafranca groups: O-002 (Trumpler 14), O-003 (Trumpler 16 W), O-025 (Trumpler 16 E), O- 027 (Trumpler 15), O-028 (Collinder 228), O-029 (Collinder 232), and O-030 (Bochum 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The V-shaped extinction lane that dominates the appearance of the nebula is clearly seen crossing the image from top to bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' size of the nebula and the high stellar density in some regions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' the core of Trumpler 14), four different GIRAFFE+UVES pointings were needed to cover a substantial fraction of the massive stars in the six central groups and part of those in Bochum 11 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The four pointings are centered at RA+δ J2000 coordinates (161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='10,−59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='430), (161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='07,−59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='695), (161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='42,−59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='835), and (160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='79,−60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='020), respectively, with the values expressed in degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The sample selection was done by compiling the available spectroscopic and photometric informa- tion at the time of the survey design (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' the Galactic O-Star Catalog, GOSSS, Ma´ız Apell´aniz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Sota et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2008) but, of course, no Gaia data existed back then.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' For that reason, we complemented our spectroscopy with GOSSS data and we have to evaluate the completeness of our sample (see below for both).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' One important difference between this one and most of the other GES data sets is the existence of a significant nebulosity in the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Furthermore, the nebular Balmer and He i emission lines not only are strong but they are placed on top of important diagnostic stellar absorption lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' For that reason, we devised a specific strategy to eliminate or at least mitigate their influence when we prepared these observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Each of the four point- ings was divided into two subpointings (for a total of eight) with half of the fibers dedicated to stars and the other half to nebulos- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Each of the two subpointings within a given pointing have identical fiber configurations but the field center is displaced by 10′′ between the two of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In that way, each star observed in a given subpointing has a nebular counterpart 10′′ away ob- served in the other subpointing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' One of us (JMA) wrote an IDL code to manually review each star/nebulosity spectral pair and use the second one to subtract the nebular contribution from the stellar fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' This strategy is the best possible one given the lim- itations of the observational setup, but it is not ideal, as in some cases nebular emission can change substantially in scales smaller than 10′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' This is one of the advantages of long-slit spectroscopy (such as that obtained by GOSSS) over its fiber-fed alternative, as the former allows a sampling of nebular emission closer to the target and at two different locations with respect to the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 3 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula In practical terms, the issue is significative only for faint stars at Hα, as the nebular contribution for bright stars and other relevant lines is usually small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' We examined all of the spectra in our GES datasets to iden- tify OB massive stars (B2 or earlier for dwarfs, B5 or earlier for giants, and all B subtypes for supergiants) and obtained a sam- ple of 234 objects, 18 of which were observed with FLAMES- UVES and 216 with FLAMES-GIRAFFE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The spectrograms are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='1 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2, in the first case at the original 47 000 spectral resolution and in the second case at the 2500 spectral resolution used for spectral classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' GOSSS spectroscopy The GOSSS project was born a decade and a half ago with the idea of obtaining mid-low resolution (R ∼ 2500) blue-violet spectroscopy of any optically-accessible Galactic object that had ever been classified as an O star to confirm its nature and to pro- vide homogeneous spectral classifications for the whole sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' While doing that, GOSSS managed not only to discover a siz- able number of new O-type stars but also to reject quite a number of them as being of B type (or, more egregiously, of even later types) and to obtain good-quality spectroscopy of several thou- sands of other early-type stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In the first three major papers, Sota et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2011) or GOSSS I, Sota et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2014) or GOSSS II, and Ma´ız Apell´aniz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2016) or GOSSS III, GOSSS pub- lished spectra for 590 O-type stars and for a few later-type ob- jects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Since that time, GOSSS has collected a large number of new spectra, some of them in the Carina Nebula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Of those, eight new GOSSS spectra for O-type stars will appear in the fourth major installment of the project (GOSSS IV, Ma´ız Apell´aniz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' in preparation) but the spectral classifications are already listed here in Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In this paper we also present GOSSS spectra for one Wolf-Rayet and 17 early-B stars in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='3 as a complement to the GES data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Spectral classifications We obtained the spectral classifications using the MGB tool (Ma´ız Apell´aniz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2012, 2015), which compares the ob- served spectra with a standard library of OB stars (in this case the GOSSS library, see GOSSS III+IV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' This interactive soft- ware allows us to vary the spectral subtype, luminosity class, line broadening, and spectral resolving power of the standard spectrum until we obtain the best match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In addition, it also al- lows us to combine two standard spectra (with different veloc- ities and flux fractions) to fit SB2 systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The spectral classi- fication was performed for the three types of spectroscopic data (UVES, GIRAFFE, and GOSSS) at the same spectral resolu- tion of the GOSSS library, 2500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The spectral classifications are given in Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' There is one specific issue with GIRAFFE spectra and spec- troscopic binaries that needs to be discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In general, each grating was observed at a different epoch and this generates a problem for spectroscopic binaries, as different lines of the same ion may be at different velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' We have dealt with this issue on a case by case basis but in some we are only able to provide a poor-quality spectral classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' See subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 for some examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Spectral types from other sources and cataloguing In addition to those from GES and GOSSS, in Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 we give spectral types from other further sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The first one is LiLiMaRlin (Library of Libraries of Massive-star High- Resolution Spectra, Ma´ız Apell´aniz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2019a) that is collect- ing multi-epoch high-resolution optical+NIR spectra of massive stars, with over 60 000 epochs to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' For the case of the Carina Nebula, the library currently has FEROS, UVES, and HARPS spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' LiLiMaRlin is especially useful for the analysis of SB2 and SB3 systems, where finding the right epoch is usually neces- sary to separate the different components in velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' One of the LiLiMaRlin spectral types had appeared before in Villafranca I but there are also nine whose spectra will appear in GOSSS IV and another 16 whose spectra will appear in Villafranca III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' For the two latter papers, those spectral types are listed here for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Multi-epoch high-resolution spectroscopy such as that from LiLiMaRlin can be used to separate in velocity spectroscopic binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' If one wants to spatially separate close visual binaries, then what is needed is the combination of high-spatial resolution with spectroscopy, either from the ground (Ma´ız Apell´aniz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2018, 2021a) or from space (Ma´ız Apell´aniz & Barb´a 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' For the case of the Carina Nebula, we give in Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 the spa- tially resolved spectral types for HD 93 129 Aa,Ab, one of the most massive systems in the region, obtained with STIS/HST (Ma´ız Apell´aniz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Another source is the already mentioned GOSC, which is a catalog that compiles information about massive stars (with an emphasis on O stars) from different sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' GOSC has a private and a (increasingly growing) public version, which will be heav- ily updated after GOSSS IV is published.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Here we have used the private version of GOSC to search for additional massive stars in the region of interest and provide spectral types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In particular, we have included the results from Smith (2006a), a previous census of the massive stars in the Carina Nebula, from Alexander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2016), a spectroscopic survey of the region, and from Preibisch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2021), a near-infrared spectroscopy survey to identify ob- scured OB stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Besides the flow of information from GOSC to this pa- per mentioned in the previous paragraph, there will be a flow of information in the opposite direction, as the spectral types here will be included in GOSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In addition to GOSC, these spectral types will be used to update the Alma Luminous Star Catalog (ALS, Reed 2003), a compilation of (originally) photo- metric and spectroscopic information for Galactic OB stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In Pantaleoni Gonz´alez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2021), the original ALS catalog was cross-matched with Gaia DR2 to eliminate the many misidenti- fications and duplicates present and to provide astrometric infor- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In a soon-to-be submitted third paper, the cross-match will be revised with Gaia DR3 information and the catalog will be expanded with new information, such as the one in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Gaia EDR3 data We have searched the Gaia EDR3 archive (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2021) for the astrometric and photometric information of the sample in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Gaia EDR3 parallaxes, ϖ, have a zero point, ZEDR3 (Lindegren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2021), that needs to be applied to yield cor- rected parallaxes, ϖc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Furthermore, the internal parallax uncer- tainties are underestimated and have to be converted into exter- nal (or true) uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Here we follow the procedure outlined in Ma´ız Apell´aniz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2021b) and Ma´ız Apell´aniz (2022) to list the corrected parallaxes with their external uncertainties in 4 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' We also list there the membership of each star to a foreground or background population according to its parallax (see Appendix A for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' As the inverse of the parallax is a biased estimator of the distance (Lutz & Kelker 1973), one has to do a proper estima- tion of the distance involving a prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The prior depends on the analyzed population itself: notoriously, OB stars do not follow the same spatial distribution in the Milky Way compared to its older populations Here we use the thin disk model and prior of Ma´ız Apell´aniz (2001, 2005) updated with the parameters of Ma´ız Apell´aniz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2008) to calculate the distances and uncer- tainties of individual stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' See Pantaleoni Gonz´alez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2021) for a comparison among different distance estimates to OB stars using Gaia parallaxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Gaia EDR3 provides photometry in three bands, G3, GBP3, and GRP3, with the two last bands being actually the result of integrating spectrophotometry in the wavelength direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The analysis of previous Gaia data releases (Ma´ız Apell´aniz 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Ma´ız Apell´aniz & Weiler 2018) revealed that the sensi- tivity curves of the Gaia instrument change with time, leading to slightly different intrinsic photometric values between data releases (each being an average over different time frames).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Furthermore, in some cases the processing introduces small trends and artifacts in the published magnitudes that require cor- rections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In a paper that will be submitted soon, a team that in- cludes some of us have computed such an analysis for G3, lead- ing to a corrected value G′ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='1 we list the G′ 3 and GBP3 − GRP3 values for our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Census Here we present the new census of massive stars in the central region of the Carina Nebula and discuss some individual stars of interest, especially if they have received little or no attention be- fore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The census itself is presented in two tables in the Appendix already introduced in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='1 lists the star identifications and coordinates, Gaia EDR3 corrected photome- try and parallaxes, and the group identification (see previous sec- tion and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 gives the spectral classifications from different sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Disagreements between spectral classifications are sometimes attributable to the nature of spectroscopic binaries caught in different orbital phases but in other cases they are due to differences in data quality (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' wavelength range, S/N, uncor- rected artifacts) or classification criteria (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' choice of lines for classification, standards used, consideration of line broadening).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' When in doubt, one should consult the published spectrograms, not the spectral types themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' That is the reason for publish- ing long appendices with figures such as the one here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Overall properties The resulting census of stars presented in this work contains 316 massive stars2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Note that, by definition and as stated be- fore, massive OB stars include all O-types and those B2-types or earlier for dwarfs, B5-types or earlier for giants, and all B sub- types for supergiants (I or II luminosity classes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Red supergiants (RSGs), Wolf-Rayet (WR) stars, and some B subtypes close to the OB-star limit (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V) are also included in the census for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' We have separated stars with distances compati- ble with Car OB1 from those in the foreground and background, 2 Note that this number does not distinguish between single and bi- nary or multiple stars, so hereinafter we refer to stellar systems when both types are included in the statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' finding four systems in the foreground (one RSG, two B dwarfs and one sdO) and 18 in the background (two O stars, five B su- pergiants and 11 B non-supergiants).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' These systems are listed in Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Of the 294 stellar systems in our census in Car OB1, 74 are of O type, 214 are of non-supergiant B type and 6 are of WR or non-O supergiant (II to Ia) spectral class (they are listed in Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Note that other WR stars in Car OB1 fall outside the surveyed area (WR 22, WR 23 and WR 27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Compared to the previous census of the massive stars in the Carina Nebula by Smith (2006a) we have significantly increased the content of known OB stars in the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Considering only the area sur- veyed in this work, the number of 105 OB stellar systems with spectral types as late as B2 reported by Smith (2006a) has been increased by a factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' There are three RSGs in the field of view: HD 93 420, HD 93 281, and HDE 303 310 (= RT Car), all of them included in the study of Humphreys et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' They are the second, fifth, and sixth G′ 3 brightest sources, as the bolometric correc- tion in that photometric band is significantly lower for RSGs than for O-type and WR stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' η Car, of course, is in a differ- ent luminosity category and is almost two magnitudes brighter in G′ 3 than the brightest RSG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' HD 93 420 is three sigmas3 closer to us in parallax than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='44 mas, the limit we are using to in- clude a star in Car OB1, and that places it in the foreground (but closer to Car OB1 than to us).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The other two stars have parallaxes compatible with being in Car OB1, HD 93 281 in Villafranca O- 028 (Collinder 228) and HDE 303 310 in Villafranca O-029 (Trumpler 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' We list in this paper their new spectral classi- fications from Villafranca III, derived from recently obtained FEROS spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The classification for HD 93 420 is identical to that of Humphreys et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (1972) but the other two are of slightly later type, with HDE 303 310 at M3 Iab and HD 93 281 at M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 Iab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' We see no sign of the alleged B-type companion for HD 93 281 (see Humphreys et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 1972) other than the strong Hα emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The three RSGs are not the only sign of the existence of previous generations of massive-star formation in Car OB1 and its immediate foreground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' We also find in our sample evolved B stars such as HDE 305 535, HDE 305 452, CPD −58 2605, CPD −59 2469, and CPD −59 2504.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' We find only one B- supergiant of luminosity class I, HDE 305 530, at the distance of Car OB1 in the footprint of this paper (the two stars by Damiani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2017) classified as B I, 2MASS J10440384−5934344 and 2MASS J10452875−5930037, are classified here as B2V and B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5Vp, respectively), but a number of B and later-type su- pergiants are observed in its vicinity (Villafranca III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' All of this establishes the existence not only of those older massive stars but also of the supernova explosions associated to those star-formation episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' It has been known for a long time that the gas in the foreground of some of the OB stars in Carina shows the most complex kinematics in any Galactic sightline (Walborn & Hesser 1975;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Walborn 1982a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Walborn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2002a), with up to 26 individual components and a range of velocities between −388 km/s and +127 km/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Those components must have been produced by supernova explosions whose progenitors were evolved massive stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The remaining three RSGs and the evolved B stars must be just the tip of the iceberg of the previous massive populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Those complex kinematics are the main reason why the interstellar lines present in the spectra of the 3 The external parallax uncertainty for such a bright star is much larger than the internal uncertainty (Ma´ız Apell´aniz 2022), so the dis- tance in sigmas would be also larger if we were to use the second one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 5 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula Carina OB stars are so strong (Penad´es Ordaz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2011, 2013), as the spread in velocity yields a more advantageous curve of growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Due to the additional Routly-Spitzer effect (Routly & Spitzer 1951;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Routly & Spitzer 1952) the Ca ii H+K lines are especially strong for these stars, making them deviate strongly from the relationship between extinction and their EW derived from other sightlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Individual stars The Carina Nebula field has a large number of interesting stars, starting with η Car, that have been analyzed in the past (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=', Damineli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2000, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Iping et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Our goal in this subsection is not to discuss such objects per se but to present new interesting objects that have received little or no attention in the past or new aspects of old objects that are mentioned for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' QZ Car Aa,Ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' This complex system (S´anchez-Berm´udez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Rainot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2020) is the brightest of O-type in the Carina Nebula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Mayer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2001) identified it as an SB1E+SB1 system and measured the two periods as 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='991 d (eclips- ing) and 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='735 96 (non-eclipsing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In GOSSS II the system was classified as O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 Ibn with no resolved components4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In GOSSS IV the system is determined to be O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 Ib + O9 II: using LiLiMaRlin but the authors note that the secondary luminosity class is poorly determined, possibly as the result of contamina- tion by one of the additional stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The high luminosity of the two primaries coupled with the smaller contribution of the sec- ondaries explains why this system seats at the top of the optical- luminosity food chain of the O-type stars in the Carina Nebula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The Gaia EDR3 parallax uncertainty is quite large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' HD 93 129 Aa,Ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' This system was spatially resolved by Ma´ız Apell´aniz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2017) using HST/STIS and determined to be composed of two O2 If* stars, with one of them having a com- panion in a tight orbit, likely a late-O star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The orbit is highly ec- centric and passed though periastron in 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='70+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='22 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='12 (del Palacio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Given that the periastron took place at a 3-D sepa- ration of just 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='6±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='0 AU (when the system was first spatially resolved in 1996 it was ∼375 AU), an order of magnitude (or even less) smaller than the expected semi-major axis of the in- ner orbit, and the high eccentricity, it is possible that the system has transitioned from an elliptic orbit to a hyperbolic trajectory and a possible ejection from Villafranca O-002 (Trumpler 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' If that had happened, this could be another example of an orphan cluster where the most (in this case, two) massive stars of a clus- ter are ejected through a dynamical interaction (Ma´ız Apell´aniz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Further observations are needed, especially with HST/STIS later in this decade (if it is still operational) when Aa and Ab are expected to reach plane-of-the-sky separations of ∼40 mas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' HD 93 403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Rauw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2000) classified this SB2 system as O5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 I + O7 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In GOSSS II the two components could not be resolved and it received a classification of O5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 III(fc) var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In GOSSS IV it is now kinematically resolved and classified as O5 Ifc + O7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V using either GOSSS or LiLiMaRlin data, that 4 Some papers quote spectral types for the four components but these are estimates: to our knowledge both spectroscopic binaries are still SB1 and no resolved (spatially or kinematically) spectral types have been determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' is, the primary is slightly earlier and the secondary slightly later compared to Rauw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The Gaia EDR3 parallax un- certainty is quite large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' HDE 305 520.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Alexander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2016) classified this system as B1 Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In Villafranca III we use LiLiMaRlin data to reclassify it as B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 Iab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' This Villafranca O-028 object is the only B super- giant at the distance of Car OB1 in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' V572 Car.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Rauw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2001) classified this SB3 system in Villafranca O-025 (Trumpler 16 E) as composed by an O7 V + O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V inner eclipsing binary and an outer B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 IV star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In GOSSS III only two components were seen and received an O7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V(n) + B0 V(n) classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' With the new data we now detect the system as an SB3: O6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 Vz + B0 V + B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 V in LiLiMaRlin data in GOSSS IV and O6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 Vz + B0 V + B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: V in UVES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' As it happened with HD 93 403, the primary is slightly earlier and the secondary slightly later compared to the original classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The outer star has been detected in NIR Long- Baseline Interferometry and is currently further monitored (see Gosset et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' CPD −59 2554.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' This system was classified as O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 IV in GOSSS II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Using LiLiMaRlin in GOSSS IV and UVES here it is now found to be an SB2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In both cases the spectral classification is O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 V + B1: V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' HD 93 342.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' This object was considered as a Villafranca O-027 (Trumpler 15) member by Smith (2006a), where it received a classification as O9 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Alexander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2016), on the other hand, classified it as B1 Ia and in Villafranca III we reclas- sify it as B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 Ib using LiLiMaRlin data, confirming it is a B- type supergiant and not an O star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Its Gaia EDR3 distance is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='58+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='41 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='33 kpc, placing it beyond Car OB1, something that it is consistent with its red color (it is the brightest OB star in our sample with GBP3 − GRP3 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' HD 93 056.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Alexander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2016) classified this system as O9 V + B2 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In Villafranca III we use LiLiMaRlin data to re- classify it as B1: V:n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Furthermore, in the UVES data no He ii is detected (either 4542, 4686, or 5412), as it should in an SB2 system composed of a late-O and an early-B stars even if caught at a disfavorable phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' HD 93 501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Alexander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2016) classified this system as B0 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In Villafranca III we use LiLiMaRlin data to reclassify it as B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: III:(n)e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Its Gaia EDR3 distance is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='87+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='12 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='11 kpc, plac- ing it in the foreground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' CPD −59 2592.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Alexander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2016) classified this object as B1 Ib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In Villafranca III we use LiLiMaRlin data to reclassify it as B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Its Gaia EDR3 distance is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='71+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='69 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='53 kpc, placing it beyond Car OB1, something that is consistent with its red color (it is the brightest OB star in our sample with GBP3 − GRP3 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' HDE 305 439 A,B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' With GIRAFFE data we classify the A com- ponent as B0 Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Its Gaia EDR3 distance is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='48+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='54 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='44 kpc, placing it beyond Car OB1, something that is consistent with its moder- 6 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula ately red color (it is the fourth brightest OB star in our sample with GBP3 − GRP3 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In Villafranca III we use LiLiMaRlin data to classify the B component, located 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='′′7 away, as B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 Ib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Its parallax is consistent with being at the same distance, making the system a likely pair of B supergiants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' HDE 305 535.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' This object was classified as B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V by Alexander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Here we derive a classification of B4 III(n) from UVES data, which leads to an absolute magni- tude more consistent with its spectral type and low extinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' HD 93 343.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Rauw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2009) classified this SB2 system as O7-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 + O8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In GOSSS III the two components could not be resolved and it received a classification of O8 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In GOSSS IV it is now kinematically resolved and classified as O7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 Vz + O7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: V(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' CPD −59 2636 A,B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' This system is a visual binary with a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='′′3 separation and ∆m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='6 mag in which both components are spectroscopic binaries (Albacete Colombo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2002): A (A+B in Albacete Colombo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2002) is an SB2 with a 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='6284 d pe- riod and B (C in Albacete Colombo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2002) is an SB1 with a 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='034 d period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Those authors gave a spectral classification of O7 V + O8 V to A and of O9 V to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In GOSSS II, the authors were only able to give two spectral types as O8 V + O8 V but with GES we are able to see the three components in an UVES single epoch and derive spectral types of O7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V + O8 V + O8 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Further epochs are needed to solve the small discrepancies with the Albacete Colombo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2002) classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Gaia EDR3 does not provide a parallax for CPD −59 2636 A,B, which is common for a visual binary of this separation and magni- tude difference, but it is a likely member of Villafranca O-025 (Trumpler 16 E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' HDE 305 534.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Alexander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2016) identified this system as a spectroscopic binary and classified it as B0 V + B0 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In Villafranca III we use LiLiMaRlin data to confirm it is an SB2 and reclassify it as B0 V + B1: V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' HDE 305 543.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Gagn´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2011) identified this system as a spectroscopic binary and classified it as B0 V + B0 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In Villafranca III we use LiLiMaRlin data to confirm it is an SB2 and reclassify it as B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 V(n) + B1: V(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' HDE 303 312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' We detect this object as a SB2 for the first time with GIRAFFE and assign it spectral types O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 III + B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In GOSSS II, where it was likely caught at a disfavorable phase, it had received the intermediate type O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' It was already known to be an eclipsing binary with a 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='4109 d period (Otero 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' CPD −58 2649 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' We classify this system as an SB2 with spec- tral types O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 III: +B0: V in GOSSS IV with GOSSS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' With GIRAFFE data, we can only give a poorer classification of O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: + B0: due to the different phases in each grating, but in any case both components are clearly later than the O7 V + O8 V of Alexander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' There is a visual companion detected in Gaia EDR3 with a separation of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='′′2 that, though relatively weak, may contaminate the GOSSS and GES spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' ALS 15 860.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' This object was considered as a Villafranca O- 027 (Trumpler 15) member by Smith (2006a), where it re- ceived a classification as O9 I-II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Using either the GOSSS or GES data here we classify it as B1 Iab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Its Gaia EDR3 dis- tance is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='31+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='23 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='20 kpc, placing it beyond Car OB1, consistent with its red color (it is the brightest OB star in our sample with GBP3 − GRP3 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' CPD −58 2634.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' We classify this object as B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V using GIRAFFE data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Its Gaia EDR3 distance is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='869+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='077 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='071 kpc, plac- ing it in the foreground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Its parallax is consistent with being at the same distance as HD 93 501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' CPD −59 2591.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' This system in Villafranca O-028 (Collinder 228) was classified as an SB2 with spectral types O8 Vz + B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: V both in GOSSS III using GOSSS spectroscopy and in GOSSS IV using LiLiMaRlin data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Here it is seen as SB2 but the classification is of poorer quality due to the multiple epochs of the GIRAFFE data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' CPD −59 2535.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' This system was classified as B2 V by Alexander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2016) but there is no GES, GOSSS, or LiLiMaRlin data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Its Gaia EDR3 distance is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='16+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='23 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='20 kpc, plac- ing it in the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2MASS J10424476−6005020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' A GIRAFFE spectrum is used to identify this system as an SB2 with the spectral types B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 V + B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In a GOSSS spectrum no double lines are seen, likely due to an unfavorable epoch, and the resulting spec- tral classification is a poorer B0: IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Its Gaia EDR3 distance is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='02+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='39 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='33 kpc, placing it in the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Its red color is con- sistent with the measured distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2MASS J10460477−5949217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' This object in Villafranca O- 030 (Bochum 11) is classified as an O star for the first time with GIRAFFE data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' It has a moderately high extinction and a spectral classification of O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 V(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The spectrum could be a composite of a late-O and an early-B stars but more epochs are needed to test that hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2MASS J10444803−5954297.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' This object is an Oe star with strong Balmer emission and no previous classification as O type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' As usual with Oe stars, spectral classification is of poor quality and we can only give O7: Ve using GIRAFFE and O8: Ve in GOSSS IV using GOSSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The Gaia EDR3 parallax indicates a background object at a distance of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='71+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='79 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='59 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' CPD −59 2618.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Alexander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2016) classified this system as B2 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' A GIRAFFE spectrum indicates it is of an earlier sub- type and with an anomalous composition, yielding a classifica- tion of B1: V(n)p He rich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In Villafranca III we use LiLiMaRlin data to conform the helium enrichment and to further discover it is an SB2, classifying it as B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7: V(n)p He rich + B1: V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' ALS 15 225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' We identify this star in Villafranca O-028 (Collinder 228) as a He-rich B star for the first time using both GIRAFFE and GOSSS spectroscopy here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 7 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula V662 Car.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Niemel¨a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2006) identified this system as an O5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 Vz + O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V SB2 and an eclipsing binary with a period of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='41355 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In GOSSS III it was classified as O5 V(n)z + B0: V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The GIRAFFE data shows there are two separate components in He ii and both narrow, with a third component clearly separated in He i, making it an SB3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' However, given the multiple epochs in the GIRAFFE data, we can only classify it as O+O+B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Both O stars have narrow lines, so the GOSSS III (n) suffix is likely due to combination of two O stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The second O star is likely a third light not participating in the orbit and appears to be of mid-O subtype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The first O star has He ii 4542 > He ii 4471 and should be close to O5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' This system needs further high-resolution spectroscopy covering the whole classification range in a single epoch at large velocity separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' ALS 15 203 A,B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In Villafranca II this Villafranca O-002 (Trumpler 14) object was identified as an SB3 with a classifica- tion of B0 V + B + B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In GIRAFFE we see some double lines but He ii lines are very weak, invalidating the Vijapurkar & Drilling (1993) classification as O7 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' As Gaia EDR3 detects two sources of similar magnitude separated by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='′′2 (confirmed by HST imag- ing), we reanalyzed the GOSSS long slit with the best seeing and proper orientation and we were able to spatially separate the two visual components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' ALS 15 203 A is an SB2 with a spectral classification of B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V + B1: V, which corresponds to those of the secondary and tertiary in Villafranca II, and ALS 15 203 B has a classification of B0 V, which corresponds to the primary in Villafranca II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' There is a hint of emission at the bottom of Hβ for ALS 15 203 B but it is unclear whether it is of stellar origin or is due to an incorrect nebular subtraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In any case, this SB3 system is now a SB2+Cas following the SBS nomenclature of Ma´ız Apell´aniz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2019b) 2MASS J10435902−5933196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' We identify this star in Villafranca O-002 (Trumpler 14) as a He-rich B star for the first time using GIRAFFE data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2MASS J10441829−5942296.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' We identify this star in Villafranca O-003 (Trumpler 16 W) as a He-rich B star for the first time using GIRAFFE data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2MASS J10453807−5944095.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' This object in Villafranca O- 025 (Trumpler 16 E) is identified as an O star for the first time here using GIRAFFE data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' It has an O8 Vz spectral classification and a high extinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2MASS J10440744−5916399.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' This is a background object caught as an SB2 but with an uncertain GIRAFFE classification of O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7: + B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5:.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' If confirmed, it would be another new O star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The derived Gaia EDR3 distance is d = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='45+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='56 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='45 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' [ESK2003] 148 = [S87b] IRS 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' This system was first iden- tified as an O-type candidate at the distance of Car OB1 by Damiani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The identification was based on a pho- tometric analysis with CHORIZOS (Ma´ız Apell´aniz 2004) and resulted in values of Teff = 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='4 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='4 kK, E(4405 − 5495) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='351 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='020 mag, and R5495 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='92 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='09, which indicates an O star with both large color excess and anomalous extinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' It is classified as O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 V(n) both in GOSSS IV and here using GIRAFFE, so the Teff appears to be slightly lower than the value measured with CHORIZOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' It is highly reddened but with a po- sition and parallax consistent with being in Villafranca O-025 (Trumpler 16 E), likely slightly behind the rest of the cluster and immersed in the molecular cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2MASS J10471498−5953374.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' A GIRAFFE spectrum yields the spectral classification B6: IIIe, with a double-peaked emis- sion line in Hα but no emission in Hγ (no other Balmer lines are covered by the GIRAFFE data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The derived Gaia EDR3 dis- tance is d = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='14+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='29 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='25 kpc, placing it in the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2MASS J10443089−5914461.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' This object is classified as a highly extinguished supergiant with spectral type O7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 II(f) us- ing either GOSSS data in GOSSS IV or GIRAFFE data here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Alexander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2016) classified it as O8 V but it is clearly not a dwarf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' It has a large parallax uncertainty: it could not be dis- carded as being in Car OB1 but it is more likely a background object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2MASS J10453185−6000293.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' This highly extinguished O star in Villafranca O-030 (Bochum 11) is identified as an O star for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' It receives a spectral classification of O7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V in GOSSS IV using GOSSS data and a slightly later one of O8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V here using GIRAFFE data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The latter is rather noisy but some lines show signs of asymmetry, indicating a possible spectro- scopic binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' [ARV2008] 217 = [S87b] IRS 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' This object is one of the most interesting discoveries in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Using GOSSS data in GOSSS IV the authors give it an O3: III: spectral classifica- tion and using GIRAFFE here we arrive at the same classifica- tion but with an (n) suffix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In both cases the O3: classification is based on the apparent absence of He i 4471 but the two spectra are too noisy to provide a more accurate classification based on the N lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Therefore, it is a new member of the limited family of Galactic O stars with spectral types earlier than O4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' It was first identified as an O-type candidate at the distance of Car OB1 by Damiani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The CHORIZOS analysis there gives Teff = 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='0 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 kK, E(4405 − 5495) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='932 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='021 mag, and R5495 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='60 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='06, that is, an early-type O star with both large color excess and anomalous extinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' That analysis is in good agreement with the spectral classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' That object po- sition and parallax are consistent with [ARV2008] 217 being in Villafranca O-025 (Trumpler 16 E), making it the earliest O-type star there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2MASS J10431945−5944488.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The Gaia EDR3 parallax for this object yields d = 794+28 −26 pc, clearly making it a foreground (and very blue) object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The existence of broad Hγ and He ii lines indicate that the spectrum is dominated by an sdO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' However, He ii 4542 and He ii 4686, observed at different epochs, have dif- ferent velocities and some lines appear to originate in a later-type star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Therefore, the system is a spectroscopic binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2MASS J10431945−5944488.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The Gaia EDR3 parallax for this object yields d = 794+28 −26 pc, clearly making it a foreground (and very blue) object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The existence of broad Hγ and He ii lines indicate that the spectrum is dominated by an sdO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' However, He ii 4542 and He ii 4686, observed at different epochs, have dif- 8 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula ferent velocities and some lines appear to originate in a later-type star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Therefore, the system is a spectroscopic binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' High-extinction population of Preibisch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' That pa- per lists several stars that were too faint to be observed with GIRAFFE in the blue-violet region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Regarding their Gaia EDR3 parallaxes, most of them are similar to or smaller than that of Car OB1 but with larger uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' There is only one with negative parallax, so it is likely a background object: 2MASS J10452648−5946188 (=[HSB2012] 3994), which was already identified as a highly-extincted B star by Damiani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2017) using CHORIZOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Results and discussion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The observed CMD and completeness We first discuss the observed Gaia EDR3 CMD for the sample of 316 objects in this paper, which is plotted on the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Of those, only four are located in the foreground but two of them are in distinct regions of the CMD: HD 93 420, a RSG in the upper right, and 2MASS J10431945−5944488, a sdO in the lower left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The main group, the 294 objects in Car OB1, do not follow the typical isochrone of a cluster or association be- cause of the strong differential extinction present in the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The majority concentrates between the extinguished isochrones that correspond to E(4405 − 5495) of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='3 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='6 (assuming an R5495 of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5) but some are significantly more extinguished than that, including the four Preibisch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2021) objects outside the frame towards the lower right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The 18 background objects have, on average, a higher extinction than the Car OB1 popu- lation, an expected effect of the extinction associated with the Carina Nebula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' They appear mixed in the vertical direction with the Car OB1 population but one should consider that if we plot- ted absolute magnitude on the vertical axis, they would move up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' For example, five of the 18 objects are B supergiants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The majority of the stars lie between the R5495 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 extinc- tion tracks for average MS stars of Teff = 20 kK and 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 kK stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' A significant fraction lies below the track of Teff = 20 kK, due to a combination of different effects: an average-age B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V star can have a Teff somewhat lower than 20 kK, ZAMS stars should be lower in the CMD than average-age ones, and the ex- tinction tracks for R5495 > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 (which is known to be appropriate for some stars in the Carina Nebula, see Ma´ız Apell´aniz & Barb´a 2018) are steeper than the plotted ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Above the extinction track of Teff = 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 kK we find ten Car OB1 stars: η Car, the two RSGs, WR 24, four O supergiants, and HD 93 250 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B (a close binary with two very early type components, see Le Bouquin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2017), that is, all of them objects that are expected to be there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The most notorious feature of the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2 is how well the Car OB1 O and B stars are separated in the CMD, with the O stars mostly above the average-age extinction track of Teff = 30 kK for R5495 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 and the B stars below it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' This is an indirect confirmation of the quality of the spectral clas- sifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The separation is not perfect but it is not expected to be for several reasons: B giants and supergiants (plus some early B + early B binaries) are expected to be above the average- age extinction track of Teff = 30 kK for R5495 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 and late O-dwarfs near the ZAMS below that track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In addition, varia- tions in R5495 among sightlines should produce some mixing, as low values of R5495 can move B stars into the O-star territory and high values of R5495 can move O stars into the B-star terri- tory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Examples of the latter possibility are two of the stars from the Preibisch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2021) sample, 2MASS J10454595−5949075 and 2MASS J10452013−5950104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' If they are confirmed to be normal O dwarfs, their value of R5495 should be high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' When building a census, one of the most important ques- tions that have to be addressed is how complete it is and that is especially important when the sample is built from multiple sources such as in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' To answer that question, we have plotted in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2 all the Gaia EDR3 sources found within the footprint that have positive corrected parallaxes consistent with being at the distance of Car OB1 and that have catalog values of GBP3 − GRP3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The right panel shows that the left panel is just the tip of the iceberg in terms of a moderately extinguished well-populated main sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In addition to that main sequence, a significant population of red stars is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' By comparison with the left panel, some of those are extin- guished OB stars but a comparison with other Galactic sight- lines indicates that most of them must be intrinsically red stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' For example, the diagonal series of stars that follows the ex- tinction track of Teff = 30 kK around GBP3 − GRP3 ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 is the red-clump extinction sequence, ubiquitously seen in the Galactic plane when plotting absolute magnitude in the vertical axis (as we are effectively doing here by selecting a population consistent with being at the same distance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' That sequence starts around GBP3 − GRP3 ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='1 for zero extinction and here we are just see- ing it with an extinction distribution not too different from that of the OB stars in Car OB1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Given the dominance of the late-type population for red col- ors (something that needs to be addressed with additional data such as NIR photometry), we do not have the means to deter- mine how complete the sample is for high extinctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Indeed, that is why a paper as recent as Preibisch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2021) was able to find several new O stars in Car OB1: thick dust clouds can eas- ily hide OB stars if one does not have access to IR data and even in that case finding the hot needle in the cool haystack is not al- ways straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Therefore, we concentrate on the low- and moderate-extinction part of the sample, defined as those OB stars with GBP3 − GRP3 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='0 (just to the left of the point where the first red clump stars are expected to appear).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Also, as for fainter stars one expects any sample to be less complete, we restrict the completeness analysis to the region above the extinction track of Teff = 20 kK in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In other words, we are assessing how complete the sample is regarding low/moderate extinction O and early-B stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' We cross-matched the two samples (the one used through- out this paper and the full Gaia EDR3 one) inside that area and found 154 coincidences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Three objects in the main sample are not present in the Gaia EDR3 sample either because they lack parallaxes (CPD −59 2636 A,B and ALS 19 740) or because they are completely absent (HD 93 129 Ab), note that η Car would be also absent if it were inside that area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Gaia EDR3 is quite complete barring a few small-separation binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' As for the other way around, 19 systems in the Gaia EDR3 sam- ple are not present in the main one (green stars in the right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Of those, five have large external parallax uncertainties (> 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='1 mas), so chances are they are not real Car OB1 mem- bers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Therefore, we estimate that our sample is around 90% complete for low/moderate extinction O and early B systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Furthermore, the location of the 19 green stars in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2, all of them below the extinction track of Teff = 30 kK, suggests that those missing objects are likely of early-B type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Therefore, we conclude that we are missing very few or even no low/moderate O-type systems in Car OB1 within our footprint in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' As mentioned above, objects with high extinc- tion may be another story.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In any case, the 74 Car OB1 O-type 9 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='0 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 30 20 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='0 GBP3−GRP3 4 5 6 7 8 9 10 11 12 13 14 15 16 G3’ foreground background Car OB1 WR or (non−O) sg Car OB1 O Car OB1 other Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' First panel, see next page for the second one: Gaia EDR3 CMD for the stars with spectral types in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Different symbols and colors are used to represent stars with parallaxes compatible with being (or otherwise assumed to be) in the foreground (4), in the background (18), or in Car OB1 (294).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Of the Car OB1 stars, 6 are of Wolf-Rayet or non-O supergiant (II to Ia) spectral class, 74 are of O type, and 214 are of non-supergiant B type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Four of the Preibisch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2021) stars are outside the frame towards the lower right due to their high extinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Black lines show the average main sequence at a distance of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='35 kpc with no extinction and with values of E(4405 − 5495) of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='0 (labelled) using the extinction law of Ma´ız Apell´aniz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2014) with a value of R5495 of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5, which is typical of the region but with a large dispersion (Ma´ız Apell´aniz & Barb´a 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Solid orange lines show the R5495 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 extinction tracks for average MS stars of Teff of 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 kK, 30 kK, and 20 kK (labelled), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The dotted orange line shows the R5495 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='0 extinction track for Teff = 30 kK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' systems in this paper are the largest nearly complete sample of objects of that spectral type in any part of a Galactic OB associ- ation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Binary fraction It is well known that multiplicity among massive stars is ubiq- uitous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Commonly, multiplicity is divided into that which is de- tected through spectroscopy (velocity changes and differences) and imaging (or visual multiplicity) and is important to indicate which one is being used, as some previous studies have conflated them and caused confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In GOSSS II we analyzed the pop- ulation of Galactic southern stars and found out that 65-91% of them are multiple stars of one type or another, with the values for spectroscopic and visual multiples being 50-60% and 30-76%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' One consequence of those numbers is that a signif- icant fraction (at least 15%) are at the same time spectroscopic and visual multiples, and most of those involved three stars, as in 2014 the number of pairs detected simultaneously with spec- troscopy and imaging was quite low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' An analysis of known mul- tiple O stars in the northern hemisphere (Ma´ız Apell´aniz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2019a) confirmed the trend towards systems of three or more stars and revealed that simple binaries are a minority once spec- troscopic and visual multiples are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' For example, hier- archical triples composed of a short-period (less than 1 month) system orbited by a companion in a long-period (years or more) orbit are quite common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In the census presented here we find 20 spectroscopic binary systems containing at least one O-type star (listed in Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5), one of them located in the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' There are six new sys- tems reported for the first time in this work, either from GES and/or GOSSS IV observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Excluding the background sys- 10 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='0 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 30 20 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='0 GBP3−GRP3 4 5 6 7 8 9 10 11 12 13 14 15 16 G3’ outside triangle inside triangle, matched inside triangle, unmatched Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Second panel, see previous page for the first one: Equivalent plot but for all Gaia EDR3 stars in the region of interest with corrected parallaxes that are compatible with the distance to Car OB1 and positive and with catalog values of GBP3 − GRP3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The plotted objects are classified according to whether they are located inside or outside of the area limited by GBP3 − GRP3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='0 and the R5495 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 extinction track for average MS stars with Teff = 20 kK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Stars inside that area are further divided into those matched with objects in the left panel (154) and those unmatched (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Note that an additional three stars inside the above mentioned area in the left panel (HD 93 129 Ab, CPD −59 2636 A,B, and ALS 19 740) plus η Car outside the that area are not shown either because they are not included in Gaia EDR3 or have no parallaxes there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' tem, the total number of O stars in the 19 systems of Car OB1 is 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' This represents a fraction of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='35 (30 out of a total of 85 O-stars, see Table 2 where binary statistics and fractions for the spectroscopic systems containing at least one O-type star in the different Villafranca groups are summarized).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' This number is still far to those reported in GOSSS II and also to the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='44 frac- tion of O-type stars in binaries quoted by Sana & Evans (2011) or even the somewhat more than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='50 indicated by Sana (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' This indicates that there is a significant number of binaries still to be identified in the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' We highlight that the multiplicity statistics reported in this work is, however, incomplete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' We only report double-line spectroscopic binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Visual binaries are not considered and only a small fraction of the sample has signif- icant multi-epoch coverage for single-line spectroscopic binary detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In spite of this, we significantly increase the fractions quoted by Sana & Evans (2011) in Trumpler 14 and Trumpler 16, for which these authors quoted fractions of zero and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='48, respectively (see Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In addition, we found 18 spectroscopic binary systems formed by early B-type stars, one of them a background sys- tem and 13 new binary detections from GES, GOSSS and LiLiMaRlin observations (see Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 3 we show an example of the new spectroscopic binary detections from GES spectra at their original resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The Si III triplet at λ4552-68- 75 Å is shown for binary late-O (a and b panels) and early-B (c and d panels) systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' See figures in Appendix A for full spectra details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The spectroscopic n-qualifier as an indicator for rotation As stated in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='3, the MGB tool has been used for the spec- tral classification of GES data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' This tool allows the user to obtain not only the spectral subtype and luminosity classification, but also spectral peculiarities and the rotation index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Broadening is denoted by (n), n, nn and nnn indexes, progressing from some- what to more and even more broadened lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Therefore, this 11 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula Wavelength (A) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='05 Normalized flux HDE 303 312 (O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 III + B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: V) (a) I P I S I P I S HDE 303 312 (O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 III + B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: V) (a) I P I S I P I S HDE 303 312 (O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 III + B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: V) (a) I P I S I P I S Wavelength (A) CPD -58 2649 (O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: + B0) (b) I P I S I P I S CPD -58 2649 (O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: + B0) (b) I P I S I P I S CPD -58 2649 (O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: + B0) (b) I P I S I P I S 4550 4555 4560 4565 4570 4575 Wavelength (A) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='05 Normalized flux CPD -59 2661 (B0: + B) (c) I S I P I S I P CPD -59 2661 (B0: + B) (c) I S I P I S I P CPD -59 2661 (B0: + B) (c) I S I P I S I P 4550 4555 4560 4565 4570 4575 Wavelength (A) HD 93 128 B (B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2: V + B1: V) (d) I P I S I P I S HD 93 128 B (B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2: V + B1: V) (d) I P I S I P I S HD 93 128 B (B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2: V + B1: V) (d) I P I S I P I S Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Example of new spectroscopic binary systems reported in this work: Si iii line profiles from GES spectra shown at their original resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' For reference, P and S letters indicate the position of the primary and secondary component, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' See figures on Appendix A for full spectra details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Binary statistics and fractions for the spectroscopic systems containing at least one O-type star present in the census.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Group Single Binary O stars Fraction O-stars O-systems in binaries O-002 (Trumpler 14) 10 3 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='33 O-003 (Trumpler 16 W) 2 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='60 O-025 (Trumpler 16 E) 8 6 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='55 O-027 (Trumpler 15) 3 0 O-028 (Collinder 228) 16 3 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='15 O-029 (Collinder 232) 2 0 O-030 (Bochum 11) 4 1 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='33 Car OB1 10 4 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='37 Whole sample 55 19 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='35 Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Background and foreground members are excluded from the statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' We separate numbers considering the different Villafranca groups of Carina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Car OB1 group refers to the stars just falling in the gaps between defined Villafranca groups (see Appendix A) for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' qualifier has been traditionally interpreted as a sign for high ro- tational velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' As a consistency check, we have determined the projected rotational velocity of stars with this qualifier, in order to know whether there is a 1:1 relationship between both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' To that aim we used iacob-broad, a user-friendly tool for the line-broadening characterization of OB stars (Sim´on- D´ıaz & Herrero 2007, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' It is based on a combined Fourier Transform (FT) and the Goodness-of-fit (GOF) method that al- lows us to determine easily the stellar projected rotational ve- locity (v sin i) and the amount of extra broadening (vmac) from a specific diagnostic line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The FT technique is based on the iden- tification of the first zero in the Fourier transform of a given line profile (Gray 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Sim´on-D´ıaz & Herrero 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The GOF technique is based on a comparison between the observed line profile and a synthetic one that is convolved with different values of v sin i and vmac to obtain the best-fit by means of a χ2 optimiza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The main advantage of this methodology is that we obtain two independent measurements of the v sin i (resulting from ei- ther the FT or the GOF analysis) whose comparison is used as a consistency check and to better understand problematic cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Since metallic lines do not suffer from strong Stark broaden- ing or nebular contamination, they are best suited for obtaining accurate v sin i values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' GIRAFFE set-ups cover the Si iii λ4552 diagnostic line, while UVES/FEROS/HARPS set-ups also cover the O iii λ5592 diagnostic line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In case none of them are present or are too weak, we then use the nebular free or weakly contam- inated He i lines (He i λ4387, λ4471, λ4713).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 4 presents the v sin i histogram of those OB stars in- cluded in our census with any broadening index in their spec- tral classification (see Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Mean v sin i values for (n), 12 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula 50 100 150 200 250 300 350 400 450 500 550 v sini (km/s) 0 5 10 15 20 25 30 35 N (n) n nn nnn Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' v sin i histogram of those OB stars with any rotation index in their spectral classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Broadening is denoted by (n), n, nn and nnn indexes, progressing from somewhat to more broad- ened lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Points and horizontal lines on the top of the figure indicate the mean v sin i value for each rotating group and the corresponding dispersion, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' n, nn and nnn are 206, 265, 317 and 392 km s−1, respectively, which confirms the trend that the higher the rotation index the higher the projected rotational velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' However, we find a sig- nificant overlap in the ranges of projected rotational velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' For example, the (n)- and n-type stars peak at the same bin: be- tween 200 and 250 km s−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' and the fastest n-star rotates 88 km s−1faster than the slowest nn-star (but the distribution of (n)-stars is slanted towards the left from that point while that of n-stars is slanted towards the right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' There are two likely explanations for the overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' On the one hand, a non-negligible fraction of these stars could end up being actually spectroscopic binaries since such broadened lines may prevent us from detecting bi- nary line profiles when multi-epoch observations are not avail- able.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' On the other hand, the sample is dominated by B1-B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 dwarfs, which have few intrinsically deep and narrow lines in the analyzed wavelength range (which is only a part of the stan- dard blue-violet classification range), making the n-type indexes more unreliable than for e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' O stars or B supergiants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Runaway candidates Benefiting from the high-precision astrometry that Gaia EDR3 provides in Carina, we have investigated the proper motions of the stars in our census to identify bona-fide runaways as a first step for future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Following the ideas of de Mink et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2013) (who propose that fast rotating stars are the product of post-interacting binaries and therefore could also have been ejected from binary systems in which the mass donor exploded as supernova) we are interested in exploring whether there is a connection between O and B-type stars with the spectroscopic n-qualifier5 and the runaway status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 5 as a proxy for fast rotation, although we emphasize that, even if there is a good correlation with the average v sin i, not all stars with this qualifier have a high projected rotational velocity, as shown in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The proper motion distribution for each Villafranca group is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' We define group centers through an iterative process assuming the average values of each group members, but excluding detected binaries, objects with a RUWE (renor- malised unit weight error) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='4 and those stars that do not com- ply with the proper motion constraint described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' For ref- erence, group proper motions in α∗ and δ derived in this work and those from the Villafranca II and III works are shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' As in Villafranca II, we find the proper motion of O- 025 not identical to that of O-003, indicating that both groups in Trumpler 16 are well separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' We iteratively filter stars with proper motions larger than the mean values for each group by more than three sigma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' To this aim, we calculated for each group σg = � σ2µα∗ + σ2µδ, deriving a final mean σg of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='342 mas a−1 and thus a three sigma value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='03 mas a−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' We find four stars with the n-qualifier in their spectral classification that do not meet the imposed constraint (those stars falling outside the circles in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Three of them (2MASS J10440866-5933488 in O-002, CPD -59 2541 in O-028 and 2MASS J10451588- 5929563 in O-029) can be considered firm runaway candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' We note, however, that the large RUWE value for CPD -58 2657 (in the O-027 group) indicates inaccurate astrometric measure- ments (see Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The rest of stars with the n-qualifier are homogeneously distributed around the core motion of each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Interestingly, the two extreme very fast B rotators of our sample, ALS 15 248 and 2MASS J10433865-5934444 rotating both at v sin i > 450 km s−1, do not show peculiar proper motions, and so are consistent with the main values of each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' We also find four further runaway candidates that are not included in the group with the n-qualifier but show peculiar proper motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Two of them are RSG stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' We note that two stars identified in Villafranca I as possible runaway stars ejected from Trumpler 14 (HDE 303 313 and ALS 16 078) spatially fall in the gaps of the redefined Villafranca groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Therefore, they have been labelled as just Car OB1 members and are not discussed here6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Thus we have four out of 90 stars with the n-qualifier iden- tified as candidate runaway objects and another four out of 168 without the n-qualifier, which means fractions of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='4% and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='4%, respectively 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' This points to a connection between runaways and fast rotators, as pointed out by other works (see f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' de Mink et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Holgado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2022, and references therein) particu- larly if we consider that the viewing angle may be affecting the projected rotational velocities, resulting in less broadened lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' However, given the limitations of our work, further research on this topic (in particular, a distribution of projected rotational ve- locities and a more detailed study of the runaway condition) is needed in order to obtain a firm conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Finally, we remark that the distribution of binary systems in the proper motion diagram (crosses in Fig 5), contrary to what might be expected, is homogeneously distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' A similar pat- tern was found in the Cygnus OB2 association (Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2020), implying that these systems may still keep their original velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 6 a direct comparison between the runaway candidates identified in both works must be done with caution since different methods have been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Note that in Villafranca works, stars are selected as candi- date runaway/walkaway objects when their proper motion points in the opposite direction to that of the center of the group (within some mar- gins, see Villafranca I-II for further details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 7 Note that detected spectroscopic binary systems, with or without the n-qualifier, have been excluded from the statistics 13 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula 8 7 6 5 (mas a 1) 1 2 3 4 (mas a 1) O-002 8 7 6 5 (mas a 1) O-003 8 7 6 5 (mas a 1) O-025 8 7 6 5 (mas a 1) O-027 8 7 6 5 (mas a 1) 1 2 3 4 (mas a 1) O-028 8 7 6 5 (mas a 1) O-029 8 7 6 5 (mas a 1) O-030 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Proper motion distribution from Gaia EDR3 astrometry for all stars of our census in each assigned Villafranca group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Orange squares indicate those OB stars analyzed in this work that are rotating at v sin i ≥ 200 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Blue crosses represent identified binary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Circles represent group proper motion constraints, whose centers µα∗,g and µδ,g are those shown in Table 3 in the central columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' For comparison, red plus symbols indicate group centers from Villafranca II and III works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Note that stars labelled as Car OB1 members are not included in the panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Group proper motions in α and δ derived in this work and those from the Villafranca II and III works, all based on Gaia EDR3 astrometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' This work Villafranca II-III Group N µα∗,g µδ,g µα∗,g µδ,g O-002 (Trumpler 14) 32 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='580 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='240 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='089 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='246 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='534 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='023 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='076 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='023 O-003 (Trumpler 16 W) 8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='179 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='102 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='730 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='103 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='128 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='024 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='670 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='024 O-025 (Trumpler 16 E) 47 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='894 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='214 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='647 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='177 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='877 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='023 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='596 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='023 O-027 (Trumpler 15) 17 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='172 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='164 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='224 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='175 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='282 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='023 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='131 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='023 O-028 (Collinder 228) 53 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='896 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='334 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='332 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='396 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='713 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='021 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='070 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='021 O-029 (Collinder 232) 12 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='667 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='414 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='238 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='294 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='552 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='023 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='142 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='023 O-030 (Bochum 11) 19 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='559 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='204 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='328 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='307 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='635 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='021 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='279 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='021 Note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Group uncertainties reported in this work refer to the standard deviation of the selected OB stars while those in Villafranca II-III correspond to the standard deviation of the mean (with the angular covariance term included) of all the stars identified as group members, a much larger number than the one used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Conclusions We present a new census of massive stars in the central part of Carina, Car OB1, based on high-quality spectroscopic data provided by GES, GOSSS, LiLiMaRlin and additional sources from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' It contains a total of 316 massive stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' We separated stars with distances compatible with Car OB1 (assign- ing group membership) from those in the foreground and back- ground, finding four systems in the foreground and 18 in the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Of the 294 stellar systems in Car OB1, 74 are of O type, 214 are of non-supergiant B type and six are of WR or non-O supergiant (II to Ia) spectral class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' We estimate that our sample is around 90% complete for low/moderate extinction O and early B systems, missing very few or even no O stars within our footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The 74 Car OB1 O-type systems quoted in this paper are the largest nearly complete sample of objects of that spectral type in any part of a Galactic OB association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Among the stellar census, we identified 20 spectroscopic bi- nary systems that contain at least one O-type star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Six of them are new identifications and one is located in the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The observed binary fraction of O stars found in the Car OB1 region is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='35, although this number only refers to double-lined spec- troscopic binaries and represents, therefore, a lower limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Visual binaries are not considered and only a small fraction of the sam- 14 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula ple has significant multi-epoch coverage for single-lined spectro- scopic binary detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Thus, this number should be considered as a lower limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' In addition, we found another 18 spectroscopic binary systems with a B-star primary, one of them being a back- ground system and 13 of them new binary detections from GES, GOSSS and LiLiMaRlin observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' We explore the correlation between the spectroscopic rota- tion index, n, and the actual projected rotational velocities of the stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' We find a good correlation of the average v sin i values with the qualitative classification of each group ((n), n, nn, nnn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' However, there is a significant overlap in their v sin i ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' We note that it is possible that a non-negligible fraction of these stars are actually spectroscopic binaries contaminating the fast rota- tors sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Finally, we investigated the proper motion distribution for the sample of those O and B-type stars with a spectroscopic n- qualifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Our results indicate a connection between runaways and fast rotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Furthermore, the distribution of binary systems in the proper motion diagram is homogeneously distributed, im- plying that these systems may still keep their original velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' This paper is based mainly on data products from spec- troscopic observations made with ESO Telescopes at the Paranal Observatory under programme ID 188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B-3002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' These data products have been processed by the Cambridge Astronomy Survey Unit (CASU) at the Institute of Astronomy, University of Cambridge, and by the FLAMES/UVES reduction team at INAF/Osservatorio Astrofisico di Arcetri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' These data have been obtained from the Gaia-ESO Survey Data Archive, prepared and hosted by the Wide Field Astronomy Unit, Institute for Astronomy, University of Edinburgh, which is funded by the UK Science and Technology Facilities Council.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' This work was partly supported by the European Union FP7 programme through ERC grant number 320360 and by the Leverhulme Trust through grant RPG-2012-541.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' We acknowledge the support from INAF and Ministero dell’ Istruzione, dell’ Universit`a’ e della Ricerca (MIUR) in the form of the grant ’Premiale VLT 2012’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The results presented here benefit from discussions held during the Gaia- ESO workshops and conferences supported by the ESF (European Science Foundation) through the GREAT Research Network Programme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Additional spectra were obtained using the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 m du Pont Telescope at the Observatorio de Las Campanas (LCO) and the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 m MPG/ESO Telescope at the Observatorio de La Silla (LSO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' This work has made use of data from the European Space Agency (ESA) mission Gaia, processed by the Gaia Data Processing and Analysis Consortium (DPAC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' This research is partially funded by the Spanish Government Ministerio de Ciencia e Innovaci´on and Agencia Estatal de Investigaci´on (MCIN/AEI/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='130 39/501 100 011 033/FEDER, UE) through grants PGC2018- 93 741-B-C21/C22, PGC2018-95 049-B-C21/C22 and PID2021-122 397NB- C21/C22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' SRB also acknowledges funding by MCIN under the Juan de la Cierva - Formaci´on grant (contract FJC 2020-45 785-I) and NextGeneration EU/PRTR and MIU (UNI/551/2021) through grant Margarita Salas-ULL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' also acknowledges support by the Severo Ochoa Program through CEX2019-000920-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='A also acknowledges financial support from the State Agency for Research of the Spanish MCIU through the “Center of Excellence Severo Ochoa” award to the Instituto de Astrof´ısica de Andaluc´ıa (SEV-2017-0709).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' MB is supported through the Lise Meitner grant from the 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2015, in HSA 8, 603–603 Ma´ız Apell´aniz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=', Alfaro, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=', & Sota, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2008, arXiv:0804.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2553 Ma´ız Apell´aniz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' & Barb´a, R.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 1982b, AJ, 87, 1300 Walborn, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 1995, in RMxAC, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2, 51–55 Walborn, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2009, in STScI Symposium Series, Massive Stars from Pop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' III and GRBs to the Milky Way (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Livio and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Villaver eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' ), Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 20, 167 Walborn, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' R.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' & Hesser, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 1975, ApJ, 199, 535 Walborn, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=', Howarth, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=', Lennon, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2002b, AJ, 123, 2754 Walborn, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' & Liller, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 1977, ApJ, 211, 181 Walsh, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 1984, A&A, 138, 380 Wolk, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=', Broos, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=', Getman, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2011, ApJS, 194, 12 Appendix A: Tables and spectrograms In this Appendix we present the tables with the information for the stars in the field of the Carina Nebula analyzed in this paper and the figures with the spectrograms that have not appeared in previous papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='1 lists the basic information for the stars: name, co- ordinates, identifications, G′ 3 magnitude, Gaia EDR3 parallax and group membership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Regarding the latter, the following al- gorithm is used: – All stars are initially labelled as Car OB1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' – A search is done to see if the star is located inside the re- gion of the sky defined by the center and radius of one of the groups defined in Villafranca II or III (Villafranca O- 002 = Trumpler 14, Villafranca O-003 = Trumpler 16 W, Villafranca O-025 = Trumpler 16 E, Villafranca O- 027 = Trumpler 15, Villafranca O-028 = Collinder 228, Villafranca O-029 = Collinder 232, and Villafranca O- 030 = Bochum 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' If found, then the membership is changed to that group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Note that, as mentioned in the Villafranca papers, the traditional division into such groups is to some point arbitrary: Villafranca O-002, Villafranca O- 025, and Villafranca O-027 are likely real clusters while the rest of the groups are just subassociations of the larger Car OB1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' As the apertures in the Villafranca papers are cir- cular, some stars just fall in the gaps and remain labelled as Car OB1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' – Stars without a parallax or with corrected parallax uncertain- ties greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='1 mas are given in bold face, to indicate that the Gaia EDR3 information is not sufficient to determine their distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Note that many of those objects are known to be located in Car OB1 for other reasons (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' η Car or HD 93 129 Ab).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' – Stars whose parallax is larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='44 mas by more than three sigmas are labelled as foreground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' – Stars whose parallax is smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='41 mas by more than three sigmas or is negative are labelled as background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The limits here and in the previous steps are determined from Villafranca II and III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 lists the spectral types for the stars in this pa- per.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Three types of spectral types are given: those derived from Gaia-ESO spectra (all new), those derived from GOSSS spectra (most from previous papers and from GOSSS IV but some new, marked as TW), and those derived from LiLiMaRlin and STIS spectra as well as from the literature (all previously published).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Tables A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='3 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='4 list those stars of the census identified in the foreground or in the background, and those of WR or non-O supergiant (II to Ia) spectral class, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Tables A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='6 list spectroscopic binary systems iden- tified in our census containing at least one O-type and those formed by early B-type stars, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 lists runaway candidates identified in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='1 shows the UVES spectra, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 the GIRAFFE spectra, and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' A.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' The photometry listed for Aa is actually the combined one for Aa,Ab, G′ 3 for Aa should be ∼7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 17 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (Continued).' metadata={'source': 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2MASS J10450879−5950537 10:45:08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='798 −59:50:53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='72 287.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='68−00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='77 01 — 5 350 308 659 792 838 016 — 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2377 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='3508 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='4992 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='8127 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='183±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='180 Car OB1 22 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Spectral classifications for the stars in the field of the Carina Nebula analyzed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Name Gaia-ESO GOSSS LiLiMaRlin + STIS + literature ST LC qual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' ST LC qual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' ST LC qual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' η Car — — — — LBV — — — M22 F: I — — W77 HD 93 420 — — — — — — — — — M4 Ib — — M23b QZ Car Aa,Ac — — — — O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 Ib n — S14 O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 Ib — O9 II: M23a WR 24 — — — — WN6 — ha — TW WN6 — ha-w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' — H06 HD 93 281 — — — — — — — — — M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 Iab — — M23b HDE 303 310 — — — — — — — — — M3 Iab — — M23b HD 93 129 Aa — — — — O2 I f* — S14 O2 I f* OB?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' M17 HD 93 403 — — — — O5 I fc O7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V M23a O5 I fc O7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V M23a HD 93 250 A,B — — — — O4 IV (fc) — M16 O3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V ((f+)) — W02 HD 93 205 — — — — O3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V ((f)) O8 V S14 O3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V ((f)) O8 V M20 HD 93 160 A,B — — — — O7 III ((f)) — S14 O6 III (f) — W72 HD 93 162 — — — — O2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 I f*/WN6 OB S14 WN6 — h O4 V01 HD 93 130 — — — — O6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 III (f) — S14 O6 III (f) — W72 HD 93 222 A,B — — — — O7 V ((f)) — M16 O7 III ((f)) — W73b HDE 303 308 A,B — — — — O4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V ((fc)) — S14 O4 V ((f+)) — W02 HD 93 249 A O9 III — — O9 III — — S14 O9 III — — W73a HD 93 028 — — — — O9 IV — — S14 O9 V — — W72 HD 93 146 A — — — — O7 V ((f)) — M16 O6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V ((f)) — W73b HD 93 204 O5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V ((f)) — O5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V ((f)) — S14 O5 V ((f)) — W02 HD 93 190 — — — — O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7: V: (n)e — S14 B0 IV pe — M55 HDE 305 523 — — — — O9 II-III — — S14 O9 II — — W73a CPD −59 2600 — — — — O6 V ((f)) — S14 O6 V ((f)) — W73b HD 93 161 B O6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 IV ((f)) — O6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 IV ((f)) — S14 O5 — — — T73 HD 93 161 A O7 V ((f)) O9 IV O7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — O9 V S14 O7 V ((f)) O9 IV M23a HD 93 129 Ab — — — — — — — — — O2 I f* — M17 HDE 305 520 — — — — — — — — — B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 Iab — — M23b HD 93 128 — — — — O3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V ((fc))z — S14 O3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V ((f+)) — W02 HD 93 027 O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 IV — — O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 IV — — S14 O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — W73b V572 Car O6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V z B0 V + B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: V O7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V (n) B0 V(n) M16 O6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V z B0 V +B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 V M23a HD 93 129 B — — — — O3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V ((fc))z — M22 O3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V ((f+)) — W02 HD 92 877 A — — — — — — — — — B2 III — — M23b HDE 305 438 — — — — O8 V z — S14 O8 — — — T74 CPD −59 2554 O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 V — B1: V O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 IV — — S14 O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 V — B1: V M23a HD 93 342 — — — — — — — — — B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 Ib — — M23b HDE 305 536 — — — — O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — S14 O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — M01 HDE 303 311 — — — — O6 V ((f))z — S14 O5 V z — W09 HD 93 056 B1: V: n — — — — — — O9 V — B2 V A16 HD 93 501 — — — — — — — — — B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: III: (n)e — M23b HDE 305 437 — — — — B0 V — — TW B0 V — — A16 CPD −59 2641 — — — — O6 V ((fc)) — S14 O5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5-6 V ((fc)) B2 V-III R09 HD 93 620 — — — — — — — — — B2 III — — M23b CPD −59 2635 O8 V z O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 V O8 V (n) O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V S14 O8 V z O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 V M23a CPD −59 2592 — — — — — — — — — B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 Ia — — M23b HDE 305 524 O6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V n((f))z — O6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V n((f))z — S14 O6 III n — V93 CPD −58 2620 — — — — O7 V ((f))z — S14 O6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V ((f)) — W73b CPD −59 2551 — — — — O9 V — — S14 O9 V — — V93 HDE 305 439 A B0 Ia — — — — — — — B0 Ia — — V93 HDE 303 299 — — — — — — — — — B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: III: (n)e — M23b HD 93 249 B B0: V (n) — B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 V (n) — TW B0 V — — A16 HDE 305 535 B4 III (n) — — — — — — B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — A16 HDE 305 452 B2 III — — — — — — — B8/9 — — — L76 HD 93 343 — — — — O8 V — — M16 O7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V z O7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: V(n) M22 CPD −58 2611 — — — — O6 V ((f))z — S14 O6 V ((f)) — W82 V573 Car — — — — O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V (n) B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V(n) S14 O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 IV — B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V M23a 23 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (Continued).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Name Gaia-ESO GOSSS LiLiMaRlin + STIS + literature ST LC qual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' ST LC qual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' ST LC qual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' CPD −59 2636 A,B O7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — O8 V + O8 V O8 V — O8 V S14 O7 V — O8 V + O9 V A02 HDE 303 316 A — — — — O7 V ((f))z — S14 O6 V — — F81 HDE 305 518 — — — — O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 III — — S14 O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — L82 HDE 305 534 — — — — — — — — — B0 V — B1: V M23b CPD −59 2624 O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 V — — O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 IV — — S14 O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — A16 HDE 305 522 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 V — B1: V — — — — — B0 V — B1 V A16 HDE 305 543 — — — — — — — — — B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 V (n) B1: V(n) M23b HDE 303 300 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — B — — — — — B1 V — — A16 HD 93 097 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 V: n — — — — — — B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 IV — — A16 HDE 305 525 — — — — O5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V ((f))z O7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V + B M16 O4 V — — V93 HDE 305 521 — — — — — — — — — B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 V (n) — M23b CPD −59 2574 — — — — — — — — — B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — M23b HDE 305 516 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V (n) B2: V — — — — — B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — G11b CPD −59 2626 A,B O7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V (n) — O7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V (n) — M16 O7 V n — L82 HDE 303 312 O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 III — B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: V O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 IV — — S14 O9 V — — F81 CPD −58 2605 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 II — — — — — — — B0 III-IV: — — M88 ALS 15 196 — — — — O8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — S14 O8 V — — M88 HD 93 146 B — — — — O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 IV — — S14 O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — L81 CPD −59 2644 O9 V — — O9 V — — S14 O8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — M93 HDE 305 532 — — — — O6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V ((f))z — S14 O6 V ((f)) — W82 CPD −58 2656 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 V — — — — — — — B1 V — — A16 CPD −58 2623 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 V — — — — — — — B0 V — — K93 CPD −59 2610 — — — — O8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — S14 O8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V ((f)) — M01 CPD −59 2627 O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 IV — — O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — S14 O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — A16 CPD −58 2649 A O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: — — B0: O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 III: — B0: V: M23a O7 V — O8 V A16 CPD −58 2627 O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V (n) — O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V (n) — S14 O9 III — — F81 CPD −59 2595 — — — — B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — TW B2 V — — A16 CPD −59 2673 O5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V (n)((f))z — O5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V (n)((f))z — S14 O5 V n — W82 ALS 15 210 O3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 I f* Nwk — O3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 I f* Nwk — S14 O3/4 I f — M93 HDE 303 313 B2 V — B2 V — — — — — B2 V — B2 V A16 CPD −59 2593 B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: IV nnn — — — — — — B2 V — — A16 HDE 305 528 — — — — — — — — — B2 V — — A16 HDE 305 439 B — — — — — — — — — B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 Ib — — M23b CPD −59 2537 — — — — — — — — — B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — M23b ALS 15 205 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 V (n) — — — — — — B2 V — — A16 ALS 15 961 — — — — B1 V — — TW B0 V — — A16 CPD −59 2469 B9 III — — — — — — — B2/5: — — — L76 HDE 305 533 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V (n) — — — — — — B0 V — — A16 ALS 15 860 B1 Iab — — B1 Iab — — TW O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 I/II — — F80 CPD −58 2625 O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 V — — S14 O9 V — — A16 HD 93 249 C B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V (n) — — — — — — B2 V n — M88 HDE 305 515 A — — — — — — — — — B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V sn: — G11b ALS 15 207 — — — — O9 V — — S14 O9 V — — M88 ALS 15 865 — — — — — — — — — B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V: b — W11 CPD −58 2634 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — — — — — — B3/6 — — — L76 HD 93 128 B B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2: V — B1: V B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 V — — M22 — — — — — CPD −59 2629 O8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V p — O8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V p — S14 O9 V — — A16 CPD −58 2657 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 V (n) — — — — — — B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V n — G11b CPD −59 2591 O8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — B0/1 O8 V z B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: V: M16 O8 V z B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: V: M23a CPD −58 2644 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — — — — — — B0 V — — F80 CPD −59 2598 — — — — B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — M22 B0 V — — A16 CPD −59 2570 — — — — — — — — — B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — A16 CPD −59 2622 — — — — — — — — — B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — W11 CPD −59 2535 — — — — — — — — — B2 V — — A16 24 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (Continued).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Name Gaia-ESO GOSSS LiLiMaRlin + STIS + literature ST LC qual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' ST LC qual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' ST LC qual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' CPD −59 2632 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V n — — — — — — B1 V — — A16 CPD −59 2495 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — — — — — — B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — A16 ALS 15 204 — — — — O7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V z O9: V M16 — — — — — CPD −58 2655 B1: V — — — — — — — B1 V — — F80 CPD −59 2614 B1 V — — — — — — — B1 III — — A16 ALS 15 219 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: V — B1: V B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 V — B1 V M22 B1 V — — M93 CPD −59 2581 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V (n) — — — — — — B1 V — — M93 [ARV2008] 206 O6 V ((f)) — O6 V ((f)) — S14 O5: V — — A16 CPD −59 2583 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: V — B2: V — — — — — B1 V — — A16 CPD −58 2640 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — B2: V — — — — — B1 V — B2 V A16 CPD −59 2606 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 V n — — — — — — B1 V — — W11 Tyc 8626-02506-1 O9 V (n) — O9 V (n) — S14 O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — A16 CPD −59 2497 B2 V — — — — — — — B7 — — — L76 ALS 15 229 B0 V — — B0 V — — S14 B0 V — — M88 CPD −59 2605 B0 V — — — — — — — B1 V — — G11b CPD −59 2640 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V (n) — B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V (n) — M22 — — — — — ALS 15 228 B1 V — — — — — — — B2: V — — W11 Gaia DR3 5350358378338864768 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: V (n) — — — — — — B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V: — — L82 ALS 15 855 B1 V (n) — — — — — — B1 V n — G11b CPD −59 2504 B7 II: nn — — — — — — B2/5 — — — L76 CPD −59 2579 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — — — — — — — — — — — CPD −59 2543 B2 V (n) — — — — — — B2 V — — A16 ALS 15 227 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 V — — — — — — — B1 V — — A16 CPD −58 2647 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — — — — — — B0 V: — — M88 ALS 15 224 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — — — — — — B1 V — — G11a CPD −59 2510 B2 V — — — — — — — B3/4 — — — L76 CPD −59 2619 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 V — — — — — — — B1 V — — A16 CPD −59 2596 B0 V — — B0 V — — TW B0 V — — M93 ALS 15 234 — — — — B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — M22 O9 V — — M88 2MASS J10424532−6012063 B2 V — — — — — — — — — — — — 2MASS J10424476−6005020 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 V — B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 V B0 IV: — — TW O: — — — P11 CPD −58 2653 A B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — — — — — — B2 V — — A16 CPD −59 2571 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V n — — — — — — B3 V — — A16 ALS 15 863 — — — — B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 V — — M22 O9: V — — M88 2MASS J10415981−5955075 B2 V — — — — — — — — — — — — 2MASS J10461906−5957543 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 V — — — — — — — B1 III — — A16 CPD −59 2661 B0: — — B B0 V (n)e B2: V TW O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — W84 ALS 15 232 B1 V — — — — — — — — — — — — ALS 15 233 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 V — — B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 V — — TW — — — — — 2MASS J10431897−5946569 B2: V (n) B — — — — — — — — — — ALS 19 739 B1: V — — — — — — — B1 V — — M93 ALS 15 235 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — — — — — — B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — A16 CPD −59 2569 B B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — — — — — — — — — — — 2MASS J10441791−5925204 B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — — — — — — — — — — — CPD −58 2667 B2 IV — — — — — — — B2 III — — A16 ALS 15 856 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — — — — — — — — — — — 2MASS J10460477−5949217 O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 V (n) — — — — — — — — — — — ALS 15 237 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: V — B2: V — — — — — — — — — — CPD −59 2617 B2 V (n)e — — — — — — — — — — — CPD −58 2648 — — — — — — — — — B1 III — — A16 2MASS J10460606−5956339 B2 V (n) — — — — — — — — — — — 2MASS J10450531−5919347 B2 V (n) — — — — — — — — — — — CPD −59 2585 B2 V — — — — — — — — — — — — [HSB2012] 1443 — — — — B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: V (n) — M22 — — — — — 25 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (Continued).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Name Gaia-ESO GOSSS LiLiMaRlin + STIS + literature ST LC qual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' ST LC qual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' ST LC qual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' CPD −59 2625 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: V — B2: V — — — — — B2 V — — A16 ALS 15 245 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — — — — — — — — — — — CPD −58 2677 B1 V — — — — — — — B2 III — — A16 CPD −59 2541 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V (n) — — — — — — — — — — — ALS 15 246 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 V — — — — — — — B1 V — — A16 CPD −59 2616 B2 V (n) — B2 V (n) — M22 B2 V — — A16 ALS 15 242 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='V ' metadata={'source': 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249 B1 V — — — — — — — B1: — — — M93 CPD −59 2539 B2 V — — — — — — — — — — — — ALS 15 248 B2: IV nnn — — — — — — — — — — — 2MASS J10440327−5919498 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — — — — — — — — — — — [HSB2012] 3192 B2: V ne — — — — — — — — — — — CPD −59 2618 B1: V (n)p He r — — — — — — B2 V — — A16 ALS 15 225 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V p He rich — B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V p He rich — TW — — — — — V662 Car O — — O+B O5 V (n)z B0: V M16 O5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V z O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V N06 HDE 303 316 B — — — — B1 V — — TW — — — — — ALS 15 958 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — — — — — — — — — — — ALS 15 864 B1 V — — — — — — — O9 V — — M88 CPD −59 2638 A B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — — — — — — — — — — — ALS 15 203 A B — — B B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — B1: V TW O7 V — — V93 ALS 19 733 B1 V (n) — B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: V — — TW B1 V — — M93 2MASS J10440973−6000432 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V e — — — — — — — — — — — ALS 19 735 B1 V — — — — — — — B2 V — — M93 2MASS J10441879−5951490 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V (n) — — — — — — — — — — — 2MASS J10443223−5933592 B2: V n — — — — — — — — — — — HD 93 129 C B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V (n) — — — — — — — — — — — 2MASS J10443139−5944080 B2 V — — — — — — — — — — — — ALS 16 082 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 V (n) — — — — — — — — — — — ALS 15 203 B — — — — B0 V — — TW — — — — — 2MASS J10435902−5933196 B2 IV p He rich — — — — — — — — — — — 2MASS J10454701−6000271 B2 V (n) — — — — — — — — — — — 2MASS J10432556−5919175 B2 V — — — — — — — — — — — — 2MASS J10451588−5929563 B2 IV n — — — — — — B — — — D17 2MASS J10422722−6000052 B2 V (n) — — — — — — — — — — — ALS 19 740 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — — — — — — B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — M93 ALS 19 746 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — — — — — — B1 V — — M93 2MASS J10452314−5940033 B2 V (n) — — — — — — — — — — — 2MASS J10440516−5921165 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — — — — — — — — — — — ALS 19 742 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — — — — — — B2 V — — M93 2MASS J10433865−5934444 B1: IV nnn — — — — — — B — — — D17 2MASS J10441729−5917154 B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — — — — — — — — — — — 2MASS J10445053−5957227 B2 IV — — — — — — — — — — — — 2MASS J10425900−6000240 B2 V — — — — — — — — — — — — 2MASS J10435207−5932401 B2 V (n) — — — — — — — — — — — 26 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (Continued).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Name Gaia-ESO GOSSS LiLiMaRlin + STIS + literature ST LC qual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' ST LC qual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' ST LC qual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2MASS J10435478−6006207 B2 V — — — — — — — — — — — — ALS 19 748 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — — — — — — B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — M93 ALS 19 732 — — — — B2: V e — TW B2 V — — A16 ALS 19 747 B2 V n — — — — — — B2 V — — M93 2MASS J10435198−6010368 B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V n — — — — — — — — — — — 2MASS J10442909−5948207 B0 V — — B0 V — — TW — — — — — ALS 17 185 B1 V — — — — — — — B2 III — — A16 2MASS J10451925−5929522 B1 V — — — — — — — B I — — D17 [HSB2012] 3545 B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: IV nn — — — — — — B — — — D17 2MASS J10440371−5948141 B1 V — — — — — — — — — — — — 2MASS J10434797−6001201 B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V (n) — — — — — — — — — — — 2MASS J10443829−6005449 B2 V (n) — — — — — — — — — — — 2MASS J10454824−5922041 B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V n — — — — — — — — — — — 2MASS J10441829−5942296 B2: V p He rich — — — — — — — — — — — 2MASS J10445080−5918005 B2 V — — — — — — — — — — — — [HSB2012] 3416 B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='nn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='— ' metadata={'source': 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— — — — — — 27 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (Continued).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Name Gaia-ESO GOSSS LiLiMaRlin + STIS + literature ST LC qual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' ST LC qual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' ST LC qual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 2MASS J10425293−6003478 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 V — — — — — — — — — — — — 2MASS J10440866−5933488 B2: V nnn — — — — — — — — — — — 2MASS J10444609−5946056 B2 V — — — — — — — — — — — — 2MASS J10454060−5937041 B2: V nn — — — — — — B — — — D17 2MASS J10470063−5957242 B2 V — — — — — — — — — — — — 2MASS J10440384−5934344 B2 V — — — — — — — B I — — D17 2MASS J10420949−6002265 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 V — — — — — — — — — — — — 2MASS J10445602−5938530 B2 V n — — — — — — — — — — — 2MASS J10443089−5914461 O7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 II (f) — O7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 II (f) — M23a O8 V — — A16 2MASS J10440683−5936116 B2 V nnn — — — — — — B2 V — — V15 2MASS J10431388−5954584 B2 V — — — — — — — — — — — — 2MASS J10434798−5933590 B2 V n 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O9-B1 — ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='P21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2MASS J10450879−5950537 — ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O6-7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='P21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2MASS J10452862−5947553 — ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O6-7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='P21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='A02: Albacete Colombo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2002), A16: Alexander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2016), D17: Damiani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2017), F80: Feinstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (1980), F81: Forte & Orsatti (1981), G11a: Gvaramadze et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2011), G11b: Gagn´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2011), H06: Hamann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2006), K93: Kilkenny (1993), L76: Loden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (1976), L81: Levato & Malaroda (1981), L82: Levato & Malaroda (1982), M01: Massey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2001), M16: Ma´ız Apell´aniz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2016), M17: Ma´ız Apell´aniz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2017), M20: Ma´ız Apell´aniz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2020), M22: Ma´ız Apell´aniz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2022a), M23a: GOSSS IV, M23b: Villafranca III, M55: Morgan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (1955), M88: Morrell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (1988), M93: Massey & Johnson (1993), N06: Niemel¨a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2006), P11: Povich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2011), P21: Preibisch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2021), R09: Rauw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2009), S14: Sota et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2014), T73: Thackeray et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (1973), T74: Thackeray & Andrews (1974), TW: This work, V01: van der Hucht (2001), V15: Vaidya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2015), V93: Vijapurkar & Drilling (1993), W02: Walborn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2002b), W09: Walborn (2009), W11: Wolk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (2011), W72: Walborn (1972), W73a: Walborn (1973b), W73b: Walborn (1973a), W77: Walborn & Liller (1977), W82: Walborn (1982b), W84: Walsh (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 28 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Stars of the census whose distance is not compatible with Car OB1 and are located in the foreground and in the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' See Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 for reference acronyms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Name Gaia-ESO GOSSS LiLiMaRlin + STIS + literature group ST LC qual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' ST LC qual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' ST LC qual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' HD 93 420 — — — — — — — — — M4 Ib — — H72 foreground HD 93 342 — — — — — — — — — B1 Ia — — A16 background HD 93 501 — — — — — — — — — B0 V — — A16 foreground CPD −59 2592 — — — — — — — — — B1 Ib — — A16 background HDE 305 439 A B0 Ia — — — — — — — B0 Ia — — V93 background HDE 305 439 B — — — — — — — — — B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 Ib — — M23b background ALS 15 860 B1 Iab — — B1 Iab — — TW O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 I/II — — F80 background CPD −58 2634 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — — — — — — B3/6 — — — L76 foreground CPD −59 2535 — — — — — — — — — B2 V — — A16 background 2MASS J10424476−6005020 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 V — B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 V B0 IV: — — TW O: — — — P11 background 2MASS J10441791−5925204 B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — — — — — — — — — — — background 2MASS J10444803−5954297 O7: V e — O8: V e — M22b — — — — — background 2MASS J10441242−5934091 B2 V n — — — — — — — — — — — background CPD −59 2539 B2 V — — — — — — — — — — — — background 2MASS J10413434−5958474 B1 V — — — — — — — — — — — — background 2MASS J10440744−5916399 O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7: — — B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: — — — — — — — — — — background 2MASS J10471498−5953374 B6: III e — — — — — — — — — — — background 2MASS J10425293−6003478 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 V — — — — — — — — — — — — background 2MASS J10420949−6002265 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 V — — — — — — — — — — — — background 2MASS J10431388−5954584 B2 V — — — — — — — — — — — — background 2MASS J10431945−5944488 sdO — — sec — — — — — — — — — — foreground 2MASS J10452648−5946188 — — — — — — — — — B1-2 — — — P21 background Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Stars identified in the Car OB1 region as of WR or non-O supergiant (II to Ia) spectral class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' See Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 for reference acronyms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Name Gaia-ESO GOSSS LiLiMaRlin + STIS + literature group ST LC qual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' ST LC qual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' ST LC qual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' η Car — — — — LBV — — — M22a F: I — — W77 O-025 WR 24 — — — — WN6 — ha — TW WN6 — ha-w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' — H06 O-028 HD 93 281 — — — — — — — — — M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 Iab: — — M23b O-028 HDE 303 310 — — — — — — — — — M3 Iab — — M23b O-027 CPD −58 2605 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 II — — — — — — — B0 III-IV: — — M88 O-002 CPD −59 2504 B7 II: nn — — — — — — B2/5 — — — L76 O-028 Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Spectroscopic binary systems identified in the census containing at least one O-type star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' See Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 for reference acronyms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Name Gaia-ESO GOSSS LiLiMaRlin + STIS + literature group Notes ST LC qual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' ST LC qual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' ST LC qual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' QZ Car Aa,Ac — — — — O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 Ib n — S14 O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 Ib — O9 II: M22b O-028 new SB2 HD 93 129 Aa — — — — O2 I f* — S14 O2 I f* OB?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' M17 O-002 HD 93 403 — — — — O5 I fc O7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V M22b O5 I fc O7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V M22b Car OB1 new SB2 HD 93 205 — — — — O3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V ((f)) O8 V S14 O3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V ((f)) O8 V M20 O-003 HD 93 162 — — — — O2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 I f*/WN6 OB S14 WN6 — h O4 V01 O-003 HD 93 161 A O7 V ((f)) O9 IV O7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — O9 V S14 O7 V ((f)) O9 IV M22b O-002 V572 Car O6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V z B0 V + B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: V O7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V (n) B0 V(n) M16 O6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V z B0 V +B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 V M22b O-025 new SB3 CPD −59 2554 O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 V — B1: V O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 IV — — S14 O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 V — B1: V M22b O-028 new SB2 CPD −59 2641 — — — — O6 V ((fc)) — S14 O5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5-6 V ((fc)) B2 V-III R09 O-025 CPD −59 2635 O8 V z O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 V O8 V (n) O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V S14 O8 V z O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 V M22b O-025 HD 93 343 — — — — O8 V — — M16 O7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V z O7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: V(n) M22a O-025 V573 Car — — — — O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V (n) B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V(n) S14 O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 IV — B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V M22b O-025 CPD −59 2636 A,B O7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — O8 V + O8 V O8 V — O8 V S14 O7 V — O8 V + O9 V A02 O-025 HDE 305 525 — — — — O5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V ((f))z O7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V + B M16 O4 V — — V93 O-030 HDE 303 312 O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 III — B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: V O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 IV — — S14 O9 V — — F81 Car OB1 new SB2 CPD −58 2649 A O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: — — B0: O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 III: — B0: V: M22b O7 V — O8 V A16 Car OB1 CPD −59 2591 O8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — B0/1 O8 V z B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: V: M16 O8 V z B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: V: M22b O-028 ALS 15 204 — — — — O7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V z O9: V M16 — — — — — O-002 V662 Car O — — O+B O5 V (n)z B0: V M16 O5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V z O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V N06 Car OB1 2MASS J10440744−5916399 O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7: — — B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: — — — — — — — — — — back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' new SB2 29 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Spectroscopic binary systems identified in the census formed by early B-type stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' See Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 for reference acronyms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Name Gaia-ESO GOSSS LiLiMaRlin + STIS + literature group Notes ST LC qual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' ST LC qual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' ST LC qual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' HDE 305 534 — — — — — — — — — B0 V — B0 V A16 O-028 HDE 305 522 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 V — B1: V — — — — — B0 V — B1 V A16 O-028 HDE 305 543 — — — — — — — — — B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 V (n) B1: V(n) M23b O-028 new SB2 HDE 303 300 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — B — — — — — B1 V — — A16 Car OB1 new SB2 HDE 305 516 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V (n) B2: V — — — — — B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — G11b O-028 new SB2 HDE 303 313 B2 V — B2 V — — — — — B2 V — B2 V A16 Car OB1 HD 93 128 B B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2: V — B1: V B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 V — — M22a — — — — — O-002 new SB2 ALS 15 219 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: V — B1: V B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 V — B1 V M22a B1 V — — M93 O-002 CPD −59 2583 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: V — B2: V — — — — — B1 V — — A16 Car OB1 new SB2 CPD −58 2640 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — B2: V — — — — — B1 V — B2 V A16 O-029 2MASS J10424476−6005020 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 V — B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 V B0 IV: — — TW O: — — — P11 back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' new SB2 CPD −59 2661 B0: — — B B0 V (n)e B2: V TW O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — — W84 O-030 new SB2 2MASS J10431897−5946569 B2: V (n) B — — — — — — — — — — O-028 new SB2 ALS 15 237 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: V — B2: V — — — — — — — — — — O-029 new SB2 CPD −59 2625 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: V — B2: V — — — — — B2 V — — A16 O-025 new SB2 ALS 15 203 A B — — B B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V — B1: V TW O7 V — — V93 O-002 new SB2 2MASS J10444550−5952537 B1: V — B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: V — — — — — — — — — — O-028 new SB2 [HSB2012] 3482 B2: V — B3: V — — — — — B3 — — — V15 O-025 new SB2 Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Proper motions from Gaia EDR3 for the identified runaway candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' Name Gaia Source µα∗ µδ RUWE group n-qualifer [mas a−1] [mas a−1] 2MASS J10440866−5933488 5350363085623074432 -7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='181 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='015 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='419 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='972 O-002 nnn CPD −58 2657 5350395383780746496 -6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='622 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='329 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='385 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='300 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='985 O-027 (n) HDE 303 310 5350389095946525568 -7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='432 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='031 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='851 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='704 O-027 CPD −59 2541 5254271056429426688 -5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='647 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='020 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='008 O-028 QZ Car Aa,Ac 5350346970905044480 -5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='750 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='123 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='455 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='108 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='471 O-028 HDE 305 523 5350347520660979840 -5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='804 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='025 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='165 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='023 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='083 O-028 2MASS J10451588−5929563 5350384629180484224 -6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='893 ± 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 IV V572 Car O6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 Vz + B0 V + B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: V CPD −59 2554 O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 V + B1: V HD 93 056 B1: V:n 4200 4300 4400 4500 4600 4700 4800 4900 5000 Wavelength (Å) 0 1 2 3 4 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' UVES spectra shown at their original resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 31 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hγ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hβ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4388 ' metadata={'source': 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4800 4900 5000 Wavelength (Å) 0 1 2 3 4 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (Continued).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 32 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hδ ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4448 / 52 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4591 / 96 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4662 / 76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4699 / 705 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4553 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4568 / 75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4089 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4116 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4631 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Mg II 4481 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4428 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4502 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4727 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Ca I is 4227 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='CH is 4300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='HDE 305 439 A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B0 Ia ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='CPD −59 2624 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 V HDE 305 522 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 V + B1: V HDE 303 300 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V + B CPD −59 2626 AB O7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V(n) HDE 303 312 O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 III + B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: V CPD −58 2605 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 II CPD −59 2644 O9 V 4100 4200 4300 4400 4500 4600 4700 Wavelength (Å) 0 1 2 3 4 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' GIRAFFE spectra shown at a resolution R = 2500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 33 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hγ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4121 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4144 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4169 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4388 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4438 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4471 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4713 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4542 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4686 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C II 4267 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4068 / 69 / 70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4187 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4326 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4647 / 50 / 51 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4530 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4602 / 07 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4614 / 21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4097 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4379 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4511 / 15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4634 / 41 / 42 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N IV 4058 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4070 / 72 / 76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4133 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4154 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4186 / 90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4254 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4276 / 85 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4317 / 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4349 / 67 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4415 / 17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4448 / 52 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4591 / 96 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4662 / 76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4699 / 705 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4553 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4568 / 75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4089 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4116 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4631 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Mg II 4481 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4428 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4502 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4727 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Ca I is 4227 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='CH is 4300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='CPD −58 2656 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 V CPD −58 2623 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 V CPD −59 2627 O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 IV CPD −58 2649 A O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: + B0: CPD −58 2627 O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V(n) CPD −59 2673 O5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V(n)((f))z ALS 15 210 O3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 If* Nwk HDE 303 313 B2 V + B2 V 4100 4200 4300 4400 4500 4600 4700 Wavelength (Å) 0 1 2 3 4 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (Continued).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 34 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hγ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4121 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4144 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4169 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4388 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4438 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4471 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4713 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4542 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4686 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C II 4267 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4068 / 69 / 70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4187 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4326 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4647 / 50 / 51 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4530 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4602 / 07 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4614 / 21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4097 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4379 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4511 / 15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4634 / 41 / 42 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N IV 4058 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4070 / 72 / 76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4133 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4154 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4186 / 90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4254 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4276 / 85 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4317 / 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4349 / 67 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4415 / 17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4448 / 52 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4591 / 96 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4662 / 76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4699 / 705 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4553 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4568 / 75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4089 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4116 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4631 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Mg II 4481 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4428 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4502 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4727 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Ca I is 4227 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='CH is 4300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='CPD −59 2593 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: IVnnn ALS 15 205 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 V(n) CPD −59 2469 B9 III HDE 305 533 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V(n) ALS 15 860 B1 Iab CPD −58 2625 O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V HD 93 249 C B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V(n) CPD −58 2634 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V 4100 4200 4300 4400 4500 4600 4700 Wavelength (Å) 0 1 2 3 4 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (Continued).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 35 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hγ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4121 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4144 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4169 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4388 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4438 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4471 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4713 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4542 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4686 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C II 4267 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4068 / 69 / 70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4187 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4326 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4647 / 50 / 51 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4530 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4602 / 07 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4614 / 21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4097 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4379 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4511 / 15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4634 / 41 / 42 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N IV 4058 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4070 / 72 / 76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4133 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4154 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4186 / 90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4254 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4276 / 85 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4317 / 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4349 / 67 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4415 / 17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4448 / 52 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4591 / 96 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4662 / 76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4699 / 705 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4553 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4568 / 75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4089 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4116 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4631 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Mg II 4481 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4428 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4502 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4727 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Ca I is 4227 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='CH is 4300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='HD 93 128 B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2: V + B1: V CPD −59 2629 O8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 Vp CPD −58 2657 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 V(n) CPD −59 2591 O8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V + B0/1 CPD −58 2644 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V CPD −59 2632 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 Vn CPD −59 2495 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V CPD −58 2655 B1: V 4100 4200 4300 4400 4500 4600 4700 Wavelength (Å) 0 1 2 3 4 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (Continued).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 36 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hγ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4121 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4144 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4169 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4388 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4438 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4471 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4713 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4542 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4686 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C II 4267 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4068 / 69 / 70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4187 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4326 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4647 / 50 / 51 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4530 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4602 / 07 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4614 / 21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4097 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4379 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4511 / 15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4634 / 41 / 42 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N IV 4058 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4070 / 72 / 76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4133 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4154 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4186 / 90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4254 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4276 / 85 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4317 / 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4349 / 67 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4415 / 17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4448 / 52 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4591 / 96 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4662 / 76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4699 / 705 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4553 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4568 / 75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4089 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4116 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4631 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Mg II 4481 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4428 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4502 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4727 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Ca I is 4227 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='CH is 4300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='CPD −59 2614 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B1 V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='ALS 15 219 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: V + B1: V CPD −59 2581 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V(n) [ARV2008] 206 O6 V((f)) CPD −59 2583 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: V + B2: V CPD −58 2640 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V + B2: V CPD −59 2606 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 Vn Tyc 8626−02506−1 O9 V(n) 4100 4200 4300 4400 4500 4600 4700 Wavelength (Å) 0 1 2 3 4 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (Continued).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 37 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hγ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4121 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4144 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4169 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4388 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4438 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4471 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4713 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4542 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4686 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C II 4267 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4068 / 69 / 70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4187 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4326 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4647 / 50 / 51 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4530 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4602 / 07 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4614 / 21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4097 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4379 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4511 / 15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4634 / 41 / 42 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N IV 4058 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4070 / 72 / 76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4133 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4154 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4186 / 90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4254 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4276 / 85 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4317 / 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4349 / 67 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4415 / 17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4448 / 52 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4591 / 96 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4662 / 76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4699 / 705 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4553 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4568 / 75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4089 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4116 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4631 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Mg II 4481 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4428 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4502 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4727 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Ca I is 4227 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='CH is 4300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='CPD −59 2497 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B2 V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='ALS 15 229 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B0 V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='CPD −59 2605 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B0 V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='CPD −59 2640 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V(n) ALS 15 228 B1 V Gaia DR3 5350358378338864768 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: V(n) ALS 15 855 B1 V(n) CPD −59 2504 B7 II:nn 4100 4200 4300 4400 4500 4600 4700 Wavelength (Å) 0 1 2 3 4 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (Continued).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 38 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hγ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4121 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4144 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4169 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4388 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4438 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4471 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4713 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4542 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4686 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C II 4267 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4068 / 69 / 70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4187 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4326 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4647 / 50 / 51 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4530 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4602 / 07 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4614 / 21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4097 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4379 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4511 / 15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4634 / 41 / 42 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N IV 4058 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4070 / 72 / 76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4133 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4154 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4186 / 90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4254 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4276 / 85 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4317 / 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4349 / 67 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4415 / 17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4448 / 52 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4591 / 96 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4662 / 76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4699 / 705 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4553 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4568 / 75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4089 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4116 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4631 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Mg II 4481 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4428 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4502 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4727 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Ca I is 4227 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='CH is 4300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='CPD −59 2579 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V CPD −59 2543 B2 V(n) ALS 15 227 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 V CPD −58 2647 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V ALS 15 224 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V CPD −59 2510 B2 V CPD −59 2619 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 V CPD −59 2596 B0 V 4100 4200 4300 4400 4500 4600 4700 Wavelength (Å) 0 1 2 3 4 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (Continued).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 39 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hγ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4121 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4144 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4169 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4388 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4438 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4471 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4713 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4542 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4686 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C II 4267 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4068 / 69 / 70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4187 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4326 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4647 / 50 / 51 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4530 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4602 / 07 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4614 / 21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4097 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4379 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4511 / 15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4634 / 41 / 42 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N IV 4058 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4070 / 72 / 76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4133 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4154 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4186 / 90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4254 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4276 / 85 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4317 / 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4349 / 67 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4415 / 17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4448 / 52 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4591 / 96 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4662 / 76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4699 / 705 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4553 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4568 / 75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4089 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4116 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4631 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Mg II 4481 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4428 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4502 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4727 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Ca I is 4227 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='CH is 4300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2MASS J10424532−6012063 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B2 V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2MASS J10424476−6005020 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 V + B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 V CPD −58 2653 A B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V CPD −59 2571 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 Vn 2MASS J10415981−5955075 B2 V 2MASS J10461906−5957543 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 V CPD −59 2661 B0: + B ALS 15 232 B1 V 4100 4200 4300 4400 4500 4600 4700 Wavelength (Å) 0 1 2 3 4 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (Continued).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 40 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hγ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4121 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4144 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4169 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4388 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4438 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4471 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4713 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4542 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4686 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C II 4267 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4068 / 69 / 70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4187 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4326 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4647 / 50 / 51 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4530 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4602 / 07 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4614 / 21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4097 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4379 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4511 / 15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4634 / 41 / 42 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N IV 4058 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4070 / 72 / 76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4133 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4154 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4186 / 90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4254 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4276 / 85 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4317 / 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4349 / 67 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4415 / 17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4448 / 52 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4591 / 96 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4662 / 76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4699 / 705 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4553 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4568 / 75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4089 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4116 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4631 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Mg II 4481 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4428 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4502 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4727 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Ca I is 4227 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='CH is 4300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='ALS 15 233 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 V 2MASS J10431897−5946569 B2: V(n) + B ALS 19 739 B1: V ALS 15 235 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V CPD −59 2569 B B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V 2MASS J10441791−5925204 B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V CPD −58 2667 B2 IV ALS 15 856 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V 4100 4200 4300 4400 4500 4600 4700 Wavelength (Å) 0 1 2 3 4 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (Continued).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 41 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hγ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4121 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4144 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4169 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4388 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4438 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4471 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4713 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4542 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4686 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C II 4267 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4068 / 69 / 70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4187 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4326 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4647 / 50 / 51 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4530 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4602 / 07 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4614 / 21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4097 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4379 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4511 / 15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4634 / 41 / 42 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N IV 4058 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4070 / 72 / 76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4133 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4154 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4186 / 90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4254 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4276 / 85 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4317 / 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4349 / 67 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4415 / 17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4448 / 52 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4591 / 96 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4662 / 76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4699 / 705 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4553 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4568 / 75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4089 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4116 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4631 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Mg II 4481 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4428 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4502 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4727 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Ca I is 4227 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='CH is 4300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2MASS J10460477−5949217 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 V(n) ALS 15 237 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: V + B2: V CPD −59 2617 B2 V(n)e 2MASS J10460606−5956339 B2 V(n) 2MASS J10450531−5919347 B2 V(n) CPD −59 2585 B2 V CPD −59 2625 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: V + B2: V ALS 15 245 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V 4100 4200 4300 4400 4500 4600 4700 Wavelength (Å) 0 1 2 3 4 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (Continued).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 42 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hγ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4121 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4144 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4169 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4388 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4438 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4471 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4713 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4542 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4686 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C II 4267 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4068 / 69 / 70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4187 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4326 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4647 / 50 / 51 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4530 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4602 / 07 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4614 / 21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4097 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4379 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4511 / 15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4634 / 41 / 42 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N IV 4058 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4070 / 72 / 76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4133 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4154 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4186 / 90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4254 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4276 / 85 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4317 / 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4349 / 67 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4415 / 17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4448 / 52 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4591 / 96 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4662 / 76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4699 / 705 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4553 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4568 / 75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4089 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4116 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4631 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Mg II 4481 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4428 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4502 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4727 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Ca I is 4227 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='CH is 4300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='CPD −58 2677 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B1 V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='CPD −59 2541 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V(n) ALS 15 246 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 V CPD −59 2616 B2 V(n) ALS 15 242 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 V CPD −58 2662 B2 V 2MASS J10444803−5954297 O7: Ve ALS 15 247 B1 V 4100 4200 4300 4400 4500 4600 4700 Wavelength (Å) 0 1 2 3 4 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (Continued).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 43 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hγ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4121 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4144 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4169 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4388 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4438 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4471 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4713 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4542 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4686 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C II 4267 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4068 / 69 / 70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4187 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4186 / 90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4254 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4276 / 85 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4317 / 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4349 / 67 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4415 / 17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4116 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4631 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Mg II 4481 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4428 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4502 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4727 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Ca I is 4227 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='CH is 4300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2MASS J10432014−5917582 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B2 V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2MASS J10441242−5934091 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B2 Vn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2MASS J10435413−5918244 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B2 Vn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='QZ Car B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V ALS 15 249 B1 V CPD −59 2539 B2 V ALS 15 248 B2: IVnnn 2MASS J10440327−5919498 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V 4100 4200 4300 4400 4500 4600 4700 Wavelength (Å) 0 1 2 3 4 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (Continued).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 44 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hγ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4121 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4144 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4169 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4388 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4438 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4471 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4713 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4542 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4686 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C II 4267 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4068 / 69 / 70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4187 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4326 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4631 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Mg II 4481 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4428 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4502 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4727 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Ca I is 4227 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='CH is 4300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='[HSB2012] 3192 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B2: Vne ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='CPD −59 2618 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B1: V(n)p He rich ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='ALS 15 225 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 Vp He rich V662 Car O + O+B ALS 15 958 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V ALS 15 864 B1 V CPD −59 2638 A B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V ALS 15 203 A B + B 4100 4200 4300 4400 4500 4600 4700 Wavelength (Å) 0 1 2 3 4 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (Continued).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 45 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hγ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4121 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4144 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4169 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4388 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4438 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4471 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4713 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4542 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4686 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C II 4267 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4068 / 69 / 70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4187 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4326 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4647 / 50 / 51 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4530 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4602 / 07 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4614 / 21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4097 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4379 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4511 / 15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4634 / 41 / 42 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N IV 4058 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4070 / 72 / 76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4133 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4154 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4186 / 90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4254 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4276 / 85 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4317 / 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4349 / 67 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4415 / 17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4448 / 52 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4591 / 96 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4662 / 76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4699 / 705 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4553 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4568 / 75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4089 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4116 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4631 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Mg II 4481 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4428 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4502 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4727 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Ca I is 4227 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='CH is 4300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='ALS 19 733 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B1 V(n) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2MASS J10440973−6000432 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 Ve ALS 19 735 B1 V 2MASS J10441879−5951490 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V(n) 2MASS J10443223−5933592 B2: Vn HD 93 129 C B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V(n) 2MASS J10443139−5944080 B2 V ALS 16 082 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 V(n) 4100 4200 4300 4400 4500 4600 4700 Wavelength (Å) 0 1 2 3 4 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (Continued).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 46 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hγ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4121 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4144 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4169 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4388 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4438 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4471 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4713 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4542 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4686 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C II 4267 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4068 / 69 / 70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4187 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4326 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4647 / 50 / 51 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4530 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4602 / 07 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4614 / 21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4097 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4379 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4511 / 15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4634 / 41 / 42 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N IV 4058 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4070 / 72 / 76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4133 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4154 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4186 / 90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4254 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4276 / 85 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4317 / 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4349 / 67 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4415 / 17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4448 / 52 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4591 / 96 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4662 / 76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4699 / 705 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4553 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4568 / 75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4089 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4116 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4631 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Mg II 4481 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4428 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4502 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4727 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Ca I is 4227 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='CH is 4300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2MASS J10435902−5933196 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B2 IVp He rich ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2MASS J10454701−6000271 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B2 V(n) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2MASS J10432556−5919175 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B2 V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2MASS J10451588−5929563 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B2 IVn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2MASS J10422722−6000052 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B2 V(n) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='ALS 19 740 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V ALS 19 746 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V 2MASS J10452314−5940033 B2 V(n) 4100 4200 4300 4400 4500 4600 4700 Wavelength (Å) 0 1 2 3 4 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (Continued).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 47 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hγ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4121 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4144 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4169 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4388 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4438 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4471 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4713 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4542 ' metadata={'source': 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+page_content='N II 4530 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4602 / 07 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4614 / 21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4097 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4379 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4511 / 15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4634 / 41 / 42 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N IV 4058 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4070 / 72 / 76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4133 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4154 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4186 / 90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4254 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4276 / 85 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4317 / 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4349 / 67 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4415 / 17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4448 / 52 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4591 / 96 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4662 / 76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4699 / 705 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4553 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4568 / 75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4089 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4116 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4631 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Mg II 4481 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4428 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4502 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4727 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Ca I is 4227 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='CH is 4300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2MASS J10440516−5921165 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V ALS 19 742 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V 2MASS J10433865−5934444 B1: IVnnn 2MASS J10441729−5917154 B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V 2MASS J10445053−5957227 B2 IV 2MASS J10425900−6000240 B2 V 2MASS J10435207−5932401 B2 V(n) 2MASS J10435478−6006207 B2 V 4100 4200 4300 4400 4500 4600 4700 Wavelength (Å) 0 1 2 3 4 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (Continued).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 48 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hγ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4121 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4144 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4169 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4388 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4438 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4471 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4713 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4542 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4686 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C II 4267 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4068 / 69 / 70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4187 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4326 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4647 / 50 / 51 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4530 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4602 / 07 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4614 / 21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4097 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4379 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4511 / 15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4634 / 41 / 42 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N IV 4058 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4070 / 72 / 76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4133 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4154 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4186 / 90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4254 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4276 / 85 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4317 / 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4349 / 67 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4415 / 17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4448 / 52 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4591 / 96 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4662 / 76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4699 / 705 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4553 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4568 / 75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4089 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4116 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4631 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Mg II 4481 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4428 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4502 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4727 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Ca I is 4227 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='CH is 4300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='ALS 19 748 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V ALS 19 747 B2 Vn 2MASS J10435198−6010368 B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 Vn 2MASS J10442909−5948207 B0 V ALS 17 185 B1 V 2MASS J10451925−5929522 B1 V [HSB2012] 3545 B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: IVnn 2MASS J10440371−5948141 B1 V 4100 4200 4300 4400 4500 4600 4700 Wavelength (Å) 0 1 2 3 4 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (Continued).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 49 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hγ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4121 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4144 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4169 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4388 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4438 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4471 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4713 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4542 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4686 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C II 4267 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4068 / 69 / 70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4187 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4326 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4647 / 50 / 51 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4530 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4602 / 07 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4614 / 21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4097 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4379 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4511 / 15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4634 / 41 / 42 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N IV 4058 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4070 / 72 / 76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4133 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4154 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4186 / 90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4254 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4276 / 85 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4317 / 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4349 / 67 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4415 / 17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4448 / 52 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4591 / 96 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4662 / 76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4699 / 705 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4553 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4568 / 75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4089 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4116 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4631 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Mg II 4481 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4428 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4502 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4727 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Ca I is 4227 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='CH is 4300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2MASS J10434797−6001201 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V(n) 2MASS J10443829−6005449 B2 V(n) 2MASS J10454824−5922041 B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 Vn 2MASS J10441829−5942296 B2: Vp He rich 2MASS J10445080−5918005 B2 V [HSB2012] 3416 B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: Vnn 2MASS J10451894−5942184 B2 V 2MASS J10442894−5943473 B2 Vn 4100 4200 4300 4400 4500 4600 4700 Wavelength (Å) 0 1 2 3 4 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (Continued).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 50 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hγ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4121 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4144 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4169 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4388 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4438 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4471 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4713 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4542 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4686 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C II 4267 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4068 / 69 / 70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4187 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4326 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4647 / 50 / 51 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4530 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4602 / 07 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4614 / 21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4097 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4379 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4511 / 15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4634 / 41 / 42 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N IV 4058 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4070 / 72 / 76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4133 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4154 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4186 / 90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4254 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4276 / 85 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4317 / 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4349 / 67 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4415 / 17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4448 / 52 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4591 / 96 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4662 / 76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4699 / 705 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4553 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4568 / 75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4089 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4116 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4631 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Mg II 4481 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4428 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4502 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4727 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Ca I is 4227 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='CH is 4300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2MASS J10413434−5958474 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B1 V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='ALS 16 078 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B2 Vn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2MASS J10444278−5921383 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B2 V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2MASS J10433443−5943264 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B0 V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2MASS J10450836−5938475 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B2 V(n) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2MASS J10453807−5944095 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O8 Vz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2MASS J10430420−5948591 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B2 V(n) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2MASS J10444235−5922029 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B2 V(n) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='4100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='4200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='4300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='4400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='4500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='4600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='4700 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Wavelength (Å) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='0 ' metadata={'source': 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4121 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4144 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4169 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4388 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4438 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4471 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4713 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4542 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4686 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C II 4267 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4068 / 69 / 70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4187 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4326 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4647 / 50 / 51 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4530 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4602 / 07 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4614 / 21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4097 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4379 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4511 / 15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4634 / 41 / 42 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N IV 4058 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4070 / 72 / 76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4133 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4154 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4186 / 90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4254 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4276 / 85 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4317 / 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4349 / 67 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4415 / 17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4448 / 52 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4591 / 96 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4662 / 76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4699 / 705 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4553 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4568 / 75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4089 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4116 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4631 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Mg II 4481 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4428 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4502 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4727 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Ca I is 4227 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='CH is 4300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2MASS J10453134−5941133 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B2 V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2MASS J10444726−5928154 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B2 V(n) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Gaia DR3 5350302097055370496 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B2 V(n) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2MASS J10434812−5950443 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B2 V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2MASS J10443591−5923356 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 Vn 2MASS J10440236−5952046 B2 V 2MASS J10460277−5950192 B2: Vnnn [HSB2012] 3314 B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: Vnn 4100 4200 4300 4400 4500 4600 4700 Wavelength (Å) 0 1 2 3 4 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (Continued).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 52 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hγ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4121 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4144 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4169 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4388 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4438 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4471 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4713 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4542 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4686 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C II 4267 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4068 / 69 / 70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4187 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4326 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4647 / 50 / 51 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4530 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4602 / 07 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4614 / 21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4097 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4379 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4511 / 15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4634 / 41 / 42 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N IV 4058 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4070 / 72 / 76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4133 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4154 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4186 / 90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4254 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4276 / 85 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4317 / 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4349 / 67 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4415 / 17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4448 / 52 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4591 / 96 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4662 / 76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4699 / 705 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4553 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4568 / 75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4089 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4116 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4631 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Mg II 4481 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4428 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4502 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4727 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Ca I is 4227 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='CH is 4300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2MASS J10440744−5916399 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7: + B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: 2MASS J10443822−5943056 B2 Vnnn 2MASS J10440576−5927078 B2 V 2MASS J10443510−5923281 B2 V 2MASS J10444550−5952537 B1: V + B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: V 2MASS J10451297−5946059 B2 Vn Gaia DR3 5350388683629610752 B2 Vn 2MASS J10450792−5939011 B2 Vn 4100 4200 4300 4400 4500 4600 4700 Wavelength (Å) 0 1 2 3 4 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (Continued).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 53 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hγ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4121 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4144 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4169 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4388 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4438 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4471 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4713 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4542 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4686 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C II 4267 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4068 / 69 / 70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4187 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4326 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4647 / 50 / 51 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4530 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4602 / 07 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4614 / 21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4097 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4379 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4511 / 15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4634 / 41 / 42 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N IV 4058 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4070 / 72 / 76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4133 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4154 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4186 / 90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4254 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4276 / 85 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4317 / 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4349 / 67 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4415 / 17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4448 / 52 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4591 / 96 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4662 / 76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4699 / 705 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4553 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4568 / 75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4089 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4116 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4631 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Mg II 4481 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4428 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4502 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4727 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Ca I is 4227 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='CH is 4300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='[HSB2012] 3211 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B2 V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='[HSB2012] 1498 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B2 V(n) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2MASS J10435501−5936242 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B2 V(n) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2MASS J10444208−5926353 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B2: Vnnn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2MASS J10443769−5929316 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 Vn [ESK2003] 148 O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 V(n) 2MASS J10471498−5953374 B6: IIIe [HSB2012] 3482 B2: V + B3: V 4100 4200 4300 4400 4500 4600 4700 Wavelength (Å) 0 1 2 3 4 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (Continued).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 54 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hγ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4121 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4144 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4169 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4388 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4438 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4471 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4713 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4542 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4686 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C II 4267 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4068 / 69 / 70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4187 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4326 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4647 / 50 / 51 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4530 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4602 / 07 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4614 / 21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4097 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4379 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4511 / 15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4634 / 41 / 42 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N IV 4058 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4070 / 72 / 76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4133 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4154 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4186 / 90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4254 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4276 / 85 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4317 / 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4349 / 67 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4415 / 17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4448 / 52 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4591 / 96 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4662 / 76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4699 / 705 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4553 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4568 / 75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4089 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4116 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4631 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Mg II 4481 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4428 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4502 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4727 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Ca I is 4227 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='CH is 4300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2MASS J10464886−5950409 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V(n) 2MASS J10453254−5942359 B2: Vnnn 2MASS J10440105−6006377 B2 V 2MASS J10425293−6003478 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 V 2MASS J10440866−5933488 B2: Vnnn 2MASS J10444609−5946056 B2 V 2MASS J10454060−5937041 B2: Vnn 2MASS J10470063−5957242 B2 V 4100 4200 4300 4400 4500 4600 4700 Wavelength (Å) 0 1 2 3 4 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (Continued).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 55 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hγ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4121 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4144 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4169 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4388 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4438 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4471 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4713 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4542 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4686 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C II 4267 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4068 / 69 / 70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4187 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4326 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4647 / 50 / 51 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4530 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4602 / 07 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4614 / 21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4097 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4379 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4511 / 15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4634 / 41 / 42 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N IV 4058 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4070 / 72 / 76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4133 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4154 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4186 / 90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4254 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4276 / 85 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4317 / 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4349 / 67 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4415 / 17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4448 / 52 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4591 / 96 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4662 / 76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4699 / 705 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4553 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4568 / 75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4089 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4116 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4631 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Mg II 4481 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4428 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4502 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4727 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Ca I is 4227 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='CH is 4300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2MASS J10440384−5934344 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B2 V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2MASS J10420949−6002265 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 V 2MASS J10445602−5938530 B2 Vn 2MASS J10443089−5914461 O7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 II(f) 2MASS J10440683−5936116 B2 Vnnn 2MASS J10431388−5954584 B2 V 2MASS J10434798−5933590 B2 Vn 2MASS J10434580−5934359 B2: Vnnn 4100 4200 4300 4400 4500 4600 4700 Wavelength (Å) 0 1 2 3 4 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (Continued).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 56 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hγ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4121 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4144 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4169 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4388 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4438 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4471 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4713 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4542 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4686 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C II 4267 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4068 / 69 / 70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4187 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4326 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4647 / 50 / 51 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4530 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4602 / 07 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4614 / 21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4097 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4379 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4511 / 15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4634 / 41 / 42 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N IV 4058 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4070 / 72 / 76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4133 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4154 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4186 / 90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4254 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4276 / 85 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4317 / 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4349 / 67 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4415 / 17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4448 / 52 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4591 / 96 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4662 / 76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4699 / 705 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4553 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4568 / 75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4089 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4116 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4631 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Mg II 4481 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4428 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4502 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4727 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Ca I is 4227 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='CH is 4300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='[HSB2012] 3150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B2 V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2MASS J10452875−5930037 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 Vp 2MASS J10445837−5932062 B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 Vn 2MASS J10460608−5957394 B2 V 2MASS J10442945−5933437 B2: Vnnn 2MASS J10460257−5957372 B2: Vne 2MASS J10452056−5942212 B2: Vnne 2MASS J10453834−5942078 Be 4100 4200 4300 4400 4500 4600 4700 Wavelength (Å) 0 1 2 3 4 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (Continued).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 57 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hγ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4121 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4144 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4169 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4388 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4438 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4471 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4713 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4542 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4686 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C II 4267 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4068 / 69 / 70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4187 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4326 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4647 / 50 / 51 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4530 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4602 / 07 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4614 / 21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4097 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4379 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4511 / 15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N III 4634 / 41 / 42 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N IV 4058 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4070 / 72 / 76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4133 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4154 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4186 / 90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4254 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4276 / 85 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4317 / 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4349 / 67 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4415 / 17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4448 / 52 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4591 / 96 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4662 / 76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4699 / 705 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4553 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4568 / 75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4089 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4116 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4631 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Mg II 4481 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4428 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4502 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4727 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Ca I is 4227 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='CH is 4300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2MASS J10444710−5939201 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B2: Vnn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='[HSB2012] 3526 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B2 V(n) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2MASS J10452415−5942313 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: Vnn 2MASS J10452190−5945249 B2 V 2MASS J10433596−5933179 B2 V 2MASS J10463643−5948048 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V 2MASS J10453185−6000293 O8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V 2MASS J10435009−5947024 B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V(n) 4100 4200 4300 4400 4500 4600 4700 Wavelength (Å) 0 1 2 3 4 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (Continued).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 58 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hγ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4121 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4144 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4169 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4388 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4438 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4471 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4713 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4542 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4686 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C II 4267 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4068 / 69 / 70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4187 ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='CH is 4300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2MASS J10462657−5956131 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5: V 2MASS J10460291−5950259 B2 V 2MASS J10453819−5942157 Be [ARV2008] 217 O3: III:(n) 2MASS J10434303−5945333 B2: Ve 2MASS J10460116−5949420 B2: V 2MASS J10454661−5948404 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7: Ve 2MASS J10431945−5944488 sdO + sec 4100 4200 4300 4400 4500 4600 4700 Wavelength (Å) 0 1 2 3 4 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (Continued).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 59 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hε ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4813 / 20 / 29 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4089 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4116 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4631 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Mg II 4481 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4428 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='WR 24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='WN6ha ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='HDE 305 437 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B0 V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='HD 93 249 B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2 V(n) CPD −59 2595 B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V ALS 15 961 B1 V 4000 4250 4500 4750 5000 Wavelength (Å) 0 1 2 3 4 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' GOSSS spectra shown at a resolution R = 2500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 60 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hε ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hγ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Hβ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4009 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4121 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4144 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4169 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4388 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4438 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4471 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4713 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 4922 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 5016 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I 5048 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4542 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He II 4686 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='He I+II 4026 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C II 4267 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4068 / 69 / 70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4187 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4326 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='C III 4647 / 50 / 51 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 3995 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4530 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='N II 4602 / 07 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4186 / 90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4254 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4276 / 85 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4317 / 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4349 / 67 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4415 / 17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4448 / 52 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4591 / 96 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4662 / 76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4699 / 705 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4907 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='O II 4943 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4553 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4568 / 75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si III 4813 / 20 / 29 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4089 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4116 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Si IV 4631 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Mg II 4481 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4428 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4502 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4727 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4762 / 65 / 80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4880 / 87 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='DIB is 4964 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Ca II is 3934 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='Ca II is 3968 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='CPD −59 2661 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B0 V(n)e + B2: V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='ALS 15 860 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B1 Iab ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='CPD −59 2596 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B0 V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='2MASS J10424476−6005020 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B0 IV: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='ALS 15 233 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='7 V ALS 15 225 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 Vp He rich ALS 15 203 A B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='5 V + B1: V HDE 303 316 B B1 V 4000 4250 4500 4750 5000 Wavelength (Å) 0 1 2 3 4 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' (Continued).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' 61 Berlanas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf'} +page_content=' : Massive stars in the Carina Nebula ' metadata={'source': 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+ +A note on the constant characteristic time of failure incubation processes under various high- +rate loads +Ivan Smirnov +Saint Petersburg State University, Universitetskaya nab. 7/9, St. Petersburg, 199034, Russia +Corresponding author: ivansmirnov.sci@gmail.com + +Abstract. The research reveals the existence of a constant characteristic time of preparatory micro- +structural processes before the onset of macro-failure at various high loading rates of brittle and +quasi-brittle materials. The presence of this characteristic is analysed based on available data in the +literature from dynamic tests for uniaxial compression and splitting. It is shown that the +characteristic time can be determined experimentally and used to calculate the strain rate +dependencies of either critical failure stresses or time to failure, at least in the case of linearly +growing loads. In addition, it is discussed that the presence of this constant parameter opens up a +prospective opportunity for research and development of new methods for assessing the structural- +temporal and scale characteristics of the strength and failure of materials under dynamic loads. + +Keywords: quasi-brittle material, high strain rate, dynamic strength, time to fracture, characteristic +time, failure criteria. + +1. Introduction +The problems of deformation and failure of materials at high-rate intensive loads have been +relevant for a significant period of time. Today, these studies continue to both certify the behaviour +of materials at a given range of strain rates [1–3], and solve fundamental issues in order to develop +approaches for qualitative forecasting the behaviour of materials, regardless of loading history +[1,4,5]. However, there are still no universally accepted test methods or parameters that could allow +us to evaluate and model the dynamic strength of materials based on simple engineering principles. + +2 + +At sufficiently slow loads, the tensile strength and yield strength can, in fact, be assumed to be +constant and considered as characteristic strength parameters of a material. This forms the basis of +systems for state and industry standards to determine the strength characteristics of brittle and +plastic materials. When the strain or loading rates exceed the standardized quasi-static range, these +strength characteristics can become unstable and often strongly depend on the load history [5,6]. +Therefore, under dynamic loads, they can no longer be considered as material parameters. To date, +there is no agreement on which characteristics can serve as the primary indicators of the properties +of materials under high-rate and shock loads. There is an urgent need to identify the characteristics, +which do not depend on the history and method (set-up) of load application. +This short communication presents an important, previously unknown effect of the +connection between failure incubation processes in the material structure and a constant time +characteristic, which are independent of strain/loading rate. It is shown that this characteristic time +can be determined experimentally and applied to criteria for evaluating the strength of a material +within a wide range of loading rates. + +2. The characteristic time of failure incubation processes +Let us consider the case of uniaxial loading for which failure begins at the stage of load +growth without the pronounced and irreversible deformation of a specimen. This situation is typical +when testing brittle and quasi-brittle materials, using the scheme of uniaxial compression or +splitting tensile tests, for example. Let the beginning of a decrease in stresses in the stress-time or +stress-strain diagram be the factor indicating the beginning of failure. The strength of a material +then corresponds to the maximum value of stress. This is a common procedure for determining the +strength of such materials with quasi-static tests. However, with an increase in the load or strain rate +above the quasi-static range, the maximum failure stresses increase. Further in this paper, the +maximum failure stresses are assumed as critical stresses. +Fig. 1 demonstrates the case described above. Since the loading area up until the moment of +failure can be considered linear, the next expression is valid for the critical stress σcr: + +3 + +𝜎𝑐𝑟 = 𝐸𝜀̇𝑡𝑐𝑟, 𝑡𝑐𝑟 > 0, (1) +where E is the modulus of elasticity, 𝜀̇ is the strain rate and tcr is the time before the start of failure. +When the critical stress σcr is greater than the quasi-static strength σst, the time to failure and +expression (1) can be written as the sum of two terms: +𝑡𝑐𝑟 = 𝑡𝑠𝑡 + 𝑡𝑑 = 𝜎𝑠𝑡 +𝐸𝜀̇ + 𝑡𝑑, (2) +𝜎𝑐𝑟 = 𝜎𝑠𝑡 + 𝐸𝜀̇𝑡𝑑, +𝑡𝑠𝑡 + 𝑡𝑑 > 0, (3) +where td is the time from time tst when the quasi-static strength of a material is reached to the time +of the actual critical stress tcr. As a rule, experimental equipment allows for recording time diagrams +of load and strain. The strain or loading rate is specified by the input test conditions. For the case +under consideration, the stress rate is 𝜎̇ = 𝐸𝜀̇. Thus, all components of expression (3) can be +determined from the experiment. + +Fig. 1. Schematic representation of loading at different rates. Designations are disclosed in the text. +Then we can estimate the time td that characterises the dynamics of micro-processes in the +structure of the material after reaching the quasi-static strength of the material σst until the onset of +‘macro’ failure tcr: +𝑡𝑑 = 𝜎𝑐𝑟 − 𝜎𝑠𝑡 +𝐸𝜀̇ +. (4) +Let us consider such an estimate regarding various experimental data. Dynamic strength is +typically evaluated by the dependence of critical stress on the strain or stress rate. An example of +(3 +cr,t3 +cr) +(1 +cr,t1 +cr) + + +st +st +st +3td +4td +2td +Stress +Time +td +st +(2 +cr,t2 +cr) + +4 + +such dependence for the case of dynamic uniaxial compression of concrete is presented in Fig. 2. +Substituting the experimental points in expression (4), we derive the dependence of time td on the +strain rate. Fig. 3 presents such calculations for the uniaxial compression and splitting test of +various materials. Usually experimental points have a spread, so it is convenient to use the equation +of the slope of a regression line to determine the time td. The results show that, regardless of the +material and loading scheme, the time td for each experiment can be considered constant. + +Fig. 2. Strain rate dependencies of critical stresses and time to failure for dynamic uniaxial +compression testing of concrete. Experimental data were obtained by [7]. The solid curves ware +calculated using expressions (2) and (3). The dashed line is a linear approximation. +Note that we use the same notation for the case of compression and splitting. This is not +only to avoid unnecessary notation, but also to show the commonality of the effect for various tests. +A caveat, however, is that when considering a specific test method, these load and material +parameters only apply to this method of testing. +Up to this point, it has been assumed that the time to failure tcr is greater than the time td. +However, the question arises: up to what value does the time to failure decrease, and would it be +less than time td? In order to answer this question, it is necessary to consider the stress impulse +before the start of failure. +0 +100 +200 +300 +400 +40 +60 +80 +100 +120 + + - cr +- tcr +- td +Strain rate (1/s) +cr (MPa) +0 +20 +40 +60 +80 +100 +tcr, td (s) + +5 + + + +Fig. 3. The characteristic times of incubation micro-processes from the moment of reaching quasi- +static strength to the moment of the onset of macro-failure of various materials under high-rate +loading. The calculations were carried out by Eq. (4) for the experimental data: a) compression tests +of concrete 1 [7], 2 [8], 3 [9], 4 [10], and granite 5 [11] and 6 [12]; b) splitting test of granite 7 [13], +ceramics 8 [14] and 9 [15] and glass 10 [16]. +A stress analysis shows that for critical stress σst < σcr < 2σst, the stress impulse applied to a +specimen is constant over a time from tcr – 2td to tcr (at the condition that td is constant). These +segments of the stress pulses Jcr are indicated by the shaded areas in Fig. 1. It is evident that the +stress impulse Jcr is equal to +𝐽𝑐𝑟 = 2𝜎𝑠𝑡𝑡𝑑. (5) +101 +102 +103 +104 +0.1 +1 +10 +100 +td (s) + + +Concrete 4 +Concrete 3 +Concrete 1 +Strain rate (1/s) +Concrete 2 +Granite 5,6 +a) +102 +103 +104 +105 +0.1 +1 +10 +100 + + +td (s) +Stress rate (GPa/s) +Glass 10 +Macor 9 +Al2O3 8 +Granite 7 +b) + +6 + +Suppose this impulse is the minimum value of the stress impulse that must be applied to the +specimen in order to initiate and prepare its macro failure. Then, if the time to failure is tcr ≤ 2td, the +failure stress impulse must also be at least 2σsttd. Since for tcr ≤ 2td 2σsttd =0.5σcrtcr, then the +expressions for the strain rate dependence of the critical stress and time to failure, taking (1) into +account, can be written in the following form: +𝜎𝑐𝑟 = √4𝐸𝜎𝑠𝑡𝑡𝑑𝜀̇, 𝑡𝑐𝑟 ≤ 2𝑡𝑑, (6) +𝑡𝑐𝑟 = √4𝜎𝑠𝑡𝑡𝑑 +𝐸𝜀̇ +, 𝑡𝑐𝑟 ≤ 2𝑡𝑑. (7) +In addition, the condition for time to failure in expression (3) should now be specified as +tcr ≥ 2td. +The characteristic time td can be easily obtained from expression (6). Fig. 4 and 5 present +these calculations for the uniaxial compression and splitting test of various materials. Thereby, the +time td was determined by formula (4) for the case σcr <2σst and from formula (6) for σcr ≥ 2σst. This +example also demonstrates that the time td can be considered constant for various strain and loading +rates. + +Fig. 4. Strain rate dependencies of critical stresses and time to failure for dynamic uniaxial +compression tests of concrete in the cases of tcr ≥ 2td and tcr ≤ 2td. Experimental data was obtained +by [17]. The solid curves were calculated using conditions (2), (3), (6) and (7). The dashed line is a +linear approximation. +0.0 +2.0x104 +4.0x104 +6.0x104 +8.0x104 +0 +200 +400 +600 +800 +−cr + − tcr +− td +Stress rate (GPa/s) +cr (MPa) +15 +30 +45 +60 +75 +tcr, td (s) + +7 + + + + +Fig. 5. The characteristic times calculated for different conditions of σcr < 2σst and σcr ≥ 2σst. The +calculations were carried out according to Eq. (4) and (6) for the experimental data: a) compression +tests of concrete 11 [18] and 12 [19], glass 10 [16] and 13 [20], brick 14 [21], and mortar 15 [22]; b) +splitting test of rocks 6 [12], 16 [23], and 17 [24], concrete 11 [18] and 12 [19], and 18 [25], mortar +19 [25], and ceramics 20 [26]. + +3. Discussion and future work +Thus, the experimental data provides reason to introduce the characteristic time of +incubation processes of failure td, which can be considered a constant value independent of a strain +rate above the quasi-static range, at least for cases of continuous linear increase in load during the +101 +102 +103 +104 +10-1 +100 +101 +102 +103 +Mortar 15 +Glass 13 + + +Glass 10 +td (s) +Strain rate (1/s) +a) +Concrete 11, 12 +Brick 14 +100 +101 +102 +103 +104 +105 +106 +107 +10-1 +100 +101 +102 +103 +Concrete 11 +Mortar 19 +Concrete 12 + + +td (s) +Stress rate (GPa/s) +b) +Ceramics 20 +Argillite 17 +Concrete 18 +Granite 6,16 + +8 + +uniaxial compression or splitting tests. This time parameter characterises the individual response of +a material to dynamic loads. Therefore, this characteristic time can be used in criteria conditions of +structural-temporal approaches to failure analysis and strength prediction. For example, similar +expressions for (3) and (6) were obtained theoretically using the incubation time criterion [5,27,28]. +The criterion assumes that the following condition must be implemented for failure to occur: +∫ 𝜎(𝑡)𝑑𝑡 ≥ 𝜎𝑠𝑡𝜏 +𝑡 +𝑡−𝜏 +, (8) +where σ(t) is the stress profile at the failure place, σst is the quasi-static strength, and τ is the +incubation time of failure. The parameters σst and τ are strength parameters of a material for a +particular test scheme, for example, compression or tension. The incubation time was introduced as +a hypothetical characteristic period of time responsible for the incubation period of macro-failure. +This was necessary to ensure the possibility of a smooth transition from pulsed loads to quasi-static +loads using the Nikiforovsky-Shemyakin integral criterion for spall fracture (full integral of the +stress over time at the spall section should reach a critical value) [5,29]. Therefore, according to (8), +failure will occur in the case that the stress impulse in the region of failure is not less than σstτ for +the time t ≤ τ (for t < 0, σ(t) = 0). A similar assumption is made in the discussion of (5). +Substituting (1) for σ(t) in (8), we can obtain simple relationships for calculating the strain +rate dependencies of critical stresses σcr and the time to failure tcr: +{ + 𝜎𝑐𝑟 = 𝜎𝑠𝑡 + 0.5𝐸𝜀̇𝜏, 𝑡𝑐𝑟 = 𝜏 +2 + 𝜎𝑠𝑡 +𝐸𝜀̇ , 𝑡𝑐𝑡 > 𝜏, +𝜎𝑐𝑟 = √2𝐸𝜎𝑠𝑡𝜏𝜀̇, + 𝑡𝑐𝑟 = √4𝜎𝑠𝑡𝑡𝑑 +𝐸𝜀̇ +, 𝑡𝑐𝑟 ≤ 𝜏. + (9) +Substituting τ = 2td, we derive the expressions in (9), which correspond exactly to +expressions (2), (3), (6) and (7). +The incubation time approach allows us to successfully solve a number of dynamic +problems related to fracture mechanics [5,27–30]. However, the question of experimental +determination of the incubation time of failure still remains open-ended. The obtained result shows + +9 + +that the incubation time of failure, at least for the case of a controlled linear increase in fast loading, +can be determined experimentally by estimating td. +The demonstrated effect opens up new possibilities for solving important problems related +to the mechanics of dynamic deformation and fracture of materials. One of these tasks relates to the +possibility of determining the parameters of the dynamic strength of materials using basic tests that +are both publicly available and generally accepted. Furthermore, it is necessary to study appropriate +methods and basic rules to determine the characteristic time td. Experimental conditions affecting +time td should also be established. +Another problem relates to the determination of the strain rate dependence of the yield +strength. The presented reasoning can be applied to consider similar characteristic time of +incubation processes involved in the plastic deformation [30]. +The found characteristic time of failure incubation processes can provide a new approach to +the determination of scale levels of failure. Since we have a constant time characteristic for the +implementation of preparatory failure processes, we can assume that there is also a constant +characteristic structural scale at which these processes are realised. For example, according to the +structural-temporal approach [29], this characteristic scale can relate to the propagation distance of +the elastic wave during time τ (d = τC, in which C is the speed of sound); as well, the scale can be +introduced as 𝑑 = (2𝐾𝐼𝐶 +2)/(𝜋𝜎𝑠𝑡 +2 ) (KIC as the stress intensity factor). However, the accuracy of +these expressions for the elementary scale of failure is still being studied. Thus, the relationship of +time td, scale factors and the dynamics of the preparatory microstructural processes of macro- +fracture should be studied. + +4. Conclusions +The presented study shows the existence of a characteristic time for preparatory micro- +processes of macro-failure of brittle and quasi-brittle materials. This time does not depend on the +strain or loading rate, at least in terms of the compression and splitting test methods under +consideration. The value of the characteristic time can be determined directly from the experiment. + +10 + +The presented results show that this integral time characteristic of the dynamic failure process can +be used for the development of experimental and theoretical foundations to determine and predict +the strength characteristics of constructional materials across a wide range of loading rates. + +Acknowledgements +This work was supported by the Russian Science Foundation, grant № 18-79-00193. + +References +[1] +Zhang QB, Zhao J. 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Structural-temporal theory of fracture as a +multiscale +process. +Phys +Mesomech +2012;15:232–7. +https://doi.org/10.1134/S1029959912020117. +[30] Borodin EN, Mayer AE, Petrov Y V., Gruzdkov AA. Maximum yield strength under quasi- +static and high-rate plastic deformation of metals. Phys Solid State 2014;56:2470–9. +https://doi.org/10.1134/S1063783414120051. + diff --git a/VNE_T4oBgHgl3EQfxhzT/content/tmp_files/load_file.txt b/VNE_T4oBgHgl3EQfxhzT/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..dd54d22b5a2111f137393928281496a190a8a2ea --- /dev/null +++ b/VNE_T4oBgHgl3EQfxhzT/content/tmp_files/load_file.txt @@ -0,0 +1,400 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf,len=399 +page_content='1 A note on the constant characteristic time of failure incubation processes under various high- rate loads Ivan Smirnov Saint Petersburg State University, Universitetskaya nab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' 7/9, St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' Petersburg, 199034, Russia Corresponding author: ivansmirnov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content='sci@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content='com Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' The research reveals the existence of a constant characteristic time of preparatory micro- structural processes before the onset of macro-failure at various high loading rates of brittle and quasi-brittle materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' The presence of this characteristic is analysed based on available data in the literature from dynamic tests for uniaxial compression and splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' It is shown that the characteristic time can be determined experimentally and used to calculate the strain rate dependencies of either critical failure stresses or time to failure, at least in the case of linearly growing loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' In addition, it is discussed that the presence of this constant parameter opens up a prospective opportunity for research and development of new methods for assessing the structural- temporal and scale characteristics of the strength and failure of materials under dynamic loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' Keywords: quasi-brittle material, high strain rate, dynamic strength, time to fracture, characteristic time, failure criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' Introduction The problems of deformation and failure of materials at high-rate intensive loads have been relevant for a significant period of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' Today, these studies continue to both certify the behaviour of materials at a given range of strain rates [1–3], and solve fundamental issues in order to develop approaches for qualitative forecasting the behaviour of materials, regardless of loading history [1,4,5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' However, there are still no universally accepted test methods or parameters that could allow us to evaluate and model the dynamic strength of materials based on simple engineering principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' 2 At sufficiently slow loads, the tensile strength and yield strength can, in fact, be assumed to be constant and considered as characteristic strength parameters of a material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' This forms the basis of systems for state and industry standards to determine the strength characteristics of brittle and plastic materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' When the strain or loading rates exceed the standardized quasi-static range, these strength characteristics can become unstable and often strongly depend on the load history [5,6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' Therefore, under dynamic loads, they can no longer be considered as material parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' To date, there is no agreement on which characteristics can serve as the primary indicators of the properties of materials under high-rate and shock loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' There is an urgent need to identify the characteristics, which do not depend on the history and method (set-up) of load application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' This short communication presents an important, previously unknown effect of the connection between failure incubation processes in the material structure and a constant time characteristic, which are independent of strain/loading rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' It is shown that this characteristic time can be determined experimentally and applied to criteria for evaluating the strength of a material within a wide range of loading rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' The characteristic time of failure incubation processes Let us consider the case of uniaxial loading for which failure begins at the stage of load growth without the pronounced and irreversible deformation of a specimen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' This situation is typical when testing brittle and quasi-brittle materials, using the scheme of uniaxial compression or splitting tensile tests, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' Let the beginning of a decrease in stresses in the stress-time or stress-strain diagram be the factor indicating the beginning of failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' The strength of a material then corresponds to the maximum value of stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' This is a common procedure for determining the strength of such materials with quasi-static tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' However, with an increase in the load or strain rate above the quasi-static range, the maximum failure stresses increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' Further in this paper, the maximum failure stresses are assumed as critical stresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' 1 demonstrates the case described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' Since the loading area up until the moment of failure can be considered linear, the next expression is valid for the critical stress σcr: 3 𝜎𝑐𝑟 = 𝐸𝜀̇𝑡𝑐𝑟, 𝑡𝑐𝑟 > 0, (1) where E is the modulus of elasticity, 𝜀̇ is the strain rate and tcr is the time before the start of failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' When the critical stress σcr is greater than the quasi-static strength σst, the time to failure and expression (1) can be written as the sum of two terms: 𝑡𝑐𝑟 = 𝑡𝑠𝑡 + 𝑡𝑑 = 𝜎𝑠𝑡 𝐸𝜀̇ + 𝑡𝑑, (2) 𝜎𝑐𝑟 = 𝜎𝑠𝑡 + 𝐸𝜀̇𝑡𝑑, 𝑡𝑠𝑡 + 𝑡𝑑 > 0, (3) where td is the time from time tst when the quasi-static strength of a material is reached to the time of the actual critical stress tcr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' As a rule, experimental equipment allows for recording time diagrams of load and strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' The strain or loading rate is specified by the input test conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' For the case under consideration, the stress rate is 𝜎̇ = 𝐸𝜀̇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' Thus, all components of expression (3) can be determined from the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' Schematic representation of loading at different rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' Designations are disclosed in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' Then we can estimate the time td that characterises the dynamics of micro-processes in the structure of the material after reaching the quasi-static strength of the material σst until the onset of ‘macro’ failure tcr: 𝑡𝑑 = 𝜎𝑐𝑟 − 𝜎𝑠𝑡 𝐸𝜀̇ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' (4) Let us consider such an estimate regarding various experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' Dynamic strength is typically evaluated by the dependence of critical stress on the strain or stress rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' An example of (\uf0733 cr,t3 cr) (\uf0731 cr,t1 cr) \uf034\uf073st \uf033\uf073st \uf032\uf073st 3td 4td 2td Stress Time td \uf073st (\uf0732 cr,t2 cr) 4 such dependence for the case of dynamic uniaxial compression of concrete is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' Substituting the experimental points in expression (4), we derive the dependence of time td on the strain rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' 3 presents such calculations for the uniaxial compression and splitting test of various materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' Usually experimental points have a spread, so it is convenient to use the equation of the slope of a regression line to determine the time td.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' The results show that, regardless of the material and loading scheme, the time td for each experiment can be considered constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' Strain rate dependencies of critical stresses and time to failure for dynamic uniaxial compression testing of concrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' Experimental data were obtained by [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' The solid curves ware calculated using expressions (2) and (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' The dashed line is a linear approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' Note that we use the same notation for the case of compression and splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' This is not only to avoid unnecessary notation, but also to show the commonality of the effect for various tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' A caveat, however, is that when considering a specific test method, these load and material parameters only apply to this method of testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' Up to this point, it has been assumed that the time to failure tcr is greater than the time td.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' However, the question arises: up to what value does the time to failure decrease, and would it be less than time td?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' In order to answer this question, it is necessary to consider the stress impulse before the start of failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' 0 100 200 300 400 40 60 80 100 120 \uf073cr tcr td Strain rate (1/s) \uf073cr (MPa) 0 20 40 60 80 100 tcr, td (\uf06ds) 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' The characteristic times of incubation micro-processes from the moment of reaching quasi- static strength to the moment of the onset of macro-failure of various materials under high-rate loading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' The calculations were carried out by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' (4) for the experimental data: a) compression tests of concrete 1 [7], 2 [8], 3 [9], 4 [10], and granite 5 [11] and 6 [12];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' b) splitting test of granite 7 [13], ceramics 8 [14] and 9 [15] and glass 10 [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' A stress analysis shows that for critical stress σst < σcr < 2σst, the stress impulse applied to a specimen is constant over a time from tcr – 2td to tcr (at the condition that td is constant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' These segments of the stress pulses Jcr are indicated by the shaded areas in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' It is evident that the stress impulse Jcr is equal to 𝐽𝑐𝑟 = 2𝜎𝑠𝑡𝑡𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' (5) 101 102 103 104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content='1 1 10 100 td (\uf06ds) Concrete 4 Concrete 3 Concrete 1 Strain rate (1/s) Concrete 2 Granite 5,6 a) 102 103 104 105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content='1 1 10 100 td (\uf06ds) Stress rate (GPa/s) Glass 10 Macor 9 Al2O3 8 Granite 7 b) 6 Suppose this impulse is the minimum value of the stress impulse that must be applied to the specimen in order to initiate and prepare its macro failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' Then, if the time to failure is tcr ≤ 2td, the failure stress impulse must also be at least 2σsttd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' Since for tcr ≤ 2td 2σsttd =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content='5σcrtcr, then the expressions for the strain rate dependence of the critical stress and time to failure, taking (1) into account, can be written in the following form: 𝜎𝑐𝑟 = √4𝐸𝜎𝑠𝑡𝑡𝑑𝜀̇, 𝑡𝑐𝑟 ≤ 2𝑡𝑑, (6) 𝑡𝑐𝑟 = √4𝜎𝑠𝑡𝑡𝑑 𝐸𝜀̇ , 𝑡𝑐𝑟 ≤ 2𝑡𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' (7) In addition, the condition for time to failure in expression (3) should now be specified as tcr ≥ 2td.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' The characteristic time td can be easily obtained from expression (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' 4 and 5 present these calculations for the uniaxial compression and splitting test of various materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' Thereby, the time td was determined by formula (4) for the case σcr <2σst and from formula (6) for σcr ≥ 2σst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' This example also demonstrates that the time td can be considered constant for various strain and loading rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' Strain rate dependencies of critical stresses and time to failure for dynamic uniaxial compression tests of concrete in the cases of tcr ≥ 2td and tcr ≤ 2td.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' Experimental data was obtained by [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' The solid curves were calculated using conditions (2), (3), (6) and (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' The dashed line is a linear approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content='0x104 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content='0x104 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content='0x104 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content='0x104 0 200 400 600 800 −\uf073cr − tcr − td Stress rate (GPa/s) \uf073cr (MPa) 15 30 45 60 75 tcr, td (\uf06ds) 7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' The characteristic times calculated for different conditions of σcr < 2σst and σcr ≥ 2σst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' The calculations were carried out according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' (4) and (6) for the experimental data: a) compression tests of concrete 11 [18] and 12 [19], glass 10 [16] and 13 [20], brick 14 [21], and mortar 15 [22];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' b) splitting test of rocks 6 [12], 16 [23], and 17 [24], concrete 11 [18] and 12 [19], and 18 [25], mortar 19 [25], and ceramics 20 [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' Discussion and future work Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' the experimental data provides reason to introduce the characteristic time of incubation processes of failure td,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' which can be considered a constant value independent of a strain rate above the quasi-static range,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' at least for cases of continuous linear increase in load during the 101 102 103 104 10-1 100 101 102 103 Mortar 15 Glass 13 Glass 10 td (\uf06ds) Strain rate (1/s) a) Concrete 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' 12 Brick 14 100 101 102 103 104 105 106 107 10 1 100 101 102 103 Concrete 11 Mortar 19 Concrete 12 td (\uf06ds) Stress rate (GPa/s) b) Ceramics 20 Argillite 17 Concrete 18 Granite 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content='16 8 uniaxial compression or splitting tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' This time parameter characterises the individual response of a material to dynamic loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' Therefore, this characteristic time can be used in criteria conditions of structural-temporal approaches to failure analysis and strength prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' For example, similar expressions for (3) and (6) were obtained theoretically using the incubation time criterion [5,27,28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' The criterion assumes that the following condition must be implemented for failure to occur: ∫ 𝜎(𝑡)𝑑𝑡 ≥ 𝜎𝑠𝑡𝜏 𝑡 𝑡−𝜏 , (8) where σ(t) is the stress profile at the failure place, σst is the quasi-static strength, and τ is the incubation time of failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' The parameters σst and τ are strength parameters of a material for a particular test scheme, for example, compression or tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' The incubation time was introduced as a hypothetical characteristic period of time responsible for the incubation period of macro-failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' This was necessary to ensure the possibility of a smooth transition from pulsed loads to quasi-static loads using the Nikiforovsky-Shemyakin integral criterion for spall fracture (full integral of the stress over time at the spall section should reach a critical value) [5,29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' Therefore, according to (8), failure will occur in the case that the stress impulse in the region of failure is not less than σstτ for the time t ≤ τ (for t < 0, σ(t) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' A similar assumption is made in the discussion of (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' Substituting (1) for σ(t) in (8), we can obtain simple relationships for calculating the strain rate dependencies of critical stresses σcr and the time to failure tcr: { 𝜎𝑐𝑟 = 𝜎𝑠𝑡 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content='5𝐸𝜀̇𝜏, 𝑡𝑐𝑟 = 𝜏 2 + 𝜎𝑠𝑡 𝐸𝜀̇ , 𝑡𝑐𝑡 > 𝜏, 𝜎𝑐𝑟 = √2𝐸𝜎𝑠𝑡𝜏𝜀̇, 𝑡𝑐𝑟 = √4𝜎𝑠𝑡𝑡𝑑 𝐸𝜀̇ , 𝑡𝑐𝑟 ≤ 𝜏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' (9) Substituting τ = 2td, we derive the expressions in (9), which correspond exactly to expressions (2), (3), (6) and (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' The incubation time approach allows us to successfully solve a number of dynamic problems related to fracture mechanics [5,27–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' However, the question of experimental determination of the incubation time of failure still remains open-ended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' The obtained result shows 9 that the incubation time of failure, at least for the case of a controlled linear increase in fast loading, can be determined experimentally by estimating td.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' The demonstrated effect opens up new possibilities for solving important problems related to the mechanics of dynamic deformation and fracture of materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' One of these tasks relates to the possibility of determining the parameters of the dynamic strength of materials using basic tests that are both publicly available and generally accepted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' Furthermore, it is necessary to study appropriate methods and basic rules to determine the characteristic time td.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' Experimental conditions affecting time td should also be established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' Another problem relates to the determination of the strain rate dependence of the yield strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' The presented reasoning can be applied to consider similar characteristic time of incubation processes involved in the plastic deformation [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' The found characteristic time of failure incubation processes can provide a new approach to the determination of scale levels of failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' Since we have a constant time characteristic for the implementation of preparatory failure processes, we can assume that there is also a constant characteristic structural scale at which these processes are realised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' For example, according to the structural-temporal approach [29], this characteristic scale can relate to the propagation distance of the elastic wave during time τ (d = τC, in which C is the speed of sound);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' as well, the scale can be introduced as 𝑑 = (2𝐾𝐼𝐶 2)/(𝜋𝜎𝑠𝑡 2 ) (KIC as the stress intensity factor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' However, the accuracy of these expressions for the elementary scale of failure is still being studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' Thus, the relationship of time td, scale factors and the dynamics of the preparatory microstructural processes of macro- fracture should be studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' Conclusions The presented study shows the existence of a characteristic time for preparatory micro- processes of macro-failure of brittle and quasi-brittle materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' This time does not depend on the strain or loading rate, at least in terms of the compression and splitting test methods under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' The value of the characteristic time can be determined directly from the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' 10 The presented results show that this integral time characteristic of the dynamic failure process can be used for the development of experimental and theoretical foundations to determine and predict the strength characteristics of constructional materials across a wide range of loading rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' Acknowledgements This work was supported by the Russian Science Foundation, grant № 18-79-00193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} +page_content='1134/S1063783414120051.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE_T4oBgHgl3EQfxhzT/content/2301.08313v1.pdf'} diff --git a/WdE_T4oBgHgl3EQfyhxs/content/tmp_files/2301.08318v1.pdf.txt b/WdE_T4oBgHgl3EQfyhxs/content/tmp_files/2301.08318v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c9d231044a77d692a0deff77b5df647e80cc74b2 --- /dev/null +++ b/WdE_T4oBgHgl3EQfyhxs/content/tmp_files/2301.08318v1.pdf.txt @@ -0,0 +1,1758 @@ +arXiv:2301.08318v1 [gr-qc] 18 Jan 2023 +Gravitational-wave equation in +effective one-body background for spinless binary +Ya Guo1, Hiroaki Nakajima2 and Wenbin Lin1, 2, ∗ +1 School of Physical Science and Technology, Southwest Jiaotong University, +Chengdu, 610031, China +2 School of Mathematics and Physics, University of South China, +Hengyang, 421001, China +∗ Email: lwb@usc.edu.cn +Abstract +We construct the gravitational-wave equation in the background of the effec- +tive one-body system for the spinless binary, which is in general available with the +spherically symmetric background as well. The gauge conditions are given in terms +of the metric perturbation. + +1 +Introduction +The direct observation of gravitational waves by LIGO and Virgo [1] has opened the +new era of cosmology. The binary system such as two black holes is a good object to +observe the gravitational waves, where the analytical calculation is also possible. One of +the calculation method of the gravitational waves radiated from the binary system is the +theory of post-Newtonian approximations [2]. On the other hand, one can also use the +black hole perturbation theory [3], which is useful for the case of the extreme mass-ratio +inspiral (EMRI). The advantage of this method is that there is no divergence and one can +calculate the observable quantity at very high post-Newtonian order [4]. +The reason why the black-hole perturbation theory is available for EMRI is because +the field of one object is regarded as the background and the other body is regarded as +the test particle. If one can map the two-body dynamics into the dynamics of the test +particle in some appropriate background exactly, then the calculation beyond the EMRI +approximation would be possible in terms of the black-hole perturbation theory. This +approach is called the effective one-body (EOB) dynamics [5, 6]. In Newtonian limit, it +is well-known that the effect of the two-body dynamics can exactly be included by just +replacing the light (heavy) mass in the EMRI approximation with the reduced (total) +mass. However when the correction of general relativity is included, the correspondence +between the two-body dynamics and the dynamics of the test particle in some background +becomes complicated. It turns out that the Hamiltonians of the two dynamics are related +by a very nontrivial way [5, 6]. +The gravitational-wave equation in the EOB background has been studied in [7] using +the Newman-Penrose formalism [8], which is a natural extension of the method to obtain +the Teukolsky equation [9] in the Schwarzschild spacetime. It is obtained for some special +cases of the background and is restricted to the even-parity mode. Later the same group +obtained the equation both for the even- and odd-parity modes using the different choices +of the gauge conditions [10]. The wave equation from the metric perturbation in the +generally spherically symmetric background has also been studied in [11, 12]. +Inspired by the work of Jing et al. [7], in this paper, we show that the gravitational- +wave equation for both the even- and odd-parity modes can be obtained using the gauge +conditions taken in their work. Moreover our formalism can be applicable for more general +spherically symmetric backgrounds. +The reminder of this work is organized as follows: in section 2, we briefly review the +EOB dynamics. +In section 3, we consider the wave equation for the perturbed Weyl +scalars. In section 4, we discuss the gauge condition. In section 5, we derive the explicit +gravitational-wave equation. Section 6 is devoted to summary and discussion. +2 +EOB dynamics +The EOB system for the spinless binary is first introduced in [5] in the post-Newtonian +formalism. The effective background metric geff +µν is taken as the spherically symmetric +1 + +form: +ds2 +eff = geff +µνdxµdxν = A(r)dt2 − B(r)dr2 − C(r)r2(dθ2 + sin2 θdϕ2), +(2.1) +where the function C(r) can be freely chosen by the coordinate transformation for the +radial coordinate r, and we here choose the Schwarzschild coordinate corresponding to +C(r) ≡ 1. The explicit forms of A(r) and B(r) can be determined by the comparison +with the two-body dynamics. For the Newman-Penrose formalism [8], we will take the +corresponding null tetrad basis +lA +µ dxµ = dt − D(r) +A(r)dr, +nA +µdxµ = A(r) +2 +dt + D(r) +2 +dr, +mA +µdxµ = − r +√ +2(dθ + i sin θdϕ), +¯mA +µdxµ = − r +√ +2(dθ − i sin θdϕ), +(2.2) +where D(r) = +� +A(r)B(r). The suffix A denotes the background quantities. The null +tetrads satisfy the orthonormal condition as +lA +µ nµ +A = 1, +mA +µ ¯mµ +A = −1, +(2.3) +and the other inner products vanish. From the tetrad basis, one can compute the spin +coefficients, the components of the Ricci tensor and the Weyl scalars as +κA = νA = σA = λA = πA = τ A = ǫA = 0, +(2.4) +ρA = − 1 +rD, +µA = − A +2rD, +γA = A′ +4D, +αA = −βA = − cot θ +2 +√ +2r, +(2.5) +ΦA +01 = ΦA +10 = ΦA +02 = ΦA +20 = ΦA +12 = ΦA +21 = 0, +(2.6) +ΦA +00 = − D′ +rD3, +ΦA +22 = −A2D′ +4rD3, +(2.7) +ΦA +11 = − +1 +8r2D3 +� +2D3 − 2AD − r2(A′D′ − A′′D) +� +, +(2.8) +ΛA = +1 +24r2D3 +� +−2D3 − r2A′D′ + 2A(D − 2rD′) + rD(4A′ + rA′′) +� +, +(2.9) +ΨA +0 = ΨA +1 = ΨA +3 = ΨA +4 = 0, +(2.10) +ΨA +2 = +1 +12r2D3 +� +2(AD + rAD′ − D3) − rA′(2D + rD′) + r2A′′D +� +, +(2.11) +where the prime denotes the ordinary derivative with respect to r. One can find that the +background belongs the petrov type D from (2.10), but is not in the vacuum since there +are nonvanishing components of the Ricci tensor. This type D property is important to +derive the gravitational-wave equation in the next section. Note that for the special case +D(r)=1, which was taken in [7, 10], we have +ΦA +00 = ΦA +22 = 0, +(2.12) +2 + +and the nonvanishing quantities in the above becomes simplified as +ρA = −1 +r, +µA = − A +2r, +γA = A′ +4 , +αA = −βA = − cot θ +2 +√ +2r, +(2.13) +ΦA +11 = − 1 +8r2(2 − 2A + r2A′′), +(2.14) +ΛA = +1 +24r2(−2 + 2A + 4rA′ + r2A′′), +(2.15) +ΨA +2 = +1 +12r2(−2 + 2A − 2rA′ + r2A′′), +(2.16) +Note that when we choose A = 1 − 2M/r and D = 1, the background (2.1) is reduced to +the Schwarzschild spacetime. +The EOB Hamiltonian Heff can be determined from the geodesic motion under the +background (2.1). The action S satisfies the Hamilton-Jacobi equation +gµν +eff PµPν − m2 +0 = 0, +(2.17) +where Pµ = ∂S/∂xµ is the momentum. m0 is the mass of the test particle and is matched +as the reduced mass in the two-body dynamics. From (2.17), Heff is computed as +Heff = m0 +� +A +� +1 + AP 2r +m2 +0D2 + +P 2ϕ +m2 +0r2 +� +, +(2.18) +where the motion plane is fixed on θ = π/2 due to the spherical symmetry. The effective +Hamiltonian Heff and the real two-body Hamiltonian Hreal are compared by matching the +masses and the action variables. It turns out that the two Hamiltonians are related by a +rather nontrivial way as [5, 6] +Hreal = M0 +� +1 + 2m0 +M0 +�Heff +m0 +− 1 +� +, +(2.19) +where M0 is the total mass in the two-body dynamics. The functions A(r) and D(r) are +obtained as [5] +A(r) = 1 − 2M0 +r ++ 2m0 +M0 +�M0 +r +�3 ++ · · · , +D(r) = 1 − 3m0 +M0 +�M0 +r +�2 ++ · · · . +(2.20) +However hereafter we leave A(r) and D(r) arbitrarily. Because of that, the metric (2.1) +takes the most general form of the spherically symmetric background, which can also be +applied to other kinds of the background. Moreover we will see later that when D(r)=1, +the gravitational-wave equation and the gauge condition becomes simplified drastically. +3 + +3 +Wave equation for perturbed Weyl scalars +It has been shown that the background (2.1) is classified as the nonvacuum Petrov type +D background, which is useful to derive the gravitational-wave equation for the perturbed +Weyl scalars using the Newman-Penrose formalism, as in the Teukolsky equation [9] in +the Schwarzschild and the Kerr background. +We begin with the following equations in Newman-Penrose formalism: +(δ + 4β − τ)Ψ4 − (∆ + 4µ + 2γ)Ψ3 + 3νΨ2 += (¯δ − ¯τ + 2¯β + 2α)Φ22 − (∆ + 2γ + 2¯µ)Φ21 − 2λΦ12 + 2νΦ11 + ¯νΦ20, +(3.1) +(D + 4ǫ − ρ)Ψ4 − (¯δ + 4π + 2α)Ψ3 + 3λΨ2 += (¯δ − 2¯τ + 2α)Φ21 − (∆ + 2γ − 2¯γ + ¯µ)Φ20 + ¯σΦ22 − 2λΦ11 + 2νΦ10, +(3.2) +(∆ + µ + ¯µ + 3γ − ¯γ)λ − (¯δ + π − ¯τ + ¯β + 3α)ν + Ψ4 = 0. +(3.3) +We split all the quantities in the above into the background part (A) and the perturbation +part (B), e. g. Ψ4 = ΨA +4 + ΨB +4 , etc. Now we have to take into account the case where ΦA +22 +is nonvanishing, then the background part of the equation becomes nontrivial, i. e. +(¯δ − ¯τ + 2¯β + 2α)AΦA +22 = 0, +(3.4) +has to be satisfied1, and one can confirm that it is indeed the case. +The part of the +first-order perturbation in (3.1)–(3.3) becomes2 +(δ + 4β − τ)AΨB +4 − (∆ + 4µ + 2γ)AΨB +3 + 3νBΨA +2 += (¯δ − ¯τ + 2¯β + 2α)BΦA +22 + (¯δ − ¯τ + 2¯β + 2α)AΦB +22 +− (∆ + 2γ + 2¯µ)AΦB +21 + 2νBΦA +11, +(3.5) +(D + 4ǫ − ρ)AΨB +4 − (¯δ + 4π + 2α)AΨB +3 + 3λBΨA +2 += (¯δ − 2¯τ + 2α)AΦB +21 − (∆ + 2γ − 2¯γ + ¯µ)AΦB +20 + ¯σBΦA +22 − 2λBΦA +11, +(3.6) +(∆ + µ + ¯µ + 3γ − ¯γ)AλB − (¯δ + π − ¯τ + ¯β + 3α)AνB + ΨB +4 = 0. +(3.7) +Again, when ΦA +22 is nonvanishing, the first term in the right hand side in (3.5) appears, +and more perturbed quantities ¯τ B, ¯βB and αB contribute to the equation, compared with +the vacuum case. In order to reduce the number of those quantities, we require that this +term should vanish by the gauge condition; +Ξ22 ≡ (¯δ − ¯τ + 2¯β + 2α)BΦA +22 = 0, +(3.8) +1Here the superscript A (B) on the parentheses denotes that all the quantities and the operators inside +the parentheses are in the background (perturbation). +2Here we will not use πA = τA = ǫA = 0 from the beginning and keep them for a while, which would +be useful to extend the result into the spinning case. +4 + +Now we will obtain the wave equation for ΨB +4 in a similar way with the method used +to derive the Teukolsky equation. First we show the following commutation relation of +the differential operators: +[∆ + (p + 1)γ − ¯γ − qµ + ¯µ]A (¯δ + pα − qπ)A +− +�¯δ + (p + 1)α + ¯β − ¯τ − qπ +�A (∆ + pγ − qµ)A += νADA − λAδA − p [(β + τ)λ − (ρ + ǫ)ν + Ψ3]A ++ q [−Dν + δλ + (¯π + τ + 3β − ¯α)λ − (3ǫ + ¯ǫ + ρ − ¯ρ)ν + 2Ψ3]A += 0, +(3.9) +where p and q are arbitrary constants and we have used νA = λA = ΨA +3 = 0. We operate +(∆ + 3γ − ¯γ + 4µ + ¯µ)A to (3.6) and (¯δ + 3α + ¯β − ¯τ + 4π)A to (3.5), and then subtract +one equation from the other. The terms with ΨB +3 cancel by (3.9) with p = 2, q = −4 and +the remaining becomes +� +(∆ + 3γ − ¯γ + 4µ + ¯µ)(D + 4ǫ − ρ) − (¯δ + 3α + ¯β − ¯τ + 4π)(δ + 4β − τ) +�A ΨB +4 ++ 3ΨA +2 +� +(∆ + 3γ − ¯γ + 4µ + ¯µ)AλB − (¯δ + 3α + ¯β − ¯τ + 4π)AνB� ++ 3λB∆AΨA +2 − 3νB¯δAΨA +2 += T4 + (∆ + 3γ − ¯γ + 4µ + ¯µ)A(¯σBΦA +22) − 2λB∆AΦA +11 − 2νB¯δAΦA +11 +− 2ΦA +11 +� +(∆ + 3γ − ¯γ + 4µ + ¯µ)AλB + (¯δ + 3α + ¯β − ¯τ + 4π)AνB� +, +(3.10) +where T4 is defined by +T4 = (∆ + 3γ − ¯γ + 4µ + ¯µ)A � +(¯δ − 2¯τ + 2α)AΦB +21 − (∆ + 2γ − 2¯γ + ¯µ)AΦB +20 +� +− (¯δ + 3α + ¯β − ¯τ + 4π)A � +(¯δ − ¯τ + 2¯β + 2α)AΦB +22 − (∆ + 2γ + 2¯µ)AΦB +21 +� +. +(3.11) +For the third line in (3.10), we have +∆AΨA +2 = −3µAΨA +2 − 2µΦA +11 − (D − ¯ρ + 2ǫ + 2¯ǫ)AΦA +22 − 2∆AΛA, +(3.12) +¯δAΨA +2 = −3πAΨA +2 + 2πAΦA +11 − 2¯δAΛA, +(3.13) +and then substituting the above into (3.10), we get +� +(∆+3γ−¯γ+4µ + ¯µ)(D+4ǫ−ρ)−(¯δ + 3α + ¯β − ¯τ + 4π)(δ + 4β − τ) +�A ΨB +4 ++ 3ΨA +2 +� +(∆ + 3γ − ¯γ + µ + ¯µ)AλB − (¯δ + 3α + ¯β − ¯τ + π)AνB� += T4 + (∆ + 3γ − ¯γ + 4µ + ¯µ)A(¯σBΦA +22) − 2λB∆AΦA +11 − 2νB¯δAΦA +11 +− 2ΦA +11 +� +(∆ + 3γ − ¯γ + µ + ¯µ)AλB + (¯δ + 3α + ¯β − ¯τ + π)AνB� ++ 3λB(D − ¯ρ + 2ǫ + 2¯ǫ)AΦA +22 + 6λB∆AΛA − 6νB¯δAΛA. +(3.14) +5 + +The second line in (3.14) becomes −3ΨA +2 ΨB +4 using (3.7), and there is also similar terms in +the fourth line but the relative sign is positive. We now require more gauge conditions as +λB = σB = 0. +(3.15) +Then the fourth line in (3.14) becomes −2ΦA +11ΨB +4 and the terms with ΦA +22 disappear. +Thus under the gauge conditions (3.8) and (3.15), the decoupled wave equation for ΨB +4 is +obtained as +� +(∆ + 3γ − ¯γ + 4µ + ¯µ)(D + 4ǫ − ρ) +− (¯δ + 3α + ¯β − ¯τ + 4π)(δ + 4β − τ) − 3Ψ2 + 2Φ11 +�AΨB +4 = T4, +(3.16) +One can also consider the wave equation for ΨB +0 , which can be obtained in a similar way. +The resultant equation becomes +� +(D − 3ǫ + ¯ǫ − 4ρ − ¯ρ)(∆ − 4γ + µ) +− (δ − 3β − ¯α + ¯π − 4τ)(¯δ − 4α + π) − 3Ψ2 + 2Φ11 +�AΨB +0 = T0. +(3.17) +The gauge conditions are (3.15) and +Ξ00 ≡ (δ + ¯π − 2¯α − 2β)BΦA +00 = 0. +(3.18) +Note that for the case D=1 we have (2.12), then (3.8) and (3.18) are obviously satisfied. +The other conditions (3.15) are also relaxed as just λB = 0 for (3.16) and just σB = 0 for +(3.17). +4 +Gauge conditions +Here we will study the consistency of the gauge conditions (3.8), (3.18) and (3.15) (or +just (3.15) for D = 1). In Newman-Penrose formalism, there are ten gauge degrees of +freedom. Six of them are the tetrad rotation (the local Lorentz transformation), which +can be decomposed as the following three kinds [13]: +lµ → lµ, +mµ → mµ + alµ, +¯mµ → ¯mµ + ¯alµ, +nµ → nµ + ¯amµ + a ¯mµ + a¯alµ, +(4.1) +nµ → nµ, +mµ → mµ + bnµ, +¯mµ → ¯mµ + ¯bnµ, +lµ → lµ + ¯bmµ + b ¯mµ + b¯bnµ, +(4.2) +lµ → e−clµ, +nµ → ecnµ, +mµ → eiϑmµ, +¯mµ → e−iϑ ¯mµ. +(4.3) +Here a and b are complex functions, and c and ϑ are real functions. The transformations of +the quantities in Newman-Penrose equations (the spin coefficients, the Weyl scalars, etc.) +under the above are shown in [14], for example. The other four are the general coordinate +transformation x′µ = xµ+ξµ(x). Under the transformation, the null tetrads behave as the +6 + +one-forms or the vector fields, and the quantities in Newman-Penrose equations behave +as the scalar fields. Because of this, the form of the transformation for the latter becomes +X → X − ξµ∂µX, +(4.4) +where X represents either one of the spin coefficients, Ψ’s, Φ’s or Λ. Since we want to +keep the background (2.4)–(2.11) (or (2.13)-(2.16) for D=1) intact, the functions a, b, c, +ϑ and ξµ have to be at the order of the perturbed quantities, and in particular the second +and the higher orders of them are neglected. +The gauge conditions (3.8), (3.15) and (3.18) are transformed under the the gauge +transformations (4.1), (4.2), (4.3) and (4.4) as3 +Ξ00 → Ξ00 + (δ + ¯π − 2¯α − 2β)A(ξµ∂µΦA +00) + DA(aΦA +00) − 2aρAΦA +00 ++ b(∆A + ¯µA − 2µA − 2¯γA − 2γA)ΦA +00 + 2ΦA +00δc, +(4.5) +Ξ22 → Ξ22 + (¯δ − ¯τ + 2α + 2¯β)A(ξµ∂µΦA +22) + ¯a(DA − ¯ρA + 2ρA)ΦA +22 ++ ∆(¯bΦA +22) + 2¯b(µA + γA + ¯γA)ΦA +22 − 2ΦA +22¯δc, +(4.6) +λB → λB + 2αA¯a + ¯δA¯a, +(4.7) +σB → σB + 2βAb − δAb. +(4.8) +One can find that there are nine real degrees (four ξ’s, a, b and c) of freedom for eight real +conditions (Ξ00, Ξ22, λB and σB). In general, it is enough to satisfy all the conditions. +In order to see the the condition explicitly, here we will use the form of the metric +perturbation, following the A-K parametrization [15, 12]. First, the metric perturbation +is given as +gµν = gA +µν + hBE +µν + hBO +µν , +(4.9) +where gA +µν = geff +µν is the background metric (2.1). hBE +µν and hBO +µν are respectively the even- +and the odd-parity part of the perturbation, which are parametrized as +hBE +µν = + + + + + +AS +−DS +−rB∂θS +−rB∂ϕS +KS +rH∂θS +rH∂ϕS +r2E+ +F +r2FPS1 +r2 sin2 θ E− +F + + + + + , +(4.10) +hBO +µν = 1 +D + + + + + +0 +0 +rC csc θ∂ϕS +−rC sin θ∂θS +0 +−rJ csc θ∂ϕS +rJ sin θ∂θS +−r2G csc θPS1 +−1 +2r2GPS2 +r2GPS3 + + + + + . +(4.11) +3Our gauge conditions are invariant under the transformation generated by ϑ. +7 + +Here the lower-left blanks have to be filled as hBE +µν and hBO +µν to be symmetric. A, B, C, +D, E, F, G, H, J and K are the functions4 of t and r, and S is the function of θ and ϕ, +which should in turn be identified as the spherical harmonics Ylm(θ, ϕ). The quantities +E± +F , PS1, PS2 and PS3 are defined by +E± +F = +� +E ± F +� +∂2 +θ + 1 +2l(l + 1) +�� +S, +(4.12) +PS1 = (∂θ∂ϕ − cot θ∂ϕ)S, +(4.13) +PS2 = (csc θ∂2 +ϕ + cos θ∂θ − sin θ∂2 +θ)S, +(4.14) +PS3 = (sin θ∂θ∂ϕ − cos θ∂ϕ)S, +(4.15) +where l is one of the label in Ylm(θ, ϕ). Other perturbed quantities are also decomposed +into the even- and the odd-parity modes, such as lB +µ = lBE +µ ++ lBO +µ , etc. +Now we have to find the null tetrads corresponding to the metric. Even though the +metric (2.1) is fixed, the tetrads are not unique due to the tetrad rotation (4.1), (4.2) and +(4.3). In other words, once a set of null tetrads is found, the others are obtained from +those tetrads by the tetrad rotation. We will fix the reference tetrads as +lE +µ dxµ ≡ (lA +µ + lBE +µ )dxµ += dt − D +A +� +1 − 1 +2A +� +A − 2A +D D + A2 +D2K +� +S +� +dr − +r +AD(DB − AH)dS, +(4.16) +nE +µ dxµ ≡ (nA +µ + nBE +µ )dxµ += A +2 +� +1+ AS +A +� +dt+ D +2 +� +1+ 1 +2A +� +A−2A +D D− A2 +D2K +� +S +� +dr− r +2D(DB+AH)dS, +(4.17) +mE +µ dxµ ≡ (mA +µ + mBE +µ )dxµ += − r +√ +2 +�� +1 − E+ +F +2 +� +dθ + i +� +1 − E− +F +2 + i csc θFPS1 +� +sin θdϕ +� +, +(4.18) +¯mE +µ dxµ ≡ ( ¯mA +µ + ¯mBE +µ )dxµ += − r +√ +2 +�� +1 − E+ +F +2 +� +dθ − i +� +1 − E− +F +2 − i csc θFPS1 +� +sin θdϕ +� +, +(4.19) +for the even-parity modes and +lO +µ dxµ ≡ (lA +µ + lBO +µ )dxµ += dt − D +Adr + +r +AD +� +C − A +DJ +� +(csc θ∂ϕS dθ − sin θ∂θS dϕ), +(4.20) +nO +µ dxµ ≡ (nA +µ + nBO +µ )dxµ +4We hope the readers may not confuse the function D here and the differential operator D in Newman- +Penrose formalism. +8 + += A +2 dt + D +2 dr + r +2D +� +C + A +DJ +� +(csc θ∂ϕS dθ − sin θ∂θS dϕ), +(4.21) +mO +µ dxµ ≡ (mA +µ + mBO +µ )dxµ += − r +√ +2 +�� +1+ csc θ +2D GPS1 +� +dθ+i sin θ +� +1 − 1 +2D csc θG(PS1 + iPS2) +� +dϕ +� +, (4.22) +¯mO +µ dxµ ≡ ( ¯mA +µ + ¯mBO +µ )dxµ += − r +√ +2 +�� +1+ csc θ +2D GPS1 +� +dθ−i sin θ +� +1 − 1 +2D csc θG(PS1 − iPS2) +� +dϕ +� +, (4.23) +for the odd-parity modes. Here dS = (∂θS)dθ + (∂ϕS)dϕ. Note that the above choice +of the reference tetrads is different from the one taken in [7] for both the even- and the +odd-parity modes even under the Regge-Wheeler gauge B = F = H = G = 0 [16, 17, 18] +or the EZ gauge B = E = F = G = 0 [15, 19]. This is again due to the different choice of +the reference and they are equivalent. The advantage of our choice is that the expressions +of λB and σB become relatively simple as +λBE = −A +4 +�� +∂2 +θ + 1 +2l(l + 1) +� +S − i csc θPS1 +� � 1 +A∂tF − 1 +D∂rF +� +, +(4.24) +σBE = 1 +2 +�� +∂2 +θ + 1 +2l(l + 1) +� +S + i csc θPS1 +� � 1 +A∂tF + 1 +D∂rF +� +, +(4.25) +for the even-parity modes and +λBO = A +8 csc θ (2PS1 − iPS2) +� 1 +A∂t +�G +D +� +− 1 +D∂r +�G +D +�� +, +(4.26) +σBO = −1 +4 csc θ (2PS1 + iPS2) +� 1 +A∂t +�G +D +� ++ 1 +D∂r +�G +D +�� +, +(4.27) +for the odd-parity modes. In particular, one can easily find that they vanish under the +Regge-Wheeler or the EZ gauge5, which also means that the above reference tetrads under +those gauges can be used as the standard form of the perturbation. +Next we will consider the gauge invariants, are obtained in [12, 15] as +α = J − r +2D∂r +�G +D +� +, +(4.28) +β = −C − r +2∂tG, +(4.29) +χ = H − D2 +2AE − l(l + 1)D2 +4A +F − r +2∂rF, +(4.30) +ψ = 1 +2K − r +4 +�D2 +A +�′ +E − D2 +2AE − rD2 +2A ∂rE +5For the even-parity modes, the choice taken in [7] can also lead to λBE = σBE = 0 under F = 0, but +not for the odd-parity mode under G = 0. +9 + +− r +8l(l + 1) +�D2 +A +�′ +F − D2 +4Al(l + 1)F − rD2 +4A l(l + 1)∂rF +(4.31) +δ = D + rD2 +2A ∂tE + +�rA′ +A − 1 +� +B − r∂rB +− r2 +2 ∂t∂rF − +� +r − r2A′ +2A − rD2 +4A l(l + 1) +� +∂tF, +(4.32) +ǫ = −1 +2A − rA′ +4 E − r∂tB − rA′ +8 l(l + 1)F − r2 +2 ∂2 +t F. +(4.33) +The perturbed Weyl scalars ΨB +4 and ΨB +0 can be written as the gauge-invariant combina- +tions of the perturbation as +ΨBE +4 += csc θ +8r2D3(PS2 + 2iPS1) +� +−A2Dψ − AD2δ + D3ǫ +−rAD2∂tχ + rA2D∂rχ + A2(D − rD′)χ +� +, +(4.34) +ΨBO +4 += −i csc θ +8rD (PS2 + 2iPS1) +� +−A +D∂tα + A2 +D2∂rα + A2(D − 2rD′) +rD3 +α ++ ∂tβ − A +D∂rβ + +�AD′ +D2 − A − rA′ +rD +� +β +� +, +(4.35) +ΨBE +0 += +csc θ +2r2A2D3(PS2 − 2iPS1) +� +−A2Dψ + AD2δ + D3ǫ ++rAD2∂tχ + rA2D∂rχ + A2(D − rD′)χ +� +, +(4.36) +ΨBO +0 += i csc θ +2rA2D(PS2 − 2iPS1) +� +A +D∂tα + A2 +D2∂rα + A2(D − 2rD′) +rD3 +α ++ ∂tβ + A +D∂rβ + +� +−AD′ +D2 + A − rA′ +rD +� +β +� +, +(4.37) +The gauge conditions (3.8), (3.18) and (3.15) can be expressed as the gauge invariants +and the set of the variables (B, F, H, G) or (B, E, F, G). The explicit form for the latter +is shown in Appendix. +5 +Explicit PDE and ODE for radial coordinate +In section 3, we have obtained the wave equation for the perturbed Weyl scalars ΨB +4 +and ΨB +0 as (3.16) and (3.17), respectively. By substituting the background (2.4)–(2.11) +(or (2.13)-(2.16) for D = 1), the partial differential equations (the master equations) are +obtained, Then by the separation of variables, the ordinary differential equation for the +radial coordinate r and that for the angular coordinates θ and ϕ are also obtained. +10 + +We define ψ(−2) as +ψ(−2) ≡ (ρA)−4ΨB +4 = (rD)4ΨB +4 . +(5.1) +From (3.16), ψ(−2) satisfies the following partial differential equation: +r2 +A +∂2ψ(−2) +∂t2 +− (r2A)2D7 ∂ +∂r +� +(r2A)−1D−9∂ψ(−2) +∂r +� ++ +�2r2A′ +AD − 4r +D +� ∂ψ(−2) +∂t ++ +� +−24r2A(D′)2 +D4 ++ 4r2AD′′ +D3 +− 3r2A′D′ +D3 +− 12rAD′ +D3 +− r2A′′ +D2 +− 2rA′ +D2 +� +ψ(−2) +− +1 +sin θ +∂ +∂θ +� +sin θ∂ψ(−2) +∂θ +� +− +1 +sin2 θ +∂2ψ(−2) +∂ϕ2 ++ 4i cot θ +sin θ +∂ψ(−2) +∂ϕ ++(4 cot2 θ+2)ψ(−2) += 2r6D4T4. +(5.2) +For homogeneous case (T4 = 0), by assuming the product form ψ(−2) = e−iωteimϕR(r)S(θ), +one can obtain the separated equations as +(r2A)2D7 d +dr +� +(r2A)−1D−9dR(r) +dr +� ++ +�r2ω2 +A ++ iω +�2r2A′ +AD − 4r +D +� ++ 24r2A(D′)2 +D4 +−4r2AD′′ +D3 ++ 3r2A′D′ +D3 ++ 12rAD′ +D3 ++ r2A′′ +D2 ++ 2rA′ +D2 − λ +� +R(r) = 0, +(5.3) +1 +sin θ +d +dθ +� +sin θdS(θ) +dθ +� +− +� m2 +sin2 θ − 4m cot θ +sin θ ++ 4 cot2 θ + 2 − λ +� +S(θ) = 0, +(5.4) +where ω gives the frequency of the gravitational wave and λ is the separation constant. +From (5.4) one can find that S(θ)eimϕ coincides with the s = −2 spin-weighted spherical +harmonics, denoted as −2Ylm(θ, ϕ), and then the separation constant λ has to be +λ = (l − 1)(l + 2), +(5.5) +where l and m take the integer value with l ≥ 2 and −l ≤ m ≤ l, respectively. For +inhomogeneous case (T4 ̸= 0), one can expand ψ(−2) and T4 as +ψ(−2) = +� +dω +� +l,m +R(−2)lω(r)−2Ylm(θ, ϕ)e−iωt, +(5.6) +2r6D4T4 = − +� +dω +� +l,m +G(−2)lω(r)−2Ylm(θ, ϕ)e−iωt. +(5.7) +Then the ordinary differential equation for the radial coordinate r is +(r2A)2D7 d +dr +� +(r2A)−1D−9dR(−2)lω(r) +dr +� ++ +�r2ω2 +A ++ iω +�2r2A′ +AD − 4r +D +� ++24r2A(D′)2 +D4 +− 4r2AD′′ +D3 ++ 3r2A′D′ +D3 ++ 12rAD′ +D3 ++ r2A′′ +D2 ++ 2rA′ +D2 +− (l − 1)(l + 2) +� +R(−2)lω(r) = G(−2)lω(r). +(5.8) +11 + +In the case of D=1, (5.8) is reduced to +(r2A)2 d +dr +� +(r2A)−1dR(−2)lω(r) +dr +� ++ +�r2ω2 +A ++ iω +�2r2A′ +A +− 4r +� ++ r2A′′ + 2rA′ − (l − 1)(l + 2) +� +R(−2)lω(r) += G(−2)lω(r), +(5.9) +Note that for the case of Schwarzschild background, i. e. A = 1 − 2M/r, the differential +equation (5.9) is reduced to the Teukolsky equation [9] with the spin weight s = −2. +The equation for ψ(2) ≡ ΨB +0 can be obtained in a similar way. The master equation +becomes +r2 +A +∂2ψ(2) +∂t2 +− (r2A)−2D−1 ∂ +∂r +� +(r2A)3D−1∂ψ(2) +∂r +� +− +�2r2A′ +AD − 4r +D +� ∂ψ(2) +∂t ++ +�3r2A′D′ +D3 +− 3r2A′′ +D2 +− 10rA′ +D2 +− 4A +D2 +� +ψ(2) +− +1 +sin θ +∂ +∂θ +� +sin θ∂ψ(2) +∂θ +� +− +1 +sin2 θ +∂2ψ(2) +∂ϕ2 −4i cot θ +sin θ +∂ψ(2) +∂ϕ +(4 cot2 θ + 2)ψ(2) += 2r2T0. +(5.10) +After the separation of the variables, for the angular part, we have the s = 2 spin-weighted +spherical harmonics 2Ylm(θ, ϕ). For the radial part, we have the following equation: +(r2A)−2D−1 d +dr +� +(r2A)3D−1dR(2)lω(r) +dr +� ++ +�r2ω2 +A +− iω +�2r2A′ +AD − 4r +D +� +− 3r2A′D′ +D3 ++ 3r2A′′ +D2 ++ 10rA′ +D2 ++ 4A +D2 +− (l − 2)(l + 3) +� +R(2)lω(r) = G(2)lω(r), +(5.11) +which is reduced to the Teukolsky equation with s = ±2 for A = 1 − 2M/r and D ≡ 1. +6 +Summary and discussion +In this paper, we have studied the wave equations for the perturbed Weyl scalars ΨB +4 and +ΨB +0 in the background of the EOB dynamics for the spinless binary. We also obtained +the Teukolsky-like equations for the function of the radial coordinate r. In the previous +work [7], the special case D=1 is studied and the (decoupled) wave equation is obtained +only for the even-parity mode. On the other hand, here we have studied the odd-parity +mode also, and have obtained the same form of the wave equation. Moreover, we have +12 + +considered a more general case including D ̸=1. The different form of the wave equation +is proposed in [10] by the use of the different gauge, although it is again restricted to the +case of D=1. It would be interesting to find the relation between our equations and their +ones. +One can easily find that the background is assumed to be just spherically symmetric +and general enough. Even in the case D = 1, it contains many kinds of the black holes. +A similar wave equations in the spherically symmetric background are obtained using +the metric perturbation in [12], which are resemble to the Regge-Wheeler and the Zerilli +equations [16, 17], and are different from the ones obtained here. This is because we +have used the different gauge and the different master variables. However in the case of +Schwarzschild background, there is a transformation originated from the connection of +the different gauges, which is called the Chandrasekhar transformation [20]. Moreover +the relation between the R(−2)lω(r) in (5.8) and R(2)lω(r) in (5.11) can also be regarded +as the special case of the Chandrasekhar transformation [9, 21]. It would be interesting +to find a similar transformation in the present case. +Another possible generalization would be to include the effect of the spin, where the +background becomes axially symmetric and contains the Kerr black hole as an example. +In this case the method of the metric perturbation is difficult to perform, because it +relies on the spherical symmetry and the expansion by the spherical harmonics. However +the Newman-Penrose formalism would still be powerful enough, and it can be expected +that the gravitational-wave equation could be obtained. Studying the equations for the +electromagnetic and the scalar waves is also interesting. +Acknowledgements +This work was supported in part by the National Natural Science Foundation of China +(Grant No. 11973025). +A +Gauge conditions with gauge invariants +The gauge conditions (3.8), (3.18) and (3.15) can be written in terms of the gauge invari- +ants and the variables (B, E, F, G). Each condition consists of the even-parity part (E), +odd-parity part (O) and the transformation part (T). +Ξ22 = ΞE +22 + ΞO +22 + ΞT +22, +(A.1) +ΞE +22 = A2(∂θS − i csc θ∂ϕS) +4 +√ +2r +� +−AD′′ +D5 +� +χ + r +2∂rF + l(l + 1) +4 +D2 +A F + D2 +2AE +� +− D′ +D3 +� +− 1 +D∂tχ+ A +2D2∂rχ+ +� +− A +2rD2 + 3A′ +2D2 −7AD′ +2D3 +� +χ+ 3 +2rAǫ− +A +2rD2ψ ++ 1 +rDδ + 2 +A∂tB − D +A∂tE + +�A′ +A − 3D′ +2D − 1 +2r +� +E + 3r +4A∂2 +t F + rA +4D2∂2 +rF +13 + ++ +� 1 +D − l(l + 1) +2 +D +A − rA′ +2AD +� +∂tF + +�3rA′ +4D2 − 7rAD′ +4D3 +� +∂rF +− +�1 +r + 3D′ +D − 2A′ +A +� l(l + 1) +4 +F +�� +, +(A.2) +ΞO +22 = −iA2(∂θS − i csc θ∂ϕS) +4 +√ +2r +� +−AD′′ +D6 +� +α + rD +2 ∂r ˜G +� ++ D′ +D3 +� 1 +D2∂tα − A +2D3∂rα ++ +� A +2rD3 − 3A′ +2D3 + 4AD′ +D4 +� +α+ +1 +2AD∂tβ− 1 +D2∂rβ+ +� A′ +AD2 − 1 +rD2 + D′ +D3 +� +β ++ r +4A∂2 +t ˜G + +� rA′ +2AD − 1 +D +� +∂t ˜G − rA +4D2∂2 +r ˜G + +� +−3rA′ +4D2 + 7rAD′ +4D3 +� +∂r ˜G +�� +, (A.3) +ΞT +22 = +A +4rD4 +�AD′ +r +− A′D′ + 3A(D′)2 +D +− AD′′ +� � +¯a − A +2 +¯b +� +−A2D′ +4rD3 +� +A′ +AD¯a− 1 +rD ¯a− A +rD +¯b+ 1 +2∂t¯b − A +2D∂r¯b − +√ +2 +r ∂θc + +√ +2i csc θ +r +∂ϕc +� +, (A.4) +Ξ00 = ΞE +00 + ΞO +00 + ΞT +00, +(A.5) +ΞE +00 = ∂θS + i csc θ∂ϕS +√ +2r +� +−AD′′ +D5 +� +χ + r +2∂rF + l(l + 1) +4 +D2 +A F + D2 +2AE +� +− D′ +D3 +� 1 +D∂tχ + A +2D2∂rχ+ +� 3A′ +2D2 − +A +2rD2 − 7AD′ +2D3 +� +χ− 5 +2rAǫ− +A +2rD2ψ +− 1 +rDδ − 2 +A∂tB + D +A∂tE − +�3D′ +2D + 1 +2r +� +E − 5r +4A∂2 +t F + rA +4D2∂2 +rF ++ +� +− 1 +D + l(l + 1) +2 +D +A + rA′ +2AD +� +∂tF + +�3rA′ +4D2 − 7rAD′ +4D3 +� +∂rF +− +�1 +r + 3D′ +D +� l(l + 1) +4 +F +�� +, +(A.6) +ΞO +00 = i(∂θS + i csc θ∂ϕS) +√ +2r +� +−AD′′ +D6 +� +α + rD +2 ∂r ˜G +� +− D′ +D3 +� 1 +D2∂tα + A +2D3∂rα ++ +� 3A′ +2D3 − +A +2rD3 −4AD′ +D4 +� +α− +1 +2AD∂tβ− 1 +D2∂rβ+ +� A′ +AD2 − 1 +rD2 + D′ +D3 +� +β +− r +4A∂2 +t ˜G + +� rA′ +2AD − 1 +D +� +∂t ˜G + rA +4D2∂2 +r ˜G + +�3rA′ +4D2 − 7rAD′ +4D3 +� +∂r ˜G +�� +, (A.7) +ΞT +00 = +1 +rD4 +�D′ +r + 3(D′)2 +D +− D′′ +� � +a − A +2 b +� +− D′ +rD3 +� +2 +rDa+ 1 +A∂ta + 1 +D∂ra + +A +2rDb − A′ +D b + +√ +2 +r ∂θc + +√ +2i csc θ +r +∂ϕc +� +, (A.8) +λB = λBE + λBO + λBT , +(A.9) +14 + +λBE = −A +4 +�� +∂2 +θ + 1 +2l(l + 1) +� +S − i csc θPS1 +� � 1 +A∂tF − 1 +D∂rF +� +, +(A.10) +λBO = A +8 csc θ (2PS1 − iPS2) +� 1 +A∂t ˜G − 1 +D∂r ˜G +� +, +(A.11) +λBT = +1 +√ +2r +(∂θ¯a − i csc θ∂ϕ¯a − ¯a cot θ), +(A.12) +σB = σBE + σBO + σBT , +(A.13) +σBE = 1 +2 +�� +∂2 +θ + 1 +2l(l + 1) +� +S + i csc θPS1 +� � 1 +A∂tF + 1 +D∂rF +� +, +(A.14) +σBO = −1 +4 csc θ (2PS1 + iPS2) +� 1 +A∂t ˜G + 1 +D∂r ˜G +� +, +(A.15) +σBT = − 1 +√ +2r +(∂θb + i csc θ∂ϕb − b cot θ). +(A.16) +Here ˜G = G/D and ξ’s are omitted since these are converted to fix the variables (B, E, F, G). +Note that in the case of D =1, the conditions Ξ22 = Ξ00 = 0 becomes trivial from (2.12) +and λB = σB = 0 can be achieved by setting F = G = a = b = 0. +References +[1] B. Abbott et al. [LIGO Scientific and Virgo], Phys. Rev. Lett. 116, no.6, 061102 +(2016) doi:10.1103/PhysRevLett.116.061102 [arXiv:1602.03837 [gr-qc]]. +[2] E. Poisson and C. M. Will, “Gravity: Newtonian, Post-Newtonian, Relativistic.” +Cambridge University Press (2014) doi:10.1017/CBO9781139507486. +[3] Y. Mino, M. Sasaki, M. Shibata, H. Tagoshi and T. Tanaka, Prog. Theor. Phys. +Suppl. 128, 1-121 (1997) doi:10.1143/PTPS.128.1 [arXiv:gr-qc/9712057 [gr-qc]]. +[4] R. +Fujita, +PTEP +2015, +no.3, +033E01 +(2015) +doi:10.1093/ptep/ptv012 +[arXiv:1412.5689 [gr-qc]]. +[5] A. +Buonanno +and +T. +Damour, +Phys. +Rev. +D +62, +064015 +(2000) +doi:10.1103/PhysRevD.62.064015 [arXiv:gr-qc/0001013 [gr-qc]]. +[6] T. Damour, Phys. Rev. D 94, no.10, 104015 (2016) doi:10.1103/PhysRevD.94.104015 +[arXiv:1609.00354 [gr-qc]]. +[7] J. Jing, S. Chen, M. Sun, X. He, M. Wang and J. Wang, Sci. China Phys. Mech. +Astron. 65, no.6, 260411 (2022) doi:10.1007/s11433-022-1885-6 [arXiv:2112.09838 +[gr-qc]]. +[8] E. 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JETP 38, 1 (1974). +16 + diff --git a/WdE_T4oBgHgl3EQfyhxs/content/tmp_files/load_file.txt b/WdE_T4oBgHgl3EQfyhxs/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..34d4feb06b2788d82812a63d4d8345d33355d495 --- /dev/null +++ b/WdE_T4oBgHgl3EQfyhxs/content/tmp_files/load_file.txt @@ -0,0 +1,511 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf,len=510 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='08318v1 [gr-qc] 18 Jan 2023 Gravitational-wave equation in effective one-body background for spinless binary Ya Guo1, Hiroaki Nakajima2 and Wenbin Lin1, 2, ∗ 1 School of Physical Science and Technology, Southwest Jiaotong University, Chengdu, 610031, China 2 School of Mathematics and Physics, University of South China, Hengyang, 421001, China ∗ Email: lwb@usc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='cn Abstract We construct the gravitational-wave equation in the background of the effec- tive one-body system for the spinless binary, which is in general available with the spherically symmetric background as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' The gauge conditions are given in terms of the metric perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' 1 Introduction The direct observation of gravitational waves by LIGO and Virgo [1] has opened the new era of cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' The binary system such as two black holes is a good object to observe the gravitational waves, where the analytical calculation is also possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' One of the calculation method of the gravitational waves radiated from the binary system is the theory of post-Newtonian approximations [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' On the other hand, one can also use the black hole perturbation theory [3], which is useful for the case of the extreme mass-ratio inspiral (EMRI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' The advantage of this method is that there is no divergence and one can calculate the observable quantity at very high post-Newtonian order [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' The reason why the black-hole perturbation theory is available for EMRI is because the field of one object is regarded as the background and the other body is regarded as the test particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' If one can map the two-body dynamics into the dynamics of the test particle in some appropriate background exactly, then the calculation beyond the EMRI approximation would be possible in terms of the black-hole perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' This approach is called the effective one-body (EOB) dynamics [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' In Newtonian limit, it is well-known that the effect of the two-body dynamics can exactly be included by just replacing the light (heavy) mass in the EMRI approximation with the reduced (total) mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' However when the correction of general relativity is included, the correspondence between the two-body dynamics and the dynamics of the test particle in some background becomes complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' It turns out that the Hamiltonians of the two dynamics are related by a very nontrivial way [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' The gravitational-wave equation in the EOB background has been studied in [7] using the Newman-Penrose formalism [8], which is a natural extension of the method to obtain the Teukolsky equation [9] in the Schwarzschild spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' It is obtained for some special cases of the background and is restricted to the even-parity mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' Later the same group obtained the equation both for the even- and odd-parity modes using the different choices of the gauge conditions [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' The wave equation from the metric perturbation in the generally spherically symmetric background has also been studied in [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' Inspired by the work of Jing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' [7], in this paper, we show that the gravitational- wave equation for both the even- and odd-parity modes can be obtained using the gauge conditions taken in their work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' Moreover our formalism can be applicable for more general spherically symmetric backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' The reminder of this work is organized as follows: in section 2, we briefly review the EOB dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' In section 3, we consider the wave equation for the perturbed Weyl scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' In section 4, we discuss the gauge condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' In section 5, we derive the explicit gravitational-wave equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' Section 6 is devoted to summary and discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' 2 EOB dynamics The EOB system for the spinless binary is first introduced in [5] in the post-Newtonian formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' The effective background metric geff µν is taken as the spherically symmetric 1 form: ds2 eff = geff µνdxµdxν = A(r)dt2 − B(r)dr2 − C(r)r2(dθ2 + sin2 θdϕ2), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='1) where the function C(r) can be freely chosen by the coordinate transformation for the radial coordinate r, and we here choose the Schwarzschild coordinate corresponding to C(r) ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' The explicit forms of A(r) and B(r) can be determined by the comparison with the two-body dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' For the Newman-Penrose formalism [8], we will take the corresponding null tetrad basis lA µ dxµ = dt − D(r) A(r)dr, nA µdxµ = A(r) 2 dt + D(r) 2 dr, mA µdxµ = − r √ 2(dθ + i sin θdϕ), ¯mA µdxµ = − r √ 2(dθ − i sin θdϕ), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='2) where D(r) = � A(r)B(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' The suffix A denotes the background quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' The null tetrads satisfy the orthonormal condition as lA µ nµ A = 1, mA µ ¯mµ A = −1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='3) and the other inner products vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' From the tetrad basis, one can compute the spin coefficients, the components of the Ricci tensor and the Weyl scalars as κA = νA = σA = λA = πA = τ A = ǫA = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='4) ρA = − 1 rD, µA = − A 2rD, γA = A′ 4D, αA = −βA = − cot θ 2 √ 2r, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='5) ΦA 01 = ΦA 10 = ΦA 02 = ΦA 20 = ΦA 12 = ΦA 21 = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='6) ΦA 00 = − D′ rD3, ΦA 22 = −A2D′ 4rD3, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='7) ΦA 11 = − 1 8r2D3 � 2D3 − 2AD − r2(A′D′ − A′′D) � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='8) ΛA = 1 24r2D3 � −2D3 − r2A′D′ + 2A(D − 2rD′) + rD(4A′ + rA′′) � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='9) ΨA 0 = ΨA 1 = ΨA 3 = ΨA 4 = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='10) ΨA 2 = 1 12r2D3 � 2(AD + rAD′ − D3) − rA′(2D + rD′) + r2A′′D � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='11) where the prime denotes the ordinary derivative with respect to r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' One can find that the background belongs the petrov type D from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='10), but is not in the vacuum since there are nonvanishing components of the Ricci tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' This type D property is important to derive the gravitational-wave equation in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' Note that for the special case D(r)=1, which was taken in [7, 10], we have ΦA 00 = ΦA 22 = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='12) 2 and the nonvanishing quantities in the above becomes simplified as ρA = −1 r, µA = − A 2r, γA = A′ 4 , αA = −βA = − cot θ 2 √ 2r, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='13) ΦA 11 = − 1 8r2(2 − 2A + r2A′′), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='14) ΛA = 1 24r2(−2 + 2A + 4rA′ + r2A′′), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='15) ΨA 2 = 1 12r2(−2 + 2A − 2rA′ + r2A′′), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='16) Note that when we choose A = 1 − 2M/r and D = 1, the background (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='1) is reduced to the Schwarzschild spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' The EOB Hamiltonian Heff can be determined from the geodesic motion under the background (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' The action S satisfies the Hamilton-Jacobi equation gµν eff PµPν − m2 0 = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='17) where Pµ = ∂S/∂xµ is the momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' m0 is the mass of the test particle and is matched as the reduced mass in the two-body dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='17), Heff is computed as Heff = m0 � A � 1 + AP 2r m2 0D2 + P 2ϕ m2 0r2 � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='18) where the motion plane is fixed on θ = π/2 due to the spherical symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' The effective Hamiltonian Heff and the real two-body Hamiltonian Hreal are compared by matching the masses and the action variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' It turns out that the two Hamiltonians are related by a rather nontrivial way as [5, 6] Hreal = M0 � 1 + 2m0 M0 �Heff m0 − 1 � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='19) where M0 is the total mass in the two-body dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' The functions A(r) and D(r) are obtained as [5] A(r) = 1 − 2M0 r + 2m0 M0 �M0 r �3 + · · · , D(r) = 1 − 3m0 M0 �M0 r �2 + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='20) However hereafter we leave A(r) and D(r) arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' Because of that, the metric (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='1) takes the most general form of the spherically symmetric background, which can also be applied to other kinds of the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' Moreover we will see later that when D(r)=1, the gravitational-wave equation and the gauge condition becomes simplified drastically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' 3 3 Wave equation for perturbed Weyl scalars It has been shown that the background (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='1) is classified as the nonvacuum Petrov type D background, which is useful to derive the gravitational-wave equation for the perturbed Weyl scalars using the Newman-Penrose formalism, as in the Teukolsky equation [9] in the Schwarzschild and the Kerr background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' We begin with the following equations in Newman-Penrose formalism: (δ + 4β − τ)Ψ4 − (∆ + 4µ + 2γ)Ψ3 + 3νΨ2 = (¯δ − ¯τ + 2¯β + 2α)Φ22 − (∆ + 2γ + 2¯µ)Φ21 − 2λΦ12 + 2νΦ11 + ¯νΦ20, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='1) (D + 4ǫ − ρ)Ψ4 − (¯δ + 4π + 2α)Ψ3 + 3λΨ2 = (¯δ − 2¯τ + 2α)Φ21 − (∆ + 2γ − 2¯γ + ¯µ)Φ20 + ¯σΦ22 − 2λΦ11 + 2νΦ10, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='2) (∆ + µ + ¯µ + 3γ − ¯γ)λ − (¯δ + π − ¯τ + ¯β + 3α)ν + Ψ4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='3) We split all the quantities in the above into the background part (A) and the perturbation part (B), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' Ψ4 = ΨA 4 + ΨB 4 , etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' Now we have to take into account the case where ΦA 22 is nonvanishing, then the background part of the equation becomes nontrivial, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' (¯δ − ¯τ + 2¯β + 2α)AΦA 22 = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='4) has to be satisfied1, and one can confirm that it is indeed the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' The part of the first-order perturbation in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='1)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='3) becomes2 (δ + 4β − τ)AΨB 4 − (∆ + 4µ + 2γ)AΨB 3 + 3νBΨA 2 = (¯δ − ¯τ + 2¯β + 2α)BΦA 22 + (¯δ − ¯τ + 2¯β + 2α)AΦB 22 − (∆ + 2γ + 2¯µ)AΦB 21 + 2νBΦA 11, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='5) (D + 4ǫ − ρ)AΨB 4 − (¯δ + 4π + 2α)AΨB 3 + 3λBΨA 2 = (¯δ − 2¯τ + 2α)AΦB 21 − (∆ + 2γ − 2¯γ + ¯µ)AΦB 20 + ¯σBΦA 22 − 2λBΦA 11, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='6) (∆ + µ + ¯µ + 3γ − ¯γ)AλB − (¯δ + π − ¯τ + ¯β + 3α)AνB + ΨB 4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='7) Again, when ΦA 22 is nonvanishing, the first term in the right hand side in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='5) appears, and more perturbed quantities ¯τ B, ¯βB and αB contribute to the equation, compared with the vacuum case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' In order to reduce the number of those quantities, we require that this term should vanish by the gauge condition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' Ξ22 ≡ (¯δ − ¯τ + 2¯β + 2α)BΦA 22 = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='8) 1Here the superscript A (B) on the parentheses denotes that all the quantities and the operators inside the parentheses are in the background (perturbation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' 2Here we will not use πA = τA = ǫA = 0 from the beginning and keep them for a while, which would be useful to extend the result into the spinning case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' 4 Now we will obtain the wave equation for ΨB 4 in a similar way with the method used to derive the Teukolsky equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' First we show the following commutation relation of the differential operators: [∆ + (p + 1)γ − ¯γ − qµ + ¯µ]A (¯δ + pα − qπ)A − �¯δ + (p + 1)α + ¯β − ¯τ − qπ �A (∆ + pγ − qµ)A = νADA − λAδA − p [(β + τ)λ − (ρ + ǫ)ν + Ψ3]A + q [−Dν + δλ + (¯π + τ + 3β − ¯α)λ − (3ǫ + ¯ǫ + ρ − ¯ρ)ν + 2Ψ3]A = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='9) where p and q are arbitrary constants and we have used νA = λA = ΨA 3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' We operate (∆ + 3γ − ¯γ + 4µ + ¯µ)A to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='6) and (¯δ + 3α + ¯β − ¯τ + 4π)A to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='5), and then subtract one equation from the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' The terms with ΨB 3 cancel by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='9) with p = 2, q = −4 and the remaining becomes � (∆ + 3γ − ¯γ + 4µ + ¯µ)(D + 4ǫ − ρ) − (¯δ + 3α + ¯β − ¯τ + 4π)(δ + 4β − τ) �A ΨB 4 + 3ΨA 2 � (∆ + 3γ − ¯γ + 4µ + ¯µ)AλB − (¯δ + 3α + ¯β − ¯τ + 4π)AνB� + 3λB∆AΨA 2 − 3νB¯δAΨA 2 = T4 + (∆ + 3γ − ¯γ + 4µ + ¯µ)A(¯σBΦA 22) − 2λB∆AΦA 11 − 2νB¯δAΦA 11 − 2ΦA 11 � (∆ + 3γ − ¯γ + 4µ + ¯µ)AλB + (¯δ + 3α + ¯β − ¯τ + 4π)AνB� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='10) where T4 is defined by T4 = (∆ + 3γ − ¯γ + 4µ + ¯µ)A � (¯δ − 2¯τ + 2α)AΦB 21 − (∆ + 2γ − 2¯γ + ¯µ)AΦB 20 � − (¯δ + 3α + ¯β − ¯τ + 4π)A � (¯δ − ¯τ + 2¯β + 2α)AΦB 22 − (∆ + 2γ + 2¯µ)AΦB 21 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='11) For the third line in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='10), we have ∆AΨA 2 = −3µAΨA 2 − 2µΦA 11 − (D − ¯ρ + 2ǫ + 2¯ǫ)AΦA 22 − 2∆AΛA, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='12) ¯δAΨA 2 = −3πAΨA 2 + 2πAΦA 11 − 2¯δAΛA, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='13) and then substituting the above into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='10), we get � (∆+3γ−¯γ+4µ + ¯µ)(D+4ǫ−ρ)−(¯δ + 3α + ¯β − ¯τ + 4π)(δ + 4β − τ) �A ΨB 4 + 3ΨA 2 � (∆ + 3γ − ¯γ + µ + ¯µ)AλB − (¯δ + 3α + ¯β − ¯τ + π)AνB� = T4 + (∆ + 3γ − ¯γ + 4µ + ¯µ)A(¯σBΦA 22) − 2λB∆AΦA 11 − 2νB¯δAΦA 11 − 2ΦA 11 � (∆ + 3γ − ¯γ + µ + ¯µ)AλB + (¯δ + 3α + ¯β − ¯τ + π)AνB� + 3λB(D − ¯ρ + 2ǫ + 2¯ǫ)AΦA 22 + 6λB∆AΛA − 6νB¯δAΛA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='14) 5 The second line in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='14) becomes −3ΨA 2 ΨB 4 using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='7), and there is also similar terms in the fourth line but the relative sign is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' We now require more gauge conditions as λB = σB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='15) Then the fourth line in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='14) becomes −2ΦA 11ΨB 4 and the terms with ΦA 22 disappear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' Thus under the gauge conditions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='8) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='15), the decoupled wave equation for ΨB 4 is obtained as � (∆ + 3γ − ¯γ + 4µ + ¯µ)(D + 4ǫ − ρ) − (¯δ + 3α + ¯β − ¯τ + 4π)(δ + 4β − τ) − 3Ψ2 + 2Φ11 �AΨB 4 = T4, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='16) One can also consider the wave equation for ΨB 0 , which can be obtained in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' The resultant equation becomes � (D − 3ǫ + ¯ǫ − 4ρ − ¯ρ)(∆ − 4γ + µ) − (δ − 3β − ¯α + ¯π − 4τ)(¯δ − 4α + π) − 3Ψ2 + 2Φ11 �AΨB 0 = T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='17) The gauge conditions are (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='15) and Ξ00 ≡ (δ + ¯π − 2¯α − 2β)BΦA 00 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='18) Note that for the case D=1 we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='12), then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='8) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='18) are obviously satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' The other conditions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='15) are also relaxed as just λB = 0 for (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='16) and just σB = 0 for (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' 4 Gauge conditions Here we will study the consistency of the gauge conditions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='8), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='18) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='15) (or just (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='15) for D = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' In Newman-Penrose formalism, there are ten gauge degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' Six of them are the tetrad rotation (the local Lorentz transformation), which can be decomposed as the following three kinds [13]: lµ → lµ, mµ → mµ + alµ, ¯mµ → ¯mµ + ¯alµ, nµ → nµ + ¯amµ + a ¯mµ + a¯alµ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='1) nµ → nµ, mµ → mµ + bnµ, ¯mµ → ¯mµ + ¯bnµ, lµ → lµ + ¯bmµ + b ¯mµ + b¯bnµ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='2) lµ → e−clµ, nµ → ecnµ, mµ → eiϑmµ, ¯mµ → e−iϑ ¯mµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='3) Here a and b are complex functions, and c and ϑ are real functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' The transformations of the quantities in Newman-Penrose equations (the spin coefficients, the Weyl scalars, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=') under the above are shown in [14], for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' The other four are the general coordinate transformation x′µ = xµ+ξµ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' Under the transformation, the null tetrads behave as the 6 one-forms or the vector fields, and the quantities in Newman-Penrose equations behave as the scalar fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' Because of this, the form of the transformation for the latter becomes X → X − ξµ∂µX, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='4) where X represents either one of the spin coefficients, Ψ’s, Φ’s or Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' Since we want to keep the background (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='4)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='11) (or (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='13)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='16) for D=1) intact, the functions a, b, c, ϑ and ξµ have to be at the order of the perturbed quantities, and in particular the second and the higher orders of them are neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' The gauge conditions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='8), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='15) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='18) are transformed under the the gauge transformations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='2), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='3) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='4) as3 Ξ00 → Ξ00 + (δ + ¯π − 2¯α − 2β)A(ξµ∂µΦA 00) + DA(aΦA 00) − 2aρAΦA 00 + b(∆A + ¯µA − 2µA − 2¯γA − 2γA)ΦA 00 + 2ΦA 00δc, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='5) Ξ22 → Ξ22 + (¯δ − ¯τ + 2α + 2¯β)A(ξµ∂µΦA 22) + ¯a(DA − ¯ρA + 2ρA)ΦA 22 + ∆(¯bΦA 22) + 2¯b(µA + γA + ¯γA)ΦA 22 − 2ΦA 22¯δc, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='6) λB → λB + 2αA¯a + ¯δA¯a, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='7) σB → σB + 2βAb − δAb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='8) One can find that there are nine real degrees (four ξ’s, a, b and c) of freedom for eight real conditions (Ξ00, Ξ22, λB and σB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' In general, it is enough to satisfy all the conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' In order to see the the condition explicitly, here we will use the form of the metric perturbation, following the A-K parametrization [15, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' First, the metric perturbation is given as gµν = gA µν + hBE µν + hBO µν , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='9) where gA µν = geff µν is the background metric (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' hBE µν and hBO µν are respectively the even- and the odd-parity part of the perturbation, which are parametrized as hBE µν = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed AS −DS −rB∂θS −rB∂ϕS KS rH∂θS rH∂ϕS r2E+ F r2FPS1 r2 sin2 θ E− F \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='10) hBO µν = 1 D \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed 0 0 rC csc θ∂ϕS −rC sin θ∂θS 0 −rJ csc θ∂ϕS rJ sin θ∂θS −r2G csc θPS1 −1 2r2GPS2 r2GPS3 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='11) 3Our gauge conditions are invariant under the transformation generated by ϑ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' 7 Here the lower-left blanks have to be filled as hBE µν and hBO µν to be symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' A, B, C, D, E, F, G, H, J and K are the functions4 of t and r, and S is the function of θ and ϕ, which should in turn be identified as the spherical harmonics Ylm(θ, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' The quantities E± F , PS1, PS2 and PS3 are defined by E± F = � E ± F � ∂2 θ + 1 2l(l + 1) �� S, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='12) PS1 = (∂θ∂ϕ − cot θ∂ϕ)S, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='13) PS2 = (csc θ∂2 ϕ + cos θ∂θ − sin θ∂2 θ)S, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='14) PS3 = (sin θ∂θ∂ϕ − cos θ∂ϕ)S, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='15) where l is one of the label in Ylm(θ, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' Other perturbed quantities are also decomposed into the even- and the odd-parity modes, such as lB µ = lBE µ + lBO µ , etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' Now we have to find the null tetrads corresponding to the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' Even though the metric (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='1) is fixed, the tetrads are not unique due to the tetrad rotation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='2) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' In other words, once a set of null tetrads is found, the others are obtained from those tetrads by the tetrad rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' We will fix the reference tetrads as lE µ dxµ ≡ (lA µ + lBE µ )dxµ = dt − D A � 1 − 1 2A � A − 2A D D + A2 D2K � S � dr − r AD(DB − AH)dS, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='16) nE µ dxµ ≡ (nA µ + nBE µ )dxµ = A 2 � 1+ AS A � dt+ D 2 � 1+ 1 2A � A−2A D D− A2 D2K � S � dr− r 2D(DB+AH)dS, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='17) mE µ dxµ ≡ (mA µ + mBE µ )dxµ = − r √ 2 �� 1 − E+ F 2 � dθ + i � 1 − E− F 2 + i csc θFPS1 � sin θdϕ � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='18) ¯mE µ dxµ ≡ ( ¯mA µ + ¯mBE µ )dxµ = − r √ 2 �� 1 − E+ F 2 � dθ − i � 1 − E− F 2 − i csc θFPS1 � sin θdϕ � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='19) for the even-parity modes and lO µ dxµ ≡ (lA µ + lBO µ )dxµ = dt − D Adr + r AD � C − A DJ � (csc θ∂ϕS dθ − sin θ∂θS dϕ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='20) nO µ dxµ ≡ (nA µ + nBO µ )dxµ 4We hope the readers may not confuse the function D here and the differential operator D in Newman- Penrose formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' 8 = A 2 dt + D 2 dr + r 2D � C + A DJ � (csc θ∂ϕS dθ − sin θ∂θS dϕ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='21) mO µ dxµ ≡ (mA µ + mBO µ )dxµ = − r √ 2 �� 1+ csc θ 2D GPS1 � dθ+i sin θ � 1 − 1 2D csc θG(PS1 + iPS2) � dϕ � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='22) ¯mO µ dxµ ≡ ( ¯mA µ + ¯mBO µ )dxµ = − r √ 2 �� 1+ csc θ 2D GPS1 � dθ−i sin θ � 1 − 1 2D csc θG(PS1 − iPS2) � dϕ � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='23) for the odd-parity modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' Here dS = (∂θS)dθ + (∂ϕS)dϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' Note that the above choice of the reference tetrads is different from the one taken in [7] for both the even- and the odd-parity modes even under the Regge-Wheeler gauge B = F = H = G = 0 [16, 17, 18] or the EZ gauge B = E = F = G = 0 [15, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' This is again due to the different choice of the reference and they are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' The advantage of our choice is that the expressions of λB and σB become relatively simple as λBE = −A 4 �� ∂2 θ + 1 2l(l + 1) � S − i csc θPS1 � � 1 A∂tF − 1 D∂rF � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='24) σBE = 1 2 �� ∂2 θ + 1 2l(l + 1) � S + i csc θPS1 � � 1 A∂tF + 1 D∂rF � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='25) for the even-parity modes and λBO = A 8 csc θ (2PS1 − iPS2) � 1 A∂t �G D � − 1 D∂r �G D �� , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='26) σBO = −1 4 csc θ (2PS1 + iPS2) � 1 A∂t �G D � + 1 D∂r �G D �� , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='27) for the odd-parity modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' In particular, one can easily find that they vanish under the Regge-Wheeler or the EZ gauge5, which also means that the above reference tetrads under those gauges can be used as the standard form of the perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' Next we will consider the gauge invariants, are obtained in [12, 15] as α = J − r 2D∂r �G D � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='28) β = −C − r 2∂tG, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='29) χ = H − D2 2AE − l(l + 1)D2 4A F − r 2∂rF, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='30) ψ = 1 2K − r 4 �D2 A �′ E − D2 2AE − rD2 2A ∂rE 5For the even-parity modes, the choice taken in [7] can also lead to λBE = σBE = 0 under F = 0, but not for the odd-parity mode under G = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' 9 − r 8l(l + 1) �D2 A �′ F − D2 4Al(l + 1)F − rD2 4A l(l + 1)∂rF (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='31) δ = D + rD2 2A ∂tE + �rA′ A − 1 � B − r∂rB − r2 2 ∂t∂rF − � r − r2A′ 2A − rD2 4A l(l + 1) � ∂tF, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='32) ǫ = −1 2A − rA′ 4 E − r∂tB − rA′ 8 l(l + 1)F − r2 2 ∂2 t F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='33) The perturbed Weyl scalars ΨB 4 and ΨB 0 can be written as the gauge-invariant combina- tions of the perturbation as ΨBE 4 = csc θ 8r2D3(PS2 + 2iPS1) � −A2Dψ − AD2δ + D3ǫ −rAD2∂tχ + rA2D∂rχ + A2(D − rD′)χ � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='34) ΨBO 4 = −i csc θ 8rD (PS2 + 2iPS1) � −A D∂tα + A2 D2∂rα + A2(D − 2rD′) rD3 α + ∂tβ − A D∂rβ + �AD′ D2 − A − rA′ rD � β � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='35) ΨBE 0 = csc θ 2r2A2D3(PS2 − 2iPS1) � −A2Dψ + AD2δ + D3ǫ +rAD2∂tχ + rA2D∂rχ + A2(D − rD′)χ � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='36) ΨBO 0 = i csc θ 2rA2D(PS2 − 2iPS1) � A D∂tα + A2 D2∂rα + A2(D − 2rD′) rD3 α + ∂tβ + A D∂rβ + � −AD′ D2 + A − rA′ rD � β � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='37) The gauge conditions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='8), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='18) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='15) can be expressed as the gauge invariants and the set of the variables (B, F, H, G) or (B, E, F, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' The explicit form for the latter is shown in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' 5 Explicit PDE and ODE for radial coordinate In section 3, we have obtained the wave equation for the perturbed Weyl scalars ΨB 4 and ΨB 0 as (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='16) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='17), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' By substituting the background (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='4)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='11) (or (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='13)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='16) for D = 1), the partial differential equations (the master equations) are obtained, Then by the separation of variables, the ordinary differential equation for the radial coordinate r and that for the angular coordinates θ and ϕ are also obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' 10 We define ψ(−2) as ψ(−2) ≡ (ρA)−4ΨB 4 = (rD)4ΨB 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='1) From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='16), ψ(−2) satisfies the following partial differential equation: r2 A ∂2ψ(−2) ∂t2 − (r2A)2D7 ∂ ∂r � (r2A)−1D−9∂ψ(−2) ∂r � + �2r2A′ AD − 4r D � ∂ψ(−2) ∂t + � −24r2A(D′)2 D4 + 4r2AD′′ D3 − 3r2A′D′ D3 − 12rAD′ D3 − r2A′′ D2 − 2rA′ D2 � ψ(−2) − 1 sin θ ∂ ∂θ � sin θ∂ψ(−2) ∂θ � − 1 sin2 θ ∂2ψ(−2) ∂ϕ2 + 4i cot θ sin θ ∂ψ(−2) ∂ϕ +(4 cot2 θ+2)ψ(−2) = 2r6D4T4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='2) For homogeneous case (T4 = 0), by assuming the product form ψ(−2) = e−iωteimϕR(r)S(θ), one can obtain the separated equations as (r2A)2D7 d dr � (r2A)−1D−9dR(r) dr � + �r2ω2 A + iω �2r2A′ AD − 4r D � + 24r2A(D′)2 D4 −4r2AD′′ D3 + 3r2A′D′ D3 + 12rAD′ D3 + r2A′′ D2 + 2rA′ D2 − λ � R(r) = 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='3) 1 sin θ d dθ � sin θdS(θ) dθ � − � m2 sin2 θ − 4m cot θ sin θ + 4 cot2 θ + 2 − λ � S(θ) = 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='4) where ω gives the frequency of the gravitational wave and λ is the separation constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' From (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='4) one can find that S(θ)eimϕ coincides with the s = −2 spin-weighted spherical harmonics, denoted as −2Ylm(θ, ϕ), and then the separation constant λ has to be λ = (l − 1)(l + 2), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='5) where l and m take the integer value with l ≥ 2 and −l ≤ m ≤ l, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' For inhomogeneous case (T4 ̸= 0), one can expand ψ(−2) and T4 as ψ(−2) = � dω � l,m R(−2)lω(r)−2Ylm(θ, ϕ)e−iωt, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='6) 2r6D4T4 = − � dω � l,m G(−2)lω(r)−2Ylm(θ, ϕ)e−iωt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='7) Then the ordinary differential equation for the radial coordinate r is (r2A)2D7 d dr � (r2A)−1D−9dR(−2)lω(r) dr � + �r2ω2 A + iω �2r2A′ AD − 4r D � +24r2A(D′)2 D4 − 4r2AD′′ D3 + 3r2A′D′ D3 + 12rAD′ D3 + r2A′′ D2 + 2rA′ D2 − (l − 1)(l + 2) � R(−2)lω(r) = G(−2)lω(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='8) 11 In the case of D=1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='8) is reduced to (r2A)2 d dr � (r2A)−1dR(−2)lω(r) dr � + �r2ω2 A + iω �2r2A′ A − 4r � + r2A′′ + 2rA′ − (l − 1)(l + 2) � R(−2)lω(r) = G(−2)lω(r), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='9) Note that for the case of Schwarzschild background, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' A = 1 − 2M/r, the differential equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='9) is reduced to the Teukolsky equation [9] with the spin weight s = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' The equation for ψ(2) ≡ ΨB 0 can be obtained in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' The master equation becomes r2 A ∂2ψ(2) ∂t2 − (r2A)−2D−1 ∂ ∂r � (r2A)3D−1∂ψ(2) ∂r � − �2r2A′ AD − 4r D � ∂ψ(2) ∂t + �3r2A′D′ D3 − 3r2A′′ D2 − 10rA′ D2 − 4A D2 � ψ(2) − 1 sin θ ∂ ∂θ � sin θ∂ψ(2) ∂θ � − 1 sin2 θ ∂2ψ(2) ∂ϕ2 −4i cot θ sin θ ∂ψ(2) ∂ϕ +(4 cot2 θ + 2)ψ(2) = 2r2T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='10) After the separation of the variables, for the angular part, we have the s = 2 spin-weighted spherical harmonics 2Ylm(θ, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' For the radial part, we have the following equation: (r2A)−2D−1 d dr � (r2A)3D−1dR(2)lω(r) dr � + �r2ω2 A − iω �2r2A′ AD − 4r D � − 3r2A′D′ D3 + 3r2A′′ D2 + 10rA′ D2 + 4A D2 − (l − 2)(l + 3) � R(2)lω(r) = G(2)lω(r), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='11) which is reduced to the Teukolsky equation with s = ±2 for A = 1 − 2M/r and D ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' 6 Summary and discussion In this paper, we have studied the wave equations for the perturbed Weyl scalars ΨB 4 and ΨB 0 in the background of the EOB dynamics for the spinless binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' We also obtained the Teukolsky-like equations for the function of the radial coordinate r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' In the previous work [7], the special case D=1 is studied and the (decoupled) wave equation is obtained only for the even-parity mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' On the other hand, here we have studied the odd-parity mode also, and have obtained the same form of the wave equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' Moreover, we have 12 considered a more general case including D ̸=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' The different form of the wave equation is proposed in [10] by the use of the different gauge, although it is again restricted to the case of D=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' It would be interesting to find the relation between our equations and their ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' One can easily find that the background is assumed to be just spherically symmetric and general enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' Even in the case D = 1, it contains many kinds of the black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' A similar wave equations in the spherically symmetric background are obtained using the metric perturbation in [12], which are resemble to the Regge-Wheeler and the Zerilli equations [16, 17], and are different from the ones obtained here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' This is because we have used the different gauge and the different master variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' However in the case of Schwarzschild background, there is a transformation originated from the connection of the different gauges, which is called the Chandrasekhar transformation [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' Moreover the relation between the R(−2)lω(r) in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='8) and R(2)lω(r) in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='11) can also be regarded as the special case of the Chandrasekhar transformation [9, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' It would be interesting to find a similar transformation in the present case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' Another possible generalization would be to include the effect of the spin, where the background becomes axially symmetric and contains the Kerr black hole as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' In this case the method of the metric perturbation is difficult to perform, because it relies on the spherical symmetry and the expansion by the spherical harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' However the Newman-Penrose formalism would still be powerful enough, and it can be expected that the gravitational-wave equation could be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' Studying the equations for the electromagnetic and the scalar waves is also interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' Acknowledgements This work was supported in part by the National Natural Science Foundation of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' 11973025).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' A Gauge conditions with gauge invariants The gauge conditions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='8), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='18) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='15) can be written in terms of the gauge invari- ants and the variables (B, E, F, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' Each condition consists of the even-parity part (E), odd-parity part (O) and the transformation part (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' Ξ22 = ΞE 22 + ΞO 22 + ΞT 22, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='1) ΞE 22 = A2(∂θS − i csc θ∂ϕS) 4 √ 2r � −AD′′ D5 � χ + r 2∂rF + l(l + 1) 4 D2 A F + D2 2AE � − D′ D3 � − 1 D∂tχ+ A 2D2∂rχ+ � − A 2rD2 + 3A′ 2D2 −7AD′ 2D3 � χ+ 3 2rAǫ− A 2rD2ψ + 1 rDδ + 2 A∂tB − D A∂tE + �A′ A − 3D′ 2D − 1 2r � E + 3r 4A∂2 t F + rA 4D2∂2 rF 13 + � 1 D − l(l + 1) 2 D A − rA′ 2AD � ∂tF + �3rA′ 4D2 − 7rAD′ 4D3 � ∂rF − �1 r + 3D′ D − 2A′ A � l(l + 1) 4 F �� , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='2) ΞO 22 = −iA2(∂θS − i csc θ∂ϕS) 4 √ 2r � −AD′′ D6 � α + rD 2 ∂r ˜G � + D′ D3 � 1 D2∂tα − A 2D3∂rα + � A 2rD3 − 3A′ 2D3 + 4AD′ D4 � α+ 1 2AD∂tβ− 1 D2∂rβ+ � A′ AD2 − 1 rD2 + D′ D3 � β + r 4A∂2 t ˜G + � rA′ 2AD − 1 D � ∂t ˜G − rA 4D2∂2 r ˜G + � −3rA′ 4D2 + 7rAD′ 4D3 � ∂r ˜G �� , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='3) ΞT 22 = A 4rD4 �AD′ r − A′D′ + 3A(D′)2 D − AD′′ � � ¯a − A 2 ¯b � −A2D′ 4rD3 � A′ AD¯a− 1 rD ¯a− A rD ¯b+ 1 2∂t¯b − A 2D∂r¯b − √ 2 r ∂θc + √ 2i csc θ r ∂ϕc � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='4) Ξ00 = ΞE 00 + ΞO 00 + ΞT 00, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='5) ΞE 00 = ∂θS + i csc θ∂ϕS √ 2r � −AD′′ D5 � χ + r 2∂rF + l(l + 1) 4 D2 A F + D2 2AE � − D′ D3 � 1 D∂tχ + A 2D2∂rχ+ � 3A′ 2D2 − A 2rD2 − 7AD′ 2D3 � χ− 5 2rAǫ− A 2rD2ψ − 1 rDδ − 2 A∂tB + D A∂tE − �3D′ 2D + 1 2r � E − 5r 4A∂2 t F + rA 4D2∂2 rF + � − 1 D + l(l + 1) 2 D A + rA′ 2AD � ∂tF + �3rA′ 4D2 − 7rAD′ 4D3 � ∂rF − �1 r + 3D′ D � l(l + 1) 4 F �� , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='6) ΞO 00 = i(∂θS + i csc θ∂ϕS) √ 2r � −AD′′ D6 � α + rD 2 ∂r ˜G � − D′ D3 � 1 D2∂tα + A 2D3∂rα + � 3A′ 2D3 − A 2rD3 −4AD′ D4 � α− 1 2AD∂tβ− 1 D2∂rβ+ � A′ AD2 − 1 rD2 + D′ D3 � β − r 4A∂2 t ˜G + � rA′ 2AD − 1 D � ∂t ˜G + rA 4D2∂2 r ˜G + �3rA′ 4D2 − 7rAD′ 4D3 � ∂r ˜G �� , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='7) ΞT 00 = 1 rD4 �D′ r + 3(D′)2 D − D′′ � � a − A 2 b � − D′ rD3 � 2 rDa+ 1 A∂ta + 1 D∂ra + A 2rDb − A′ D b + √ 2 r ∂θc + √ 2i csc θ r ∂ϕc � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='8) λB = λBE + λBO + λBT , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='9) 14 λBE = −A 4 �� ∂2 θ + 1 2l(l + 1) � S − i csc θPS1 � � 1 A∂tF − 1 D∂rF � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='10) λBO = A 8 csc θ (2PS1 − iPS2) � 1 A∂t ˜G − 1 D∂r ˜G � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='11) λBT = 1 √ 2r (∂θ¯a − i csc θ∂ϕ¯a − ¯a cot θ), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='12) σB = σBE + σBO + σBT , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='13) σBE = 1 2 �� ∂2 θ + 1 2l(l + 1) � S + i csc θPS1 � � 1 A∂tF + 1 D∂rF � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='14) σBO = −1 4 csc θ (2PS1 + iPS2) � 1 A∂t ˜G + 1 D∂r ˜G � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='15) σBT = − 1 √ 2r (∂θb + i csc θ∂ϕb − b cot θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='16) Here ˜G = G/D and ξ’s are omitted since these are converted to fix the variables (B, E, F, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' Note that in the case of D =1, the conditions Ξ22 = Ξ00 = 0 becomes trivial from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='12) and λB = σB = 0 can be achieved by setting F = G = a = b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' References [1] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' [LIGO Scientific and Virgo], Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' He, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' Wang and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' Wang, Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' China Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' 65, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='6, 260411 (2022) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='1007/s11433-022-1885-6 [arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content='09838 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' [8] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' Newman and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE_T4oBgHgl3EQfyhxs/content/2301.08318v1.pdf'} +page_content=' Penrose, J.' metadata={'source': 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a/WtFQT4oBgHgl3EQfcDb0/content/tmp_files/2301.13326v1.pdf.txt b/WtFQT4oBgHgl3EQfcDb0/content/tmp_files/2301.13326v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..58f7cdb8b974d90f58605e8279abeec9853787e1 --- /dev/null +++ b/WtFQT4oBgHgl3EQfcDb0/content/tmp_files/2301.13326v1.pdf.txt @@ -0,0 +1,2908 @@ +A Framework for Adapting Offline Algorithms to Solve +Combinatorial Multi-Armed Bandit Problems with Bandit +Feedback +Guanyu Nie +nieg@iastate.edu +Yididiya Y Nadew +yididiya@iastate.edu +Yanhui Zhu +yanhui@iastate.edu +Vaneet Aggarwal +vaneet@purdue.edu +Christopher John Quinn +cjquinn@iastate.edu +Editor: +Abstract +We investigate the problem of stochastic, combinatorial multi-armed bandits where the +learner only has access to bandit feedback and the reward function can be non-linear. We +provide a general framework for adapting discrete offline approximation algorithms into +sublinear α-regret methods that only require bandit feedback, achieving O +� +T +2 +3 log(T) +1 +3 +� +expected cumulative α-regret dependence on the horizon T. The framework only requires +the offline algorithms to be robust to small errors in function evaluation. The adaptation +procedure does not even require explicit knowledge of the offline approximation algorithm +— the offline algorithm can be used as black box subroutine. +To demonstrate the utility of the proposed framework, the proposed framework is +applied to multiple problems in submodular maximization, adapting approximation algo- +rithms for cardinality and for knapsack constraints. The new CMAB algorithms for knap- +sack constraints outperform a full-bandit method developed for the adversarial setting in +experiments with real-world data. +1. Introduction +Many real world sequential decision problems can be modeled using the framework of +stochastic multi-armed bandits (MAB), such as scheduling, assignment problems, ad-campaigns, +and product recommendations, among others. The decision maker sequentially selects ac- +tions and receives stochastic rewards from an unknown distribution. The goal of the decision +maker is to maximize the expected cumulative reward over a (possibly unknown) time hori- +zon. Actions result both in the immediate reward and, more importantly, information about +that action’s reward distribution. Such problems result in a trade-off between trying actions +the agent is uncertain of (exploring) or only taking the action that is empirically the best +seen so far (exploiting). +In the classic MAB setting, the number of possible actions is small relative to the time +horizon, meaning each action can be taken at least once, and there is no assumed relationship +between the reward distributions of different arms. The combinatorial multi-armed bandit +(CMAB) setting involves a large but structured action space. +For example, in product +recommendation problems, the decision maker may select a subset of products (base arms) +1 +arXiv:2301.13326v1 [cs.LG] 30 Jan 2023 + +from among a large set. There are several aspects that can affect the difficulty of these +problems. First, MAB methods are typically compared against a learner with access to +a value oracle of the reward function (an offline problem). For some problems, it is NP- +hard for the baseline learner with value oracle access to optimize. An example is if the +expected/averaged reward function is submodular and actions are subsets constrained by +cardinality. At best, for these problems, approximation algorithms may exist. Thus, unless +the time horizon is large (exponentially long in the number of base arms, for instance), +it would be more reasonable to compare the CMAB agent against the performance of the +approximation algorithm for the related offline problem. Likewise, one could apply state of +the art methods for (unstructured) MAB problems treating each subset as a separate arm, +and obtain ˜O(T +1 +2 ) dependence on the horizon T for the subsequent regret bound. However, +that dependence would only apply for exponentially large T. +Feedback plays an important role in how challenging the problem is. When the decision +maker only observes a (numerical) reward for the action taken, that is known as bandit +or full-bandit feedback. When the decision maker observes additional information, such as +contributions of each base arm in the action, that is semi-bandit feedback. Semi-bandit +feedback greatly facilitates learning. Suppose for instance that the reward function (on +average) was monotone increasing over the inclusion lattice and there was a cardinality +constraint of size k. The agent would know from the start that no set of size smaller than +k could be optimal (or could even be the near-optimal solution the baseline learning using +a value oracle would find). However, there would be +�n +k +� +sets of size k. For n = 100 and +k = 10, the agent would need a horizon T > 1012 to try each cardinality k set even just +once. If the reward function belongs to a certain class, such as the class of submodular +functions, then one approach would be to use a greedy procedure based on base arm values. +With semi-bandit feedback, the agent could on the one hand only take actions of cardinality +k (putatively optimal actions), gain the subsequent rewards, and yet also observe samples +of the base arms’ values to improve future actions. +Bandit feedback is much more challenging, as only the joint reward is observed. In +general, for non-linear reward functions, the individual values or marginal gains of base +arms can only be loosely bounded if actions only consist of maximal subsets. Thus, to +estimate values or marginal gains of base arms, the agent would need to deliberately spend +time sampling actions (such as smaller sets) that are known to be sub-optimal in order to +estimate their values to later better select actions of cardinality k. Standard MAB methods +like UCB or TS based methods by design do not take actions known to be sub-optimal. +Thus, while such strategies could be used when semi-bandit feedback is available, it is less +clear whether they can be effectively used when only bandit feedback is available. +There are important applications where semi-bandit feedback may not be available, such +as in influence maximization and recommender systems. Influence maximization models +the problem of identifying a low-cost subset (seed set) of nodes in a (known) social network +that can influence the maximum number of nodes in a network (Nguyen and Zheng, 2013; +Leskovec et al., 2007; Bian et al., 2020). Recent research has generalized the problem to +online settings where the knowledge of the network and diffusion model is not required +(Wang et al., 2020; Perrault et al., 2020a) but extra feedback is assumed. However, for +many networks the user interactions and user accounts are private; only aggregate feedback +2 + +(such as the count of individuals using a coupon code or going to a website) might be visible +to the decision maker. +In this work, we seek to address these challenges by proposing a general framework for +adapting offline approximation algorithms into algorithms for stochastic CMAB problems +when only bandit feedback is available. We identify that a single condition related to the +robustness of the approximation algorithm to erroneous function evaluations is sufficient to +guarantee that a simple explore-then-commit (ETC) procedure accessing the approximation +algorithm as a black box results in a sublinear α-regret CMAB algorithm despite having +only bandit feedback available. The approximation algorithm does not need to have any +special structure (such as an iterative greedy design). Importantly, no effort is needed on +behalf of the user in mapping steps in the offline method into steps of the CMAB method. +We demonstrate the utility of this framework by assessing the robustness of several +approximation algorithms in the submodular optimization literature (three approximation +algorithms designed for knapsack constraints and one designed for cardinality constraints) +which immediately result in sublinear α-regret CMAB algorithms that only rely on bandit- +feedback, the first such algorithms for CMAB problems with submodular rewards and knap- +sack constraints. We also show that despite the simplicity and universal design of the adap- +tation, the resulting CMAB algorithms work well on budgeted influence maximization and +song recommendation problems using real world data. +The main contributions of this paper can be summarized as: 1. We provide a general +framework for adapting discrete offline approximation algorithms into sublinear α-regret +methods for stochastic CMAB problems where only bandit feedback is available. +The +framework only requires the offline algorithms to be robust to small errors in function +evaluation, a property important in its own right for offline problems. The algorithms are +not required to have a special structure — instead they are used as black boxes. +Our +procedure has minimal storage and time-complexity overhead, and achieves a regret bound +with ˜O(T +2 +3 ) dependence on the horizon T. +2. We illustrate the utility of the proposed framework by assessing the robustness of several +approximation algorithms for (offline) constrained submodular optimization, a class of re- +ward functions lacking simplifying properties of linear or Lipschitz reward functions. Specifi- +cally, we prove the robustness of approximation algorithms given in Nemhauser et al. (1978); +Badanidiyuru and Vondr´ak (2014); Sviridenko (2004); Khuller et al. (1999); Yaroslavtsev +et al. (2020) with cardinality or knapsack constraints, and use the general framework to +give regret bounds for the stochastic CMAB. In particular, we note that this paper gives +the first regret bounds for stochastic submodular CMAB with knapsack constraints under +bandit feedback. +3. We evaluate the performance of proposed framework through the stochastic submod- +ular CMAB with knapsack constraints problem for two applications: Budgeted Influence +Maximization, and Song Recommendation. The evaluation results demonstrate that the +proposed approach significantly outperforms a full-bandit method for a related problem in +the adversarial setting. +3 + +2. Related Work +We now briefly discuss only the most closely related works. See the supplementary material +for more discussion. +Adversarial CMAB +The closest related works are on adversarial CMAB. In (Niazadeh +et al., 2021), the authors propose a framework for transforming greedy α-approximation +algorithms for offline problems to online methods in an adversarial bandit setting, for +both semi-bandit (achieving �O(T 1/2) α−regret) and full-bandit feedback (achieving �O(T 2/3) +α−regret). Their framework requires the offline approximation algorithm to have an iter- +ative greedy structure (unlike ours), satisfy a robustness property (like ours), and satisfy +a property referred to as Blackwell reducibility (unlike ours). In addition to these condi- +tions, the adaptation depends on the number of subproblems (greedy iterations) which for +some algorithms can be known ahead of time (such as with cardinality constraints) but for +other algorithms can only be upper-bounded. (Our adaptation uses the offline algorithm +as a black box.) The authors check those conditions and explicitly adapt several offline +approximation algorithms. In this paper, we consider an approach for converting offline +approximation algorithm to online for stochastic CMAB, while requiring less assumptions. +We also note that (Niazadeh et al., 2021) do not consider submodular CMAB with +knapsack constraints, and thus do not verify whether any approximation algorithms for +the offline problem satisfy the required properties (of sub-problem structure or robustness +or Blackwell reducibility) to be transformed, and this is an example we consider for our +general framework. Consequently, in our experiments for submodular CMAB with knapsack +constraints in Section 7, we use the algorithm in (Streeter and Golovin, 2008) designed for +a knapsack constraint (in expectation) as representative of methods for the adversarial +setting. Other related works for adversarial stochastic CMAB are described in Appendix +H. +Stochastic Submodular CMAB with Full Bandit Feedback +Recently, Nie et al. +(2022) propose an algorithm for stochastic MAB with submodular rewards, when there is a +cardinality constraint. Their algorithm is a specific adaptation of an offline greedy method. +In our work, we propose a general framework that employs the offline algorithm as a black +box (and this result becomes a special case of our approach). While there are multiple +results for semi-bandit feedback (see Appendix I), this paper considers full bandit feedback. +3. Problem Statement +We consider sequential, combinatorial decision-making problems over a finite time horizon +T. Let Ω denote the ground set of base elements (arms). Let n = |Ω| denote the number +of arms. Let D ⊆ 2Ω denote the subset of feasible actions (subsets), for which we presume +membership can be efficiently evaluated. We will later consider applications with cardinality +and knapsack constraints, though our methods are not limited to those. We will use the +terminologies subset and action interchangeably throughout the paper. +At each time step t, the learner selects a feasible action At ∈ D. After the subset At is +selected, the learner receives reward ft(At). We assume the reward ft is stochastic, bounded +in [0, 1], and i.i.d. conditioned on a given subset. Define the expected reward function as +f(A) = E[ft(A)]. +4 + +The goal of the learner is to maximize the cumulative reward �T +t=1 ft(At). To measure +the performance of the algorithm, one common metric is to compare the learner to an agent +with access to a value oracle for f. However, if optimizing f over D is NP-hard, such a +comparison would not be meaningful unless the horizon is exponentially large in the problem +parameters. +If there is a known approximation algorithm A with approximation ratio α ∈ (0, 1] for +optimizing f over D, a more natural alternative is to evaluate the performance of a CMAB +algorithm against what A could achieve. Thus, we consider the the expected cumulative +α-regret Rα,T , which is the difference between α times the cumulative reward of the optimal +subset’s expected value and the average received reward, (we write RT when α is understood +from context) +E[RT ] = αTf(OPT) − E +� T +� +t=1 +ft(At) +� +, +(1) +where OPT is the optimal solution, i.e., OPT ∈ arg maxA∈D f(A) and the expectations are +over both the random rewards and the sequence of actions. +4. Robustness of Offline Algorithms +In this section, we introduce a criterion for an offline approximation algorithm’s sensitivity +to (bounded) additive perturbations to function evaluations. Investigating robustness of +approximation algorithms in offline settings is valuable in its own right. Importantly, we +will show that this property alone is sufficient to guarantee that the offline algorithm can be +adapted to solve analogous combinatorial multi-armed bandit (CMAB) problems with just +bandit feedback and yet achieve sub-linear regret. Furthermore, the CMAB adaptation will +not rely on any special structure of the algorithm design, instead employing it as a black +box. +Definition 1 ((α, δ)-Robust Approximation) An algorithm A is an (α, δ)-robust ap- +proximation algorithm for the combinatorial optimization problem of maximizing a function +f : D → R over a finite domain D ⊆ 2Ω if its output S∗ using a value oracle for ˆf satisfies +the relation below with the optimal solution OPT under f, provided that for any ϵ > 0 that +|f(S) − ˆf(S)| < ϵ for all S ∈ D, +f(S∗) ≥ αf(OPT) − δϵ. +Note that the perturbed ˆf is not required to be in the same class as f (linear, quadratic, +submodular, etc.). Thus, this definition is a stronger notion of robustness than one limited +to ˆf in the same class that have bounded L∞ distance from f. +For (unstructured) k armed bandit problems, one can view the analogous offline algo- +rithm with access to a value oracle for the elements as first evaluating each arm (D = +{{1}, {2}, . . . , {k}}), so N = k queries total, and then evaluating arg max over the k values. +That algorithm trivially is a (1, 2)-robust approximation algorithm. +Remark 2 In Niazadeh et al. (2021), there is a related definition of robustness for offline +approximation algorithms. That definition and the subsequent offline-to-online adaptation +5 + +Algorithm 1 Combinatorial Explore-then-Commit +Input: horizon T, set of base elements Ω, an offline (α, δ)-robust algorithm A, and an +upper-bound N on the number of A’s queries to the value oracle +Initialize m ← +� +δ2/3T 2/3 log(T)1/3 +2N2/3 +� +// Exploration Phase // +while A queries the value of some A ⊆ Ω do +For m times, play action A +Calculate the empirical mean ¯f +Return ¯f to A +end while +// Exploitation Phase // +for remaining time do +Play action S output by algorithm A. +end for +procedure is restricted to approximation algorithms with an iterative greedy structure. The +criterion Theorem 1 we consider does not require the approximation algorithm to have an +iterative greedy structure. +To illustrate the utility of our proposed framework, in Section 6 we will show that +several approximation algorithms from the constrained submodular maximization literature +are (α, δ)-robust, leading to new sublinear α-regret algorithms for related stochastic CMAB +problems with submodular rewards. +5. C-ETC Algorithm: Offline to Stochastic +In this section, we present our proposed algorithm for adapting offline approximation to algo- +rithms for stochastic CMAB, Combinatorial Explore-Then-Commit (C-ETC). The pseudo- +code is shown in Algorithm 1. The algorithm takes an offline (α, δ) robust algorithm A +with an upper bound N on the number of oracle queries by A. In the exploration phase, +when the offline algorithm queries the value oracle for action A, C-ETC will play action A +for m times, where m is a constant chosen to minimizing regret. C-ETC then computes +the empirical mean ¯f of rewards for A and feeds ¯f back to the offline algorithm A. In the +exploitation phase, C-ETC keeps playing the solution S output from algorithm A. Thus, +the CMAB procedure does not need A to have any special structure. No careful construc- +tion is needed for the CMAB procedure beyond running A. All that is needed is checking +robustness (Theorem 1). Also, there is no over-heard in terms of storage and per-round +time complexities— C-ETC is as efficient as the offline algorithm A itself. +Now we analyze the α-regret for C-ETC (Algorithm 1). +Theorem 3 For the sequential decision making problem defined in Section 2 and T ≥ +2 +√ +2N +δ +, the expected cumulative α-regret of C-ETC using an (α, δ)-robust approximation al- +6 + +gorithm as subroutine is at most O +� +δ +2 +3 N +1 +3 T +2 +3 log(T) +1 +3 +� +, where N upper-bounds the number +of value oracle queries made by the offline algorithm A. +The detailed proof is in the supplementary material. We highlight some key steps. +We show that with high probability, the empirical means of all actions taken during +exploration phase will be within rad = +� +log T +2m +of their corresponding statistical means. +As is common in proofs for ETC methods, we refer to this occurrence as the clean event +E. Then, using an (α, δ)-robust approximation algorithm as subroutine will guarantee the +quality of of the set S used in the exploitation phase of Algorithm 1: +f(S) ≥ αf(OPT) − δ · rad. +(2) +We then break up the expected cumulative α-regret conditioned on the clean event E, +E[R(T)|E] = +N +� +i=1 +m (αf(S∗) − E[f(St)]) +� +�� +� +exploration phase ++ +T +� +t=TN+1 +(αf(S∗) − E[f(S)]) +� +�� +� +exploitation phase +. +(3) +Using the fact that the reward is bounded between [0, 1], we have +E[R(T)|E] ≤ Nm + Tδrad. +Optimizing over m then results in +E[R(T)|E] = O +� +δ +2 +3 N +1 +3 T +2 +3 log(T) +1 +3 +� +. +We then show that because the clean event E happens with high probability, the expected +cumulative regret E[R(T)] is dominated by E[R(T)|E], which concludes the proof. +Lower bounds: +For the general setting we explore in this paper, with stochastic (or +even adversarial) combinatorial MAB and only bandit feedback, it is unknown whether +˜O(T 1/2) expected cumulative α-regret is possible (ignoring problem parameters like n). For +special cases, such as linear reward functions, ˜O(T 1/2) is known to be achievable even with +bandit feedback. Even for the special case of submodular reward functions and a cardinality +constraint, it remains an open question. Niazadeh et al. (2021) obtain ˜Ω(T 2/3) lower bounds +for the harder setting where feedback is only available during “exploration” rounds chosen +by the agent, who incurs an associated penalty. +Remark 4 C-ETC uses knowledge of the horizon T to optimize the number m of samples +per action. When the time horizon T is not known, we can use geometric doubling trick +to extend our result to an anytime algorithm. We refer to the general detailed procedure +in (Besson and Kaufmann, 2018). From Theorem 4 in (Besson and Kaufmann, 2018), we +can show that the regret bound conserves the original T 2/3 log(T)1/3 dependence with only +changes in constant factors. +7 + +6. Applications on Submodular Maximization +In this section, we apply our general framework to stochastic CMAB problems with mono- +tone submodular rewards where only bandit feedback is available. This application results +in the first sublinear α-regret CMAB algorithms for knapsack constraints under bandit feed- +back. We begin with a brief background, and analyze the robustness of offline approximation +algorithms, and then obtain problem independent regret bounds. +6.1 Background and Definitions +Denote the marginal gain f(e|A) = f(A ∪ e) − f(A) and the marginal density ρ(e|A) = +f(A∪e)−f(A) +c(e) +for any subset A ⊆ Ω and element e ∈ Ω \ A. A set function f : 2Ω → R +defined on a finite ground set Ω is said to be submodular if it satisfies the diminishing +return property: for all A ⊆ B ⊆ Ω, and e ∈ Ω \ B, it holds that f(e|A) ≥ f(e|B). A set +function is said to be monotonically non-decreasing if f(A) ≤ f(B) for all A ⊆ B ⊆ Ω. Our +aim is to find a set S such that f(S) is maximized subject to some constraints. +For knapsack constraints, we assume that the cost function c : Ω → R>0 is known +and linear, so the cost of a subset is be the sum of the costs of individual items: c(A) = +� +v∈A c(v). +To simplify the presentation, we avoid the cases of trivially large budgets +B > � +v∈Ω c(v) and assume all items have non-trivial costs 0 < c(v) ≤ B. A cardinality +constraint is a special case with unit costs. +In the following, we consider both types of those constraints: cardinality and knapsack. +Maximizing a monotone submodular set function under a k-cardinality constraint is NP- +hard even with a value oracle Nemhauser et al. (1978). The best achievable approximation +ratio with a polynomial time algorithm is 1−1/e Nemhauser et al. (1978) using O(nk) oracle +calls. In Badanidiyuru and Vondr´ak (2014), 1−1/e−ϵ′ is achieved within O( n +ϵ′ log n +ϵ′ ) time, +where ϵ′ is a user selected parameter to balance accuracy and time complexity. +Maximizing a monotone submodular set function under a knapsack constraint is conse- +quently also NP-hard Khuller et al. (1999). The best achievable approximation ratio with +a polynomial time algorithm is 1 − 1/e (Sviridenko, 2004; Khuller et al., 1999), but that +requires O(n5) function evaluations, making it prohibitive for many applications. There +are other offline algorithms that achieve worse approximation ratios but are much more ef- +ficient. We adapt a 1 +2 approximation algorithm (Yaroslavtsev et al., 2020) and a 1 +2(1 − 1/e) +approximation algorithm (Khuller et al., 1999), both of which use O(n2) function evalua- +tions. There is another algorithm proposed recently in Li et al. (2022), but since it queries +infeasible sets, we do not consider it. +6.2 Offline Approximation Algorithms – Robustness +For an overview of offline approximation algorithms for submodular optimization, please +refer Appendix A. We next state our results on (α, δ)-robustness of the offline algorithms +considered. The assumption of complete/noiseless access to a value oracle is often a strong +assumption for real world applications. Thus, even for offline applications, it is worthwhile +knowing how robust an algorithm is. So the following results are relevant even in the offline +setting. +For the CMAB setting we consider, robustness is also a sufficient property to +8 + +guarantee a no-regret adaptation of the offline algorithm. Detailed proofs are included in +Appendix B in the supplementary material. +Theorem 5 (Corollary 4.3 of Nie et al. (2022)) Greedy in Nemhauser et al. (1978) +is a (1 − 1 +e, 2k)-robust approximation algorithm for submodular maximization under a k- +cardinality constraint. +Theorem 6 ThresholdGreedy Badanidiyuru and Vondr´ak (2014) is a (1− 1 +e −ϵ′, 2(2− +ϵ′)k)-robust approximation algorithm for submodular maximization under a k-cardinality +constraint. +Theorem 7 PartialEnumeration Sviridenko (2004); Khuller et al. (1999) is a (1− 1 +e, 4+ +2 ˜K + 2β)-robust approximation algorithm for submodular maximization under a knapsack +constraint. +Theorem 8 Greedy+Max Yaroslavtsev et al. (2020) is a ( 1 +2, 1 +2 + ˜K + 2β)-robust approx- +imation algorithm for submodular maximization problem under a knapsack constraint. +Theorem 9 Greedy+ Khuller et al. (1999) is a ( 1 +2(1− 1 +e), 2+ ˜K+β)-robust approximation +algorithm for submodular maximization problem under a knapsack constraint. +Remark 10 For the offline setting, Greedy+Max is superior to Greedy+, as it achieves +a better α approximation ratio with the same calls to the value oracle. However, their (α, δ) +pairs are incomparable, as for β > 1.5 (with β = 1 corresponding to a cardinality con- +straint), Greedy+ has a smaller δ (thus more robust) which affects exploration time in +their adaptations and in turn affects their regret. +To illustrate the robustness analysis, we highlight some key steps for the proof of Theo- +rem 8 for Greedy+Max. Let o1 ∈ arg maxe:e∈OPT c(e) denote the most expensive element +in OPT. Inspired by the proof techniques in (Yaroslavtsev et al., 2020), we consider the +last item added by the greedy solution (based on noisy evaluation) before the cost of this +solution exceeds B −c(o1). Let Gi denote the set selected by Greedy that has cardinality i +and denote the constituent elements as Gi = {g1, · · · , gi}. Denote Gℓ as the largest greedy +sequence that consumes less than B−c(o1) of the budget B, so c(Gℓ) ≤ B−c(o1) < c(Gℓ+1). +Let Si denote the augmented set at i-th iteration and S denote the final output of the algo- +rithm. Denote ˆf(e|S) := ˆf(S ∪ e) − ˆf(S) and ˆρ(e|S) := +ˆf(S∪e)− ˆf(S) +c(e) +. We prove the following +lemma. +Lemma 11 (Greedy+Max inequality) For i ∈ {0, 1, · · · , ℓ}, the following inequality +holds: +ˆf(Gi ∪ o1)+ max{0, ˆρ(gi+1|Gi)}(B − c(o1)) +≥ f(OPT) − (2 ˜K − 1)ϵ. +For i = ℓ, Theorem 11 tells us that there can be two cases: +ˆf(Gℓ ∪ o1) ≥ 1 +2f(OPT) − +� +˜K − 1 +2 + γ +� +ϵ, or +9 + +ˆρ(gℓ+1|Gℓ)(B − c(o1)) ≥ 1 +2f(OPT) − +� +˜K − 1 +2 − γ +� +ϵ, +where γ will be selected later to minimize the additive error δ coefficient. +If ˆf(Gℓ ∪ o1) ≥ 1 +2f(OPT) − +� +˜K − 1 +2 + γ +� +ϵ, then denote aℓ = arg maxe∈Ω\Gℓ ˆf(e|Gℓ), +which is the element selected to augment Gℓ. We have +ˆf(Gℓ ∪ aℓ) ≥ ˆf(Gℓ ∪ o1) +≥ 1 +2f(OPT) − +� +˜K − 1 +2 + γ +� +ϵ. +(4) +Then the final output of the algorithm S will satisfy +f(S) ≥ ˆf(S) − ϵ +≥ ˆf(Gℓ ∪ aℓ) − ϵ +≥ 1 +2f(OPT) − +� +˜K + 1 +2 + γ +� +ϵ. +(using (4)) +If ˆρ(gℓ+1|Gℓ)(B − c(o1)) ≥ 1 +2f(OPT) − ( ˜K − 1 +2 − γ)ϵ, rearranging we have +ˆρ(gℓ+1|Gℓ) ≥ +f(OPT) +2(B − c(o1)) − ( ˜K − 1 +2 − γ)ϵ +B − c(o1) +. +(5) +Moreover, +ˆf(Gℓ) = +l−1 +� +j=0 +ˆρ(gj+1|Gj)c(gj+1) +≥ +l−1 +� +j=0 +ˆρ(gℓ+1|Gj)c(gj+1) +(6) +≥ +l−1 +� +j=0 +� +ρ(gℓ+1|Gj) − +2ϵ +c(gℓ+1) +� +c(gj+1) +≥ +l−1 +� +j=0 +� +ρ(gℓ+1|Gℓ) − +2ϵ +c(gℓ+1) +� +c(gj+1) +(7) += +� +ρ(gℓ+1|Gℓ) − +2ϵ +c(gℓ+1) +� +c(Gℓ) +≥ +� +ˆρ(gℓ+1|Gℓ) − +4ϵ +c(gℓ+1) +� +c(Gℓ) +≥ ˆρ(gℓ+1|Gℓ)c(Gℓ) − 4βϵ, +(8) +10 + +where (6) follows from the greedy selection rule, the (7) follows from submodularity of f, +and (8) follows from the definition of β. We then have +ˆf(Gℓ+1) += ˆf(Gℓ) + c(gℓ+1)ˆρ(gℓ+1|Gℓ) +≥ +� +ˆρ(gℓ+1|Gℓ)c(Gℓ) − 4βϵ +� ++ c(gℓ+1)ˆρ(gℓ+1|Gℓ) +(9) += ˆρ(gℓ+1|Gℓ)c(Gℓ+1) − 4βϵ +≥ +1 +2f(OPT) − ( ˜K − 1 +2 − γ)ϵ +B − c(o1) +c(Gℓ+1) − 4βϵ +(10) +≥ 1 +2f(OPT) − ( ˜K − 1 +2 − γ)ϵ − 4βϵ +(11) += 1 +2f(OPT) − +� +˜K − 1 +2 − γ + 4β +� +ϵ, +(12) +where (9) follows from (8), (10) follows from (5), and (11) follows from the chosen ℓ satisfies +c(Gℓ+1) > B − c(o1). Thus, the final output of the algorithm S will satisfy +f(S) ≥ ˆf(S) − ϵ +≥ ˆf(Gℓ+1) − ϵ +≥ 1 +2f(OPT) − +� +˜K + 1 +2 − γ + 4β +� +ϵ. +Finally, combining both cases and selecting γ = 2β completes the proof. +6.3 CMAB algorithms for Submodular Rewards with Knapsack Constraints +Now that we have analyzed the robustness of several offline algorithms, we can invoke +Theorem 3 to bound the expected cumulative α regret for stochastic CMAB adaptations +that rely only on bandit feedback. We name the adapted algorithms as C-ETC-N, C-ETC- +B for cardinality constraint, C-ETC-S C-ETC-K and C-ETC-Y for knapsack constraint, +respectively, based on which offline algorithm it is adapted from (using the first author’s +last name); which are in order Nemhauser et al. (1978); Badanidiyuru and Vondr´ak (2014); +Sviridenko (2004); Khuller et al. (1999); Yaroslavtsev et al. (2020). PartialEnumeration +was first proposed and analyzed by Khuller et al. (1999) for maximum coverage problems +and then analyzed by Sviridenko (2004) for monotone submodular functions. To distinguish +CMAB adaptations of Greedy+ and C-ETC-K, both proposed in Khuller et al. (1999), we +use C-ETC-S for the adaption of PartialEnumeration. The following corollaries hold +immediately: +Corollary 12 For an online submodular maximization under a cardinality constraint, the +expected cumulative (1 − 1/e)-regret of C-ETC-N is at most O +� +kn +1 +3 T +2 +3 log(T) +1 +3 +� +for T ≥ +√ +2n. +Remark 13 This result improves upon the result from Nie et al. (2022) by a factor of k +1 +3 +despite our use of a generic framework. +11 + +Corollary 14 For an online submodular maximization under a cardinality constraint, the +expected cumulative (1−1/e−ϵ′)-regret of C-ETC-B is at most O +� +k +2 +3 n +1 +3 (ϵ′) +1 +3 (log n +ϵ′ ) +1 +3 T +2 +3 log(T) +1 +3 +� +for T ≥ +√ +2n +(2−ϵ′)ϵ′k log n +ϵ′ . +Corollary 15 For an online submodular maximization under a knapsack constraint, the +expected cumulative (1 − 1/e)-regret of C-ETC-S is at most O +� +β +2 +3 ˜K +1 +3 n +4 +3 T +2 +3 log(T) +1 +3 +� +for +T ≥ +√ +2 ˜ +Kn4 +2+ ˜ +K+β. +Corollary 16 For an online submodular maximization under a knapsack constraint, the +expected cumulative +1 +2-regret of C-ETC-Y is at most O +� +β +2 +3 ˜K +1 +3 n +1 +3 T +2 +3 log(T) +1 +3 +� +for T ≥ +2 +√ +2 ˜ +Kn +1 +2 + ˜ +K+2β. +Corollary 17 For an online submodular maximization under a knapsack constraint, the +expected cumulative 1 +2(1 − 1 +e)-regret of C-ETC-K is at most O +� +β +2 +3 ˜K +1 +3 n +1 +3 T +2 +3 log(T) +1 +3 +� +for +T ≥ 2 +√ +2 ˜ +Kn +2+ ˜ +K+β. +Storage and Per-Round Time Complexities: C-ETC-Y and C-ETC-K have low +storage complexity and per-round time-complexity. During exploitation, only the indices of +at most ˜K base arms are needed in memory and does not need any computation. During +exploration, they just need to update the empirical mean for the current action at time +t, which can be done in O(1) time. It additionally stores the highest empirical density so +far in the current iteration of the greedy routine and its associated base arm (C-ETC-K +needs to store one more arm and C-ETC-Y an additional O( ˜K) storage is needed to store +the augmented set). Thus, C-ETC-Y and C-ETC-K have O( ˜K) storage complexity and +O(1) per-round time complexity. For comparison, the algorithm proposed by Streeter and +Golovin (2008) for an averaged knapsack constraint in the adversarial setting uses O(n ˜K) +storage complexity and O(n) per-round time complexity. Some comments on lower bound +are given in Appendix E. +7. Experiments +In this section, we conduct experiments on real world data with a Budgeted Influence Maxi- +mization (BIM). We also conduct experiments on Song Recommendation (SR) in Appendix +J. Both of these are applications of stochastic CMAB with submodular rewards under a +knapsack constraint. There are three adaptions we considered in Section 6 for knapsack +constraint. Since the time complexity for PartialEnumeration is much larger than the +other two offline algorithms we consider, it will use at least T ≈ 108 for C-ETC-S to finish +exploration. +For this reason, we do not consider C-ETC-S in the experiments. +To our +knowledge, our work is the first to consider these applications with only bandit feedback +available. +Baseline: +The only other algorithm designed for combinatorial MAB with general sub- +modular rewards, under a knapsack constraint, and using full-bandit feedback is Online +Greedy with opaque feedback model (OGo) proposed by Streeter and Golovin (2008) +12 + +(a) +(b) +(c) +(d) +Figure 1: Plots for budgeted influence maximization (BIM) example. (a) and (b) are comparison +results for cumulative regret as a function of time horizon T. (c) and (d) are the moving average +plot with window size 100 of instantaneous reward as a function of t. The gray dashed lines in (a) +and (b) represent y = aT 2/3 for various values of a for visual reference. The gray dashed lines in (c) +and (d) represent expected rewards for the action chosen by an offline greedy algorithm. +for the adversarial setting. However, OGo only satisfies the knapsack constraint in expecta- +tion, while our algorithms C-ETC-K ands C-ETC-Y satisfies a strict constraint (i.e. every +action At must be under budget). See Appendix D for more details about OGo and its +implementation. +In Section 6, we used N = ˜Kn as an upper bound on the number of function evaluations +for both C-ETC-K and C-ETC-Y, where n is the number of base arms and ˜K is an upper +bound of the cardinality of any feasible set. When the time horizon T is small, it is possible +that the exploration phase will not finish due to the formula being optimized for m (the +number of plays for each action queried by A) uses a loose bound on the exploitation time. +When this is the case, we select the largest m (closest to the formula) for which we can +guarantee that exploration will finish. For details, see Appendix F. +13 + +BIM B=6 +Cumulative Regret +10 +4 +C-ETC-K +C-ETC-Y +103 +3 +OGo +104 +105 +Horizon TBIM B=8 +Cumulative Regret +10 +4 +C-ETC-K +C-ETC-Y +103 +3 +OGo +104 +105 +Horizon TBIM B=6 +1e-1 +Instantaneous Reward +3 +2 +C-ETC-K +C-ETC-Y +OGo +0.00 +0.25 +0.50 +0.75 +1.00 +1e5 +Time-step tBIM B=8 +1e-1 +Instantaneous Reward +3 +2 +C-ETC-K +C-ETC-Y +OGo +0.00 +0.25 +0.50 +0.75 +1.00 +Time-step t +1e5We first conduct experiments for the application of budgeted influence maximization +(BIM) on a portion of the Facebook network graph. BIM models the problem of identifying +a low-cost subset (seed set) of nodes in a (known) social network that can influence the +maximum number of nodes in a network. While there are prior works proposing algorithms +for budgeted online influence maximization problems, the state of the art (e.g., Perrault et al. +(2020b)) presumes knowledge of the diffusion model (such as independent cascade) and, +more importantly, extensive semi-bandit feedback on individual diffusions, such as which +specific nodes became active or along which edges successful infections occurred, in order +to estimate diffusion parameters. For social networks with user privacy, this information is +not available. +Data Set Description and Experiment Details: The Facebook network dataset +was introduced in Leskovec and Mcauley (2012). To facilitate running multiple experiments +for different horizons, we used the community detection method proposed by Blondel et al. +(2008) to detect a community with 354 nodes and 2853 edges. We further changed the +network to be directed by replacing every undirected edge by two directed edge with opposite +directions, yielding a directed network with 5706 edges. The diffusion process is simulated +using the independent cascade model (Kempe et al., 2003), where in each discrete step, an +active node (that was inactive at the previous time step) independently attempts to infect +each of its inactive neighbors. Following existing work of Tang et al. (2015, 2018); Bian et al. +(2020), we set the probability of each edge (u, v) as 1/din(v), where din(v) is the in-degree of +node v. Moreover, we consider a user u is more influential if the user has more out-degrees, +dout(u). In our experiment, we only consider influential users to spend our budget more +efficiently. We pick the users with out-degrees that are above 95th percentile (18 users). +Denote this set as I, then for a user u ∈ I, the cost is defined as c(u) = 0.01dout(u) + 1, +similar to (Wu et al., 2022). For each time horizon that was used, we ran each method ten +times. +For this set of experiments, instead of cumulative +1 +2-regret, which requires knowing +OPT, we compare the cumulative rewards achieved by C-ETC and OGo against Tf(Sgrd), +where Sgrd is the solution returned by the offline 1 +2-approximation algorithm proposed by +Yaroslavtsev et al. (2020). +Tf(Sgrd) ≥ +1 +2Tf(OPT), so Tf(Sgrd) is a more challenging +reference value. +Results and Discussion: +Figures 1a and 1b show average cumulative regret curves +for C-ETC-K (in blue), C-ETC-Y (in orange) and OGo (in green) for different horizon T +values when the budget constraint B is 6 and 8, respectively. For B = 8, the turning point +is T = 21544. Standard errors of means are presented as error bars, but might be too small +to be noticed. Figures 1c and 1d are the instantaneous reward plots. The peaks at the +very beginning of exploration phase correspond to the time step that the single person with +highest influence is sampled. +C-ETC significantly outperforms OGo for all time horizons and budget considered. To +evaluate the gap between the empirical performance and the theoretical guarantee, we +estimated the slope for both methods on log-log scale plots. +Over the horizons tested, +OGo’s cumulative regret (averaged over ten runs) has a growth rate of 0.98. The growth +rates of C-ETC-K for budgets 6 and 8 are 0.76 and 0.68, respectively. The growth rates of +C-ETC-Y for budgets 6 and 8 are 0.75 and 0.69, respectively. The slopes are close to the +2/3 ≈ 0.67 theoretical guarantee, and notably, the performance for larger B is better. +14 + +References +Sanjeev Arora, Elad Hazan, and Satyen Kale. 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Pereira, and K.Q. +Weinberger, editors, Advances in Neural Information Processing Systems, volume 24. +Curran Associates, Inc., 2011a. URL https://proceedings.neurips.cc/paper/2011/ +file/33ebd5b07dc7e407752fe773eed20635-Paper.pdf. +Yisong Yue and Carlos Guestrin. +Linear submodular bandits and their application to +diversified retrieval. Advances in Neural Information Processing Systems, 24, 2011b. +17 + +A. Offline Approximation Algorithms – Overview +We give a brief overview of the offline approximation algorithms which we will analyze (α, δ) +robustness for. +For a k-cardinality constraint, the greedy algorithm Greedy proposed in Nemhauser +et al. (1978) starts from an empty set G ← ∅. Then it repeatedly add the element with +highest marginal gain f(e|G) until the cardinality |G| reaches k. ThresholdGreedy, pro- +posed in Badanidiyuru and Vondr´ak (2014), considers a sequence of decreasing thresholds: +{τ = d; τ ≥ ϵ′ +n d; τ ← (1−ϵ′)τ} where d = maxe∈Ω f(e). Then starting from empty set G = ∅, +the algorithm includes any element e /∈ G such that f(e|G) ≥ τ whenever the cardinality +is smaller than k. The algorithm then repeats using a lower threshold. Badanidiyuru and +Vondr´ak (2014) showed that ThresholdGreedy can achieve 1 − 1/e − ϵ′ approximation. +For a knapsack constraint, several algorithms run the following greedy subroutine, which +we refer to as Greedy (cardinality is a special case of this routine with budget k and +unit cost, so we keep the same name without confusion). Start with empty set G ← ∅. +Repeatedly add the element e with the highest marginal density ρ(e|G) that fits into the +budget. Let Gi denote the set selected by Greedy that has cardinality i and denote the +constituent elements as Gi = {g1, · · · , gi}. Let L denote the cardinality of the final greedy +set (i.e. when no more elements remain that are under budget), so GL is output by Greedy. +Note that L can only be bounded ahead of time—there could be maximal subsets (to which +no other elements could be added without violating the budget) of different cardinalities. +Greedy can have an unbounded approximation ratio Khuller et al. (1999) for knapsack +constraint. Khuller et al. (1999) proposed Greedy+, which outputs the better of the best +individual element a∗ ∈ arg maxe∈Ω f(e) and the output of Greedy, arg maxS∈{GL,a∗} f(S). +Khuller et al. (1999) proved that Greedy+ achieves a 1 +2(1− 1 +e) approximation ratio. Then, +Sviridenko (2004); Khuller et al. (1999) proposed PartialEnumeration. It first enumerate +all sets with cardinality up to three. For each enumerated triplets, it build the rest of the +solution set greedily. Then it outputs the set with largest value among all evaluated sets. +They showed that PartialEnumeration can achieve 1 − 1/e approximation ratio. +Greedy+Max generalizes Greedy+ by augmenting each set {Gi}L +i=1 in the nested se- +quence produced by Greedy with another element. +For 0 ≤ i ≤ L − 1, define G′ +i ← +Gi ∪ arg maxe∈Ω:c(Gi)+c(e)≤B f(Gi ∪ e). By construction, G′ +0 = {a∗}, the best individual +element. For i = L, G′ +L ← GL. Greedy+Max then outputs the best set in the augmented +sequence, arg maxS∈{G′ +0,...,G′ +L} f(S). +Yaroslavtsev et al. (2020) proposed Greedy+Max +and proved it achieves an approximation ratio of 1 +2. +A bound on the number of value oracle calls will be important in adapting offline meth- +ods. Denote β := B/cmin and ˜K := min{n, β} as an upper bound of the number of items +in any feasible set. We note here that while PartialEnumeration uses O( ˜Kn4) function +evaluations, both Greedy+Max and Greedy+ use O( ˜Kn) oracle calls, same as Greedy. +We use N = ˜Kn in the analysis for Greedy+Max and Greedy+. +B. Proof for Robustness of Offline Algorithms +In this section, we prove the (α, δ) robustness of algorithms considered in Section 6 of the +main paper. +18 + +B.1 Notation +We first review notations used in the analysis. Recall that we are only able to evaluate the +surrogate function ˆf such that | ˆf(S) − f(S)| ≤ ϵ for any feasible set S and some ϵ > 0, +we further denote ˆf(e|S) = ˆf(S ∪ e) − ˆf(S) and ˆρ(e|S) = +ˆf(S∪e)− ˆf(S) +c(e) +. Let Gi denote the +set selected by basic Greedy (based on surrogate function ˆf) as described in Section 3 up +until ith item and Gi = {g1, · · · , gi} in the order of each item is selected. Without loss of +generality, define G0 = ∅ and f(G0) = ˆf(G0) = 0. Denote cmin = mine∈Ω c(e) be the item +with lowest individual cost. Let β = B/cmin and ˜K = min{n, β} being an upper bound of +the number of items in any feasible set. Since all selected actions should be feasible, for +ease of notation, we omit denoting that condition throughout the proof. For example, we +write arg maxe∈Ω\A f(e|A) to simplify the notation of arg maxe:e∈Ω\A and A∪e∈D f(e|A). Let +S be the set returned by modified algorithms in corresponding context. +B.2 Robustness of Offline Methods for Submodular Maximization under +Cardinality Constraint +B.2.1 Greedy +We consider the original greedy algorithm Greedy proposed in Nemhauser et al. (1978), +which gives a (1 − 1 +e)-approximation guarantee for submodular maximization under a k- +cardinality constraint. To restate Theorem 5 in the main paper, Greedy is a (1 − 1 +e, 2k)- +robust approximation algorithm for submodular maximization under a k-cardinality con- +straint. The result follows from Corollary 4.3 of Nie et al. (2022), part of the regret analysis +for a CMAB adaptation of Greedy. +B.2.2 ThresholdGreedy +We then consider the threshold greedy algorithm ThresholdGreedy proposed in Badani- +diyuru and Vondr´ak (2014), which gives a (1− 1 +e −ϵ′)-approximation guarantee for submod- +ular maximization under a k-cardinality constraint, where ϵ′ is a user specified parameter +to balance accuracy and run time. +Restating Theorem 6 in the main paper, Thresh- +oldGreedy is a (1 − 1 +e − ϵ′, 2(2 − ϵ′)k)-robust approximation algorithm for submodular +maximization under a k-cardinality constraint. +Proof From the assumption of the surrogate function ˆf we know +f(e|S) − 2ϵ ≤ ˆf(e|S) ≤ f(e|S) + 2ϵ +for any e ∈ Ω \ S and S ⊆ Ω. Now assume the the next chosen element is a and the current +partial solution is S. On one hand, we have +ˆf(a|S) ≥ w =⇒ f(a|S) ≥ w − 2ϵ, +(13) +on the other hand, for every e ∈ OPT \ S, +ˆf(e|S) ≤ +w +1 − ϵ′ =⇒ f(e|S) ≤ +w +1 − ϵ′ + 2ϵ. +(14) +Combining and manipulating (13) and (14) we have for any e ∈ OPT \ S: +f(a|S) + 2ϵ ≥ (f(e|S) − 2ϵ)(1 − ϵ′) =⇒ f(a|S) ≥ (1 − ϵ′)f(e|S) − 2(2 − ϵ′)ϵ. +(15) +19 + +Taking an average over all e ∈ OPT \ S, +f(a|S) ≥ +1 − ϵ′ +|OPT \ S| +� +e∈OPT\S +f(e|S) − 2(2 − ϵ′)ϵ +≥ 1 − ϵ′ +k +� +e∈OPT\S +f(e|S) − 2(2 − ϵ′)ϵ. +(16) +Now consider after i ∈ [k − 1] steps, we get a partial solution Si = {a1, · · · , ai}. By (16), +we have +f(ai+1|Si) ≥ 1 − ϵ′ +k +� +e∈OPT\S +f(e|Si) − 2(2 − ϵ′)ϵ +≥ 1 − ϵ′ +k +f(OPT|Si) − 2(2 − ϵ′)ϵ +(submodularity) +≥ 1 − ϵ′ +k +(f(OPT) − f(Si)) − 2(2 − ϵ′)ϵ, +(monotonicity) +and hence for i ∈ [k − 1], +f(Si+1) − f(Si) = f(ai+1|Si) ≥ 1 − ϵ′ +k +(f(OPT) − f(Si)) − 2(2 − ϵ′)ϵ. +(17) +Using (17) as induction hypothesis, we then prove by induction (omitted) that for i ∈ [k−1], +f(Si+1) ≥ +� +1 − +� +1 − 1 − ϵ′ +k +�i+1� +f(OPT) − 2(i + 1)(2 − ϵ′)ϵ, +and plugging in i = k − 1 we get +f(Sk) ≥ +� +1 − +� +1 − 1 − ϵ′ +k +�k� +f(OPT) − 2k(2 − ϵ′)ϵ +≥ (1 − e−(1−ϵ′))f(OPT) − 2k(2 − ϵ′)ϵ +≥ (1 − 1/e − ϵ′)f(OPT) − 2k(2 − ϵ′)ϵ. +We finish the proof by observing that Sk is the output. +B.3 Proof for Robustness of Greedy+Max +In this section, we give a detailed proof for Theorem 8 in Section 6 of the main paper. Recall +the statement is that Greedy+Max is a ( 1 +2, 1 +2 + ˜K + 2β)-robust approximation algorithm +for submodular maximization problem under a knapsack constraint. +Let o1 ∈ arg maxe:e∈OPT c(e) denote the most expensive element in OPT. During the ith +iteration of the greedy process, having previously selected the set Gi−1 with i − 1 elements, +20 + +it will select the element gi with highest marginal density (based on surrogate function ˆf) +among feasible elements, +gi = +arg max +e: e∈Ω\Gi−1 +ˆρ(e|Gi−1). +(18) +Inspired by the proof techniques in Yaroslavtsev et al. (2020), we consider the last item +added by the greedy solution (based the surrogate function ˆf) before the cost of this solution +exceeds B − c(o1). +Denote Gℓ as the largest greedy sequence that consumes less than +B − c(o1) budgets, c(Gℓ) ≤ B − c(o1) < c(Gℓ+1). +Let ai denote the element selected +to augment with the greedy solution Gi, i.e., ai = arg maxe∈Ω\Gi ˆf(e|Gi), and Si denote +the augmented set at i-th iteration. Before proving the theorem, we show Theorem 11 in +Section 6 of the main paper, that for i ∈ {0, 1, · · · , ℓ}, the following inequality holds: +ˆf(Gi ∪ o1) + max{0, ˆρ(gi+1|Gi)}(B − c(o1)) ≥ f(OPT) − (2 ˜K − 1)ϵ. +Proof +Recall that from the definition of ˆf, we have | ˆf(S) − f(S)| ≤ ϵ for any evaluated +set S and some ϵ > 0. Consequently, we have for any i ∈ {0, 1, · · · , ℓ}, +| ˆf(Gi) − f(Gi)| ≤ ϵ. +(19) +Now we evaluate the set Gi ∪ o1. +• Case 1: If o1 has already been added, o1 ∈ Gi, then +| ˆf(Gi ∪ o1) − f(Gi ∪ o1)| = | ˆf(Gi) − f(Gi)| ≤ ϵ. +• Case 2: If o1 /∈ Gi, then ˆf(Gi ∪ o1) is evaluated in iteration i + 1. This iteration i + 1 +does exist1 because for any i ∈ {0, 1, · · · , ℓ}, we only used less than B − c(o1) budget. +For the remaining budget, at least o1 can still fit into the budget so Gi ∪ o1 will be +evaluated in iteration i + 1. In this case, we still have +| ˆf(Gi ∪ o1) − f(Gi ∪ o1)| ≤ ϵ. +Combining these two cases, we have +| ˆf(Gi ∪ o1) − f(Gi ∪ o1)| ≤ ϵ. +(20) +Also, for any evaluated action in iteration i + 1, namely the actions {Gi ∪ e|e ∈ Ω \ +Gi and c(e) + c(Gi) ≤ B}, we have +ρ(e|Gi) = f(Gi ∪ e) − f(Gi) +c(e) +≤ +ˆf(Gi ∪ e) − ˆf(Gi) +c(e) ++ 2ϵ +c(e) += ˆρ(e|Gi) + 2ϵ +c(e). +(21) +1. For (α, δ) robustness alone, this point is not necessary due to the assumption of |f(S)− ˆf(S)| ≤ ϵ for all +S ⊆ Ω. For the regret bound proof of our proposed C-ETC method in Appendix C.4, the “clean event” +(corresponding to concentration of empirical mean of set values around their statistical means) will only +imply concentration for those actions taken and thus for which empirical estimates exist. +21 + +Then we have +f(OPT) ≤ f(Gi ∪ OPT) +(Monotonicity of f) +≤ f(Gi ∪ o1) + f(OPT \ (Gi ∪ o1)|Gi ∪ o1) +≤ f(Gi ∪ o1) + +� +e∈OPT\(Gi∪o1) +f(e|Gi ∪ o1) +(Submodularity of f) +≤ ˆf(Gi ∪ o1) + ϵ + +� +e∈OPT\(Gi∪o1) +c(e)ρ(e|Gi ∪ o1). +(22) +where (22) uses (20). +Since we picked iteration i such that c(Gi) ≤ B −c(o1), then all items in OPT\(Gi ∪o1) +still fit, as o1 is the largest item in OPT. Since the greedy algorithm always selects the +item with the largest marginal density with respect to the surrogate function ˆf, gi = +arg maxe∈Ω\Gi ˆρ(e|Gi), thus we have +ˆρ(gi+1|Gi) = max +e∈Ω\Gi +ˆρ(e|Gi) ≥ +max +e∈Ω\(Gi∪o1) ˆρ(e|Gi). +(23) +Hence, continuing with (22), +f(OPT) ≤ ˆf(Gi ∪ o1) + ϵ + +� +� +� +e∈OPT\(Gi∪o1) +c(e)ρ(e|Gi ∪ o1) +� +� +≤ ˆf(Gi ∪ o1) + ϵ + +� +e∈OPT\(Gi∪o1) +c(e)ρ(e|Gi) +(Submodularity) +≤ ˆf(Gi ∪ o1) + ϵ + +� +e∈OPT\(Gi∪o1) +c(e) +� +ˆρ(e|Gi) + 2ϵ +c(e) +� +(using (21)) +≤ ˆf(Gi ∪ o1) + ϵ + +� +e∈OPT\(Gi∪o1) +� +c(e)ˆρ(e|Gi) +� ++ 2ϵ|OPT \ (Gi ∪ o1)| +≤ ˆf(Gi ∪ o1) + ϵ + ˆρ(gi+1|Gi) +� +e∈OPT\(Gi∪o1) +� +c(e) +� ++ 2ϵ|OPT \ (Gi ∪ o1)| +(Using (23)) +≤ ˆf(Gi ∪ o1) + ϵ + ˆρ(gi+1|Gi)c(OPT \ (Gi ∪ o1)) + 2ϵ|OPT \ (Gi ∪ o1)| +≤ ˆf(Gi ∪ o1) + ϵ + max{0, ˆρ(gi+1|Gi)}c(OPT \ (Gi ∪ o1)) + 2ϵ|OPT \ (Gi ∪ o1)| +≤ ˆf(Gi ∪ o1) + ϵ + max{0, ˆρ(gi+1|Gi)}(gi+1|Gi)(B − c(o1)) + 2ϵ|OPT \ (Gi ∪ o1)| +≤ ˆf(Gi ∪ o1) + max{0, ˆρ(gi+1|Gi)}(gi+1|Gi)(B − c(o1)) + (2 ˜K − 1)ϵ. +Rearranging terms gives the desired result. +Now we are ready to prove Theorem 8 (robustness of Greedy+Max algorithm). Applying +Theorem 11 (Greedy+Max inequality) for i = ℓ, and recalling that ℓ is chosen as the +index of the last greedy set such that c(Gℓ) ≤ B − c(o1) < c(Gℓ+1), +ˆf(Gℓ ∪ o1) + max{0, ˆρ(gℓ+1|Gℓ)}(B − c(o1)) ≥ f(OPT) − (2 ˜K − 1)ϵ. +(24) +22 + +From (24), we will next argue at least one of the terms in the left hand side must be large. +We will consider cases for the two terms being large. To minimize the worst-case additive +error term from the cases, we will split the cases into whether ˆf(Gℓ ∪ o1) is larger than or +equal to 1 +2f(OPT) − ( ˜K − 1 +2 + γ)ϵ, or max{0, ˆρ(gℓ+1|Gℓ}(B − c(o1)) is larger than or equal +to 1 +2f(OPT) − ( ˜K − 1 +2 − γ)ϵ, where γ will be selected later to minimize the additive error δ +coefficient. +Case 1: If ˆf(Gℓ ∪ o1) ≥ 1 +2f(OPT) − ( ˜K − 1 +2 + γ)ϵ, recall that aℓ as the element selected +to augment with the greedy solution Gℓ, aℓ = arg maxe∈Ω\Gℓ ˆf(e|Gℓ), then +ˆf(Gℓ ∪ aℓ) ≥ ˆf(Gℓ ∪ o1) +≥ 1 +2f(OPT) − +� +˜K − 1 +2 + γ +� +ϵ. +(25) +The set S that the algorithm selects in the end will be the set with the highest mean (based +on surrogate function ˆf) among all those evaluated (both sets in the greedy process and +their augmentations). Also, its observed value ˆf(Sℓ) is at most ϵ above f(S). Thus +f(S) ≥ ˆf(S) − ϵ +≥ ˆf(Gℓ ∪ aℓ) − ϵ +≥ 1 +2f(OPT) − +� +˜K + 1 +2 + γ +� +ϵ. +(using (25)) +Case 2(a): If max{0, ˆρ(gℓ+1|Gℓ)}(B−c(o1)) ≥ 1 +2f(OPT)−( ˜K− 1 +2−γ)ϵ and ˆρ(gℓ+1|Gℓ) > +0, rearranging we have +ˆρ(gℓ+1|Gℓ) ≥ +f(OPT) +2(B − c(o1)) − ( ˜K − 1 +2 − γ)ϵ +B − c(o1) +. +(26) +23 + +Then, +ˆf(Gℓ) = ˆf(Gℓ) − ˆf(Gℓ−1) + ˆf(Gℓ−1) + · · · − ˆf(G1) + ˆf(G1) − ˆf(G0) +(telescoping sum; G0 = ∅, ˆf(G0) := 0) += +l−1 +� +j=1 +ˆf(gj+1|Gj) +(Definition of ˆf(·|·)) += +l−1 +� +j=0 +ˆρ(gj+1|Gj)c(gj+1) +(Definition of ˆρ(·|·)) +≥ +l−1 +� +j=0 +ˆρ(gℓ+1|Gj)c(gj+1) +(greedy choice of gj+1) +≥ +l−1 +� +j=0 +� +ρ(gℓ+1|Gj) − +2ϵ +c(gℓ+1) +� +c(gj+1) +≥ +l−1 +� +j=0 +� +ρ(gℓ+1|Gℓ) − +2ϵ +c(gℓ+1) +� +c(gj+1) +(submodularity of f) += +� +ρ(gℓ+1|Gℓ) − +2ϵ +c(gℓ+1) +� +c(Gℓ) +(simplifying) +≥ +� +ˆρ(gℓ+1|Gℓ) − +4ϵ +c(gℓ+1) +� +c(Gℓ) +≥ ˆρ(gℓ+1|Gℓ)c(Gℓ) − 4βϵ. +(27) +Recalling that ℓ is chosen as the index of the last greedy set that has a remaining budget +as big as the cost of the heaviest element in OPT, c(Gℓ) ≤ B − c(o1) < c(Gℓ+1), +ˆf(Gℓ+1) = ˆf(Gℓ ∪ gℓ+1) += ˆf(Gℓ) + c(gℓ+1)ˆρ(gℓ+1|Gℓ) +≥ +� +ˆρ(gℓ+1|Gℓ)c(Gℓ) − 4βϵ +� ++ c(gℓ+1)ˆρ(gℓ+1|Gℓ) +(from (27)) += ˆρ(gℓ+1|Gℓ)c(Gℓ+1) − 4βϵ +(simplifying) +≥ +1 +2f(OPT) − ( ˜K − 1 +2 − γ)ϵ +B − c(o1) +c(Gℓ+1) − 4βϵ +(case 2 condition) +≥ 1 +2f(OPT) − ( ˜K − 1 +2 − γ)ϵ − 4βϵ +(ℓ chosen so that c(Gℓ+1) > B − c(o1)) += 1 +2f(OPT) − +� +˜K − 1 +2 − γ + 4β +� +ϵ. +(28) +The set S that the algorithm selects at the end of the exploitation phase will be the set +with the highest empirical mean among all those explored (both sets in the greedy process +24 + +and augmented sets). Thus its empirical mean is at most ϵ above f(S). +f(S) ≥ ˆf(S) − ϵ +≥ ˆf(Gℓ+1) − ϵ +≥ 1 +2f(OPT) − +� +˜K + 1 +2 − γ + 4β +� +ϵ. +(using (28)) +Case 2(b): If max{0, ˆρ(gℓ+1|Gℓ)}(B−c(o1)) ≥ 1 +2f(OPT)−( ˜K− 1 +2−γ)ϵ and ˆρ(gℓ+1|Gℓ) ≤ +0, then the set S that the algorithm selects at the end satisfies +f(S) ≥ 0 +≥ 1 +2f(OPT) − ( ˜K − 1 +2 − γ)ϵ +(Case 2(b) condition) +≥ 1 +2f(OPT) − ( ˜K − 1 +2 − γ + 4β)ϵ. +Thus, combining cases 1 and 2, and selecting γ = 2β, the additive 1 +2-approximation error +we get by the modified Greedy+Max algorithm is at most (1 +2 + ˜K + 2β)ϵ, which concludes +the proof. +B.4 Proof for Robustness of Greedy+ +In this section, we prove Theorem 9 in Section 6 of the main paper. The following state- +ments, Lemmas 18,19 and 21, and their proofs are adapted from the proof of 1 +2(1 − 1 +e) +approximation ratio in the offline setting Khuller et al. (1999) using a value oracle. Krause +and Guestrin (2005) adapted the proof of Khuller et al. (1999) to an offline setting where +the greedy process relies on an exact oracle to evaluate individual element values and to +compare the best individual element to the set output by the greedy process, but use an +inexact value oracle (within ϵ of the correct value) to evaluate marginal densities. +The main differences arise from (i) the algorithms of Khuller et al. (1999); Krause +and Guestrin (2005) evaluate densities before checking for feasibility,2 leading to different +definitions of the augmented greedy sequence, necessitating us to use more care to show +analogous properties, (ii) exact value oracles for best individual elements and for selecting +OPT are used in Khuller et al. (1999); Krause and Guestrin (2005), simplifying work to +conclude the final bound for the approximation ratio α = 1 +2(1− 1 +e) and leading to a different +δ. +Recall that Theorem 9 in Section 6 of the main paper states that Greedy+ is a ( 1 +2(1 − +1 +e), 2+ ˜K +β)-robust approximation algorithm for submodular maximization problem under +a knapsack constraint. +We define Gi and gi the same as previous section. Recall that the greedy process (using +a surrogate ˆf) produces a nested sequence of subsets ∅ = G0 ⊂ G1 ⊂ · · · ⊂ GL, where +L denotes the cardinality of the set final output of the greedy process. For the proof, we +describe the greedy process as running for L + 1 iterations, though on the final iteration no +elements are added. +2. As noted in Footnote 1, concentration of estimates (i.e. the surrogate ˆf) used by C-ETC in the bandit +setting will only be for evaluated subsets, which by restriction will all be feasible. +25 + +For any action Gi−1 ∪a evaluated in iteration i of the greedy process, its marginal gains +are upper bounded by that of the best subset based on surrogate function ˆf, +f(Gi−1 ∪ a) − f(Gi−1) − 2ϵ +c(a) +≤ +ˆf(Gi−1 ∪ a) − ˆf(Gi−1) +c(a) +≤ +ˆf(Gi−1 ∪ gi) − ˆf(Gi−1) +c(gi) +(gi selected by greedy rule based on ˆf) +≤ f(Gi−1 ∪ gi) − f(Gi−1) + 2ϵ +c(gi) += f(Gi) − f(Gi−1) + 2ϵ +c(gi) +, +(29) +where (29) just uses the definition of Gi ← Gi−1 ∪ gi. We will use (29) to lower bound the +true marginal gains (i.e. in terms of f) achieved for each iteration of the greedy process. +Let ℓ ∈ {1, . . . , L + 1} denote the first iteration for which there was an element a′ ∈ +Ω\Gℓ−1 whose cost exceeds the remaining budget (c(a′)+c(Gℓ−1) > B) (thus subset Gℓ−1∪a′ +was not sampled), yet whose marginal density was higher than the marginal density of the +chosen element gℓ up to ±2ϵ normalized by the cost, specifically, for ℓ ≤ L, +f(Gℓ−1 ∪ a′) − f(Gℓ−1) − 2ϵ +c(a′) +> f(Gℓ−1 ∪ aℓ) − f(Gℓ−1) + 2ϵ +c(ar) +. +(30) +If there is no such iteration ℓ < L+1, then for ℓ = L+1, we take the element a′ maximizing +the term on the left hand side of (30), +a′ = arg max +a∈Ω\Gℓ−1 +f(Gℓ−1 ∪ a) − f(Gℓ−1) − 2ϵ +c(a) +. +(31) +Likewise, if there is more than one element satisfying (30) for some (earliest) iteration r, +then we also take the maximizer (31). +We define an “augmented” greedy sequence of length ℓ which matches the greedy se- +quence up to the set of cardinality ℓ, where the element a′ is selected despite violating the +budget, +{ �G0 = G0 = ∅, �G1 = G1, . . . , �Gℓ−1 = Gℓ−1, �Gℓ = Gℓ−1 ∪ {a′}} +(32) +and correspondingly enumerate the elements of �Gℓ in the order they were selected, +{�g1 = g1, . . . , �gℓ−1 = gℓ−1, �gℓ = g′}. +(33) +We first prove the following lemma, bounding the marginal gains of the augmented +greedy sequence { �G0, . . . , �Gℓ}. +Lemma 18 For all i ∈ {1, 2, · · · , ℓ}, the following inequality holds: +f( �Gi) − f( �Gi−1) ≥ c(�gi) +B +� +f(OPT) − f( �Gi−1) +� +− 2 +� +1 + +˜Kc(�gi) +B +� +ϵ. +26 + +Proof +Set any i ∈ {1, 2, · · · , ℓ}. +Let {v1, v2, . · · · , vk} = OPT \ �Gi−1. +Note that by +construction (32), we have �Gi−1 = Gi−1. +The difference f(OPT) − f( �Gi−1) can be bounded by the marginal gains of elements in +the set difference, +f(OPT) − f( �Gi−1) ≤ +k +� +j=1 +� +f( �Gi−1 ∪ vj) − f( �Gi−1) +� +(Fact 1) += +k +� +j=1 +� +f( �Gi−1 ∪ vj) − f( �Gi−1) − 2ϵ + 2ϵ +� += +k +� +j=1 +c(vj)f( �Gi−1 ∪ vj) − f( �Gi−1) − 2ϵ +c(vj) ++ 2kϵ +≤ +k +� +j=1 +c(vj)f( �Gi−1 ∪ �gi) − f( �Gi−1) + 2ϵ +c(�gi) ++ 2kϵ +(34) += +k +� +j=1 +c(vj)f( �Gi) − f( �Gi−1) + 2ϵ +c(�gi) ++ 2kϵ +(35) +where (34) holds by following. We consider four cases, depending on whether or not ˆf(Gi−1∪ +vj) was evaluated during the iteration i. +• Case 1 ( ˆf(Gi−1 ∪ vj) was evaluated and i < ℓ): At iteration i (necessarily i ≤ L +since no subsets were evaluated in iteration L + 1) with current greedy set Gi−1, +adding the element vj to the current greedy set was feasible, c(vj) ≤ B − c(Gi−1). +Then Greedy+ would have evaluated ˆf(Gi−1 ∪ vj). Since vj was not selected, the +chosen element gi = Gi\Gi−1 must have had a higher surrogate density ˆf(Gi−1∪vj) > +ˆf(Gi−1 ∪ gi), so for i < ℓ, for which �gi = gi by construction (33), (29) implies (34). +• Case 2 ( ˆf(Gi−1 ∪ vj) was evaluated and i = ℓ): By the reasoning in the previous +case, for the item aℓ chosen at iteration ℓ by the greedy process (due to feasibility and +having the highest surrogate density), we still have the bound (29) on true values, +which coupled with our specific construction of �gℓ (30) means +f( �Gℓ−1 ∪ vj) − f( �Gℓ−1) − 2ϵ +c(vj) +≤ f( �Gℓ−1 ∪ ar) − f( �Gℓ−1) + 2ϵ +c(ar) +(by (29)) +< f( �Gℓ−1 ∪ �gr) − f( �Gℓ−1) − 2ϵ +c(�gr) +(by construction (30)) +< f( �Gℓ−1 ∪ �gr) − f( �Gℓ−1) + 2ϵ +c(�gr) +. +• Case 3 ( ˆf(Gi−1 ∪ vj) was not evaluated and i < ℓ): At iteration i < ℓ ≤ L + 1 +with the current greedy set Gi−1, adding the element vj to the current greedy set was +27 + +not feasible, c(vj) > B −c(Gi−1). By construction of the augmented greedy sequence, +only at iteration ℓ was there an infeasible element whose surrogate marginal density +satisfied the inequality (30). Thus, for iterations i < ℓ, Gi−1 = �Gi−1 and Gi = �Gi, so +(34) holds. +• Case 4 ( ˆf(Gi−1 ∪ vj) was not evaluated and i = ℓ): For iteration i = ℓ, with +current greedy set Gi−1, the augmented greedy sequence construction implies (34). +Namely, with i = ℓ, +f( �Gℓ−1 ∪ vj) − f( �Gℓ−1) − 2ϵ +c(vj) +< f( �Gℓ−1 ∪ �gr) − f( �Gℓ−1) − 2ϵ +c(�gr) +(by (31)) +< f( �Gℓ−1 ∪ �gr) − f( �Gℓ−1) + 2ϵ +c(�gr) +. +menaing (34) holds. +We now continue lower bounding f(OPT) − f( �Gi−1), +f(OPT) − f( �Gi−1) ≤ +� +� +k +� +j=1 +c(vj)f( �Gi) − f( �Gi−1) + 2ϵ +c(�gi) +� +� + 2kϵ +(copying (35)) += +� +� +k +� +j=1 +c(vj) +� +� f( �Gi) − f( �Gi−1) + 2ϵ +c(�gi) ++ 2kϵ +≤ B f( �Gi) − f( �Gi−1) + 2ϵ +c(�gi) ++ 2kϵ +(OPT is feasible, so �k +j=1 c(vj) ≤ B) +≤ +B +c(�gi) +� +f( �Gi) − f( �Gi−1) +� ++ 2 +� B +c(�gi) + ˜K +� +ϵ. +(rearranging; k ≤ ˜K) +Multiplying both sides by c(�gi) +B +and rearranging finishes the proof. +We unravel the recurrence in Theorem 18 to lower bound f( �Gi). +Lemma 19 For all i ∈ {1, 2, · · · , ℓ}, +f( �Gi) ≥ +� +�1 − +i� +j=1 +(1 − c(�gj) +B +) +� +� f(OPT) − 2(β + ˜K)ϵ. +Remark 20 The steps to unravel the recurrence to obtain the first term (coefficient of +f(OPT)) is the same as the proof for the analogous result in the offline setting Khuller +et al. (1999). The second term (with ϵ) is due to working with marginal densities of a +28 + +surrogate function ˆf. The basic steps for working with that second term is the same as +Krause and Guestrin (2005), though we use a looser bound β; in Krause and Guestrin +(2005) we think there may be a mistake in applying the induction step (with “c(Xi)” fixed +for different i in the proof), though they were loosely bounded with β later on. +Proof +The proof will follow by induction. +We first show the base case i = 1 using +Theorem 18. +f( �G1) = f( �G1) − f( �G0) +(f is normalized; �G0 = ∅) +≥ c(�g1) +B +� +f(OPT) − f( �G0) +� +− 2 +� +1 + +˜Kc(�g1) +B +� +ϵ +(using Theorem 18) += +� +1 − +� +1 − c(�g1) +B +�� +f(OPT) − 2 +� +1 + +˜Kc(�g1) +B +� +ϵ +(36) +where (36) follows from rearranging. For the second term in (36), using that +1 + +˜Kc(�g1) +B +≤ +B +c(�g1) +� +1 + +˜Kc(�g1) +B +� +(since +B +c(�g1) ≥ 1) += +B +c(�g1) + ˜K +≤ +B +cmin ++ ˜K += β + ˜K, +(37) +then +f( �G1) ≥ +� +1 − +� +1 − c(�g1) +B +�� +f(OPT) − 2 +� +1 + +˜Kc(�g1) +B +� +ϵ +(copying (36)) +≥ +� +1 − +� +1 − c(�g1) +B +�� +f(OPT) − 2(β + ˜K)ϵ. +(using (37)) +This completes the base case of i = 1. +29 + +We next consider i > 1. Unraveling the recurrence shown in Theorem 18, +f( �Gi) = f( �Gi) − f( �Gi−1) + f( �Gi−1) +≥ +� +c(�gi) +B +� +f(OPT) − f( �Gi−1) +� +− 2 +� +1 + +˜Kc(�gi) +B +� +ϵ +� ++ f( �Gi−1) +(using Theorem 18) += +�c(�gi) +B +� +f(OPT) − 2 +� +1 + +˜Kc(�gi) +B +� +ϵ + +� +1 − c(�gi) +B +� +f( �Gi−1) +(rearranging) += +� +1 − (1 − c(�gi) +B ) +� +f(OPT) − 2 +� +1 + +˜Kc(�gi) +B +� +ϵ ++ +� +1 − c(�gi) +B +� +f( �Gi−1) +(rearranging) +≥ +� +1 − (1 − c(�gi) +B ) +� +f(OPT) − 2 +� +1 + +˜Kc(�gi) +B +� +ϵ ++ +� +1 − c(�gi) +B +� � +� +� +�1 − +i−1 +� +j=1 +(1 − c(�gj) +B +) +� +� f(OPT) − 2(β + ˜K)ϵ +� +� +(induction step) += +� +�1 − (1 − c(�gi) +B ) + +� +1 − c(�gi) +B +� � +�1 − +i−1 +� +j=1 +(1 − c(�gj) +B +) +� +� +� +� f(OPT) +− 2 +� +1 + +˜Kc(�gi) +B ++ +� +1 − c(�gi) +B +� +(β + ˜K) +� +ϵ +(rearranging) += +� +�1 − +i� +j=1 +(1 − c(�gj) +B +) +� +� f(OPT) +− 2 +� +1 + β − β c(�gi) +B ++ ˜K +� +ϵ. +(38) +For the second term in (38), using that +β c(�gi) +B += +B +cmin +c(�gi) +B +(def. of β) += c(�gi) +cmin +≥ 1, +(39) +then +−2 +� +1 + β − β c(�gi) +B ++ ˜K +� +ϵ = −2 +� +β + ˜K +� +ϵ + 2 +� +β c(�gi) +B +− 1 +� +ϵ +(rearranging) +≥ −2 +� +β + ˜K +� +ϵ. +(using (39)) +30 + +Applying this to (38) completes the proof. +The inequality in Theorem 19 for the augmented greedy set of cardinality ℓ can be +further simplified. We will use the following observations. +Lemma 21 The following inequality holds: +f( �Gℓ) ≥ (1 − 1 +e)f(OPT) − 2(β + ˜K)ϵ. +Proof Applying i = ℓ to Theorem 19 and bounding the coefficient for f(OPT), +f( �Gℓ) ≥ +� +�1 − +ℓ� +j=1 +(1 − c(�gj) +B +) +� +� f(OPT) − 2(β + ˜K)ϵ +≥ +� +�1 − +ℓ� +j=1 +(1 − c(�gj) +c( �Gℓ) +) +� +� f(OPT) − 2(β + ˜K)ϵ +(by construction, c( �Gℓ) > B) +≥ +� +�1 − +ℓ� +j=1 +(1 − c( �Gℓ)/ℓ +c( �Gℓ) +) +� +� f(OPT) − 2(β + ˜K)ϵ +(using Fact 2) += +� +1 − (1 − 1 +ℓ )ℓ +� +f(OPT) − 2(β + ˜K)ϵ +(simplifying) +≥ +� +1 − 1 +e +� +f(OPT) − 2(β + ˜K)ϵ. +(using Fact 3) +Using the aforementioned lemmas, we are now ready to complete the proof for Theorem +3 (robustness of Greedy+ algorithm). We will bound the value of set GL using the results +on the augmented greedy set (32) of cardinality ℓ, and in turn bound the value of the set +S, the final output of Greedy+. +Recall that Greedy+ chooses the set S to be either the best individual element +(based on ˆf) a∗ ← arg maxe∈Ω ˆf(e) or the output of the greedy process GL. Let aOPT = +arg maxe∈Ω f(e) denote the element with the highest value under f. Then +f(a∗) ≥ ˆf(a∗) − ϵ +≥ ˆf(aOPT) − ϵ +(by definition of a∗) +≥ f(aOPT) − 2ϵ. +(40) +By construction (32), �Gℓ includes one more element a′ than �Gℓ−1 (and a′ maximizes +(31)). By submodularity, the marginal gain of a′ is bounded by f(a′) and in turn by the +31 + +best individual element based on surrogate function ˆf, +f( �Gℓ−1) + f(aOPT) ≥ f( �Gℓ−1) + f(a′) +(by definition of aOPT) +≥ f( �Gℓ−1) + +� +f( �Gℓ−1 ∪ a′) − f( �Gℓ−1) +� +(by submodularity) += f( �Gℓ−1 ∪ a′) += f( �Gℓ) +(by construction (32)) +≥ (1 − 1 +e)f(OPT) − 2(β + ˜K)ϵ, +(41) +where (41) follows from Theorem 21. +Also by construction (32), the greedy and augmented greedy processes match up to and +including the set of cardinality ℓ − 1, so +f(GL) ≥ f(Gℓ−1) +(monotonicity) += f( �Gℓ−1). +(By construction (32)) +Thus, +f(GL) + f(aOPT) ≥ f( �Gℓ−1) + f(aOPT) +≥ (1 − 1 +e)f(OPT) − 2(β + ˜K)ϵ. +(using (41)) +At least one of f(GL) and f(aOPT) is at least half of the value of the right hand side, +max{f(GL), f(aOPT)} ≥ 1 +2(1 − 1 +e)f(OPT) − (β + ˜K)ϵ +(42) +Thus, for the chosen set S +f(S) ≥ ˆf(S) − ϵ += max{ ˆf(GL), ˆf(a∗)} − ϵ +≥ max{ ˆf(GL), ˆf(aOPT)} − ϵ +(a∗ is the element with largest ˆf value) +≥ max{f(GL) − ϵ, f(aOPT) − ϵ} − ϵ +(element-wise dominance) += max{f(GL), f(aOPT)} − 2ϵ +≥ 1 +2(1 − 1 +e)f(OPT) − (β + ˜K)ϵ − 2ϵ +(from (42)) += 1 +2(1 − 1 +e)f(OPT) − (2 + β + ˜K)ϵ. +which completes the proof. +B.5 Proof for Robustness of PartialEnumeration +Now we analyze the PartialEnumeration algorithm for submodular maximization under +a knapsack constraint proposed in Sviridenko (2004); Khuller et al. (1999). Recall that +Theorem 7 in Section 6 of the main paper states PartialEnumeration is a (1 − 1 +e, 4 + +32 + +2 ˜K + 2β)-robust approximation algorithm for submodular maximization under a knapsack +constraint. +Proof Assume |OPT| > 3, otherwise the algorithm finds a (1, 2)-robust approximation, so it +is also a (1− 1 +e, 2( ˜K+β))-robust approximation for non-trivial cases where ˜K ≥ 1 and β ≥ 1. +Enumerate the elements of the optimal solution as OPT = {Y1, · · · , Ym}, corresponding to +the order they would be selected by the simple greedy algorithm (iteratively selecting the +element with the largest marginal gain, not the largest marginal density) +Yi+1 = arg max +Y ∈OPT +f({Y1, · · · , Yi, Y }) − f({Y1, · · · , Yi}), +(43) +and let R = {Y1, Y2, Y3}. Consider the iteration where the algorithm considers R. Define +the function +f′(A) = f(A ∪ R) − f(R). +(44) +f′ is a non-decreasing submodular set function with f′(∅) = 0, and the optimal solution +(with budget B − c(R)) is OPT \ R since for any set S with cost c(S) ≤ B − c(R), +f′(OPT \ R) = f(OPT ∪ R) − f(R) +(def of f′) += f(OPT) − f(R) +(R ⊆ OPT by construction) +≥ f(S ∪ R) − f(R) += f′(S). +Hence we can apply Greedy+ algorithm to f′ (based on noisy evaluations). Let gℓ be the +first element from OPT \ R which could not be added due to budget constraints, and let +A = {g1, · · · , gℓ−1} be first ℓ−1 elements selected by Greedy+ algorithm. Let G = A∪R. +Using Theorem 21, we get +f′(A ∪ gℓ) ≥ (1 − 1 +e)f′(OPT \ R) − 2(β′ + ˜K′)ϵ, +where β′ = B−c(R) +c′ +min , ˜K′ = min{n − 3, β′} and c′ +min = mine∈Ω\R c(e). Simple calculation can +show that β′ ≤ β and ˜K′ ≤ ˜K. Thus, +f′(A ∪ gℓ) ≥ (1 − 1 +e)f′(OPT \ R) − 2(β + ˜K)ϵ, +From the definition of f′, we have f(G) = f′(A) + f(R). Let ∆ = f′(A ∪ gℓ) − f′(A). We +have +f′(A) + ∆ ≥ (1 − 1 +e)f′(OPT \ R) − 2(β + ˜K)ϵ. +(45) +Further observe that elements in OPT are ordered that for all 1 ≤ i ≤ 3, +f({Y1, · · · , Yi}) − f({Y1, · · · , Yi−1}) +≥f({Y1, · · · , Yi−1, gℓ}) − f({Y1, · · · , Yi−1}) +(ordering rule) +≥f(R ∪ A ∪ gℓ) − f(R ∪ A) +({Y1, · · · , Yi−1} ⊆ R when 1 ≤ i ≤ 3 and submodularity) +=f(R ∪ A ∪ gℓ) − f(R) − (f(R ∪ A) − f(R)) +=f′(A ∪ gℓ) − f′(A) +=∆. +33 + +By telescoping sum, f(R) ≥ 3∆. Now we get +f(G) = f(R) + f′(A) +≥ f(R) + (1 − 1 +e)f′(OPT \ R) − 2(β + ˜K)ϵ − ∆ +≥ f(R) + (1 − 1 +e)f′(OPT \ R) − 2(β + ˜K)ϵ − f(R)/3 +≥ (1 − 1 +3)f(R) + (1 − 1 +e)f′(OPT \ R) − 2(β + ˜K)ϵ +≥ (1 − 1 +e) +� +f′(OPT \ R) + f(R) +� +− 2(β + ˜K)ϵ +(e ≤ 3) += (1 − 1 +e)f(OPT) − 2(β + ˜K)ϵ. +(definition of f′) +The output of the algorithm is not necessarily G because the values of the evaluated triplets +are based on surrogate function ˆf. Denote O as the output of the algorithm and denote G′ +as the best evaluated set (with respect to ˆf) with size ℓ + 2 (same as G). We must have +that ˆf(G′) ≥ ˆf(G). Also denote the final set (until violating budget) continuing G′ as G′′. +We have, +f(O) ≥ ˆf(O) − ϵ +≥ ˆf(G′′) − ϵ +(selection rule of the algorithm) +≥ f(G′′) − 2ϵ +≥ f(G′) − 2ϵ +(G′ ⊆ G′′ and monotonicity of f) +≥ ˆf(G′) − 3ϵ +≥ ˆf(G) − 3ϵ +≥ f(G) − 4ϵ +≥ (1 − 1 +e)f(OPT) − (4 + 2β + 2 ˜K)ϵ, +finishing the proof. +C. Proof for Regret of C-ETC +In this section, we prove Theorem 3 in Section 4 of the main paper. We restate the theorem: +For the sequential decision making problem defined in Section 2 and T ≥ 2 +√ +2N +δ +, the expected +cumulative α-regret of C-ETC using an (α, δ)-robust approximation algorithm as subroutine +is at most O +� +δ +2 +3 N +1 +3 T +2 +3 log(T) +1 +3 +� +, where N upper-bounds the number of value oracle queries +made by the offline algorithm A. +C.1 Overview and Notations +We will separate the proof into two cases. The first case is for when the clean event E +happens, which we will show in Theorem 24 happens with high probability. Under the +34 + +clean event, using the fact that the offline algorithm is an (α, δ)-robust approximation, C- +ETC’s chosen set S for the exploitation phase will nonetheless be near-optimal. The second +case is when the complementary event happens, which occurs with low probability. +The proof structure analyzing a high-probability “clean event” where empirical estimates +are sufficiently concentrated around their means is analogous to that for the unstructured +non-combinatorial setting (see for instance, Section 1.2 in (Slivkins, 2019)). However, un- +like the ETC procedure for non-combinatorial MAB problems, C-ETC makes sequences of +decisions during exploration. Furthermore, the combinatorial action space, non-linearity +of the reward function, and lack of extra feedback (like marginal gains) make the problem +challenging. Even in the special setting of deterministic rewards, the standard MAB prob- +lem becomes trivial (finding the largest of n base arms) while the problem we considered +are NP-hard. +Recap that for any (feasible) action A, ft(A) denotes a (random) reward at time t for the +agent taking that action, f(A) denotes the expected value for action A. Let ¯ft(A) denote +the empirical mean of rewards received from playing action A up to and including time t. In +the following, we will drop the subscript t from the empirical mean, writing ¯f(A) when it is +clear from context that action A has been played m times. Also, we use Ai, i ∈ {1, · · · , N} +denotes the i-th action the algorithm samples. We further denote Ti, i ∈ {1, . . . , N} as the +time step when the sampling of the i-th action has been determined, or Ai has been played +m times. For notation consistency, we also denote T0 = 0 and TN+1 = T. +C.2 Probability of the Clean Event +Now we define events that are important in our analysis. Recall that for each action A being +explored, the m rewards are i.i.d. with mean f(A) and bounded in [0, 1]. Thus, we can +bound the deviation of the (unbiased) empirical mean ¯f(Ai) from the expected value f(Ai) +for each action played. Specifically, we can use a two-sided Hoeffding bound for bounded +variables. +Remark 22 For convenience, we assume the reward function bounded in [0, 1], but the +result can be generalized to the case where the deviation of the true reward and the expected +reward has a light tailed distribution (e.g., sub-Gaussian). +Lemma 23 (Hoeffding’s inequality) Let X1, · · · , Xn be independent random variables +bounded in the interval [0, 1], and let ¯X denote their empirical mean. Then we have for any +ϵ > 0, +P +��� ¯X − E[ ¯X] +�� ≥ ϵ +� +≤ 2exp +� +−2nϵ2� +. +(46) +By C-ETC, each sampled action will be played the same number of times, denoted by m, +so we consider bounding the probabilities of equal-sized confidence radii rad := +� +log(T)/2m +for all the actions played during exploration. +We next analyze the probability of the event that the empirical means of all actions +played during exploration are concentrated around their statistical means within a radius +rad. Denote the corresponding events for each action played having empirical means con- +centrated around their respective statistical means as Ei, +Ei := +� +{ +�� ¯f(Ai) − f(Ai) +�� < rad}, +i ∈ {1, · · · , N}. +(47) +35 + +Define the clean event E to be the event that the empirical means of all actions played in +the exploration phase are within rad of their corresponding statistical means: +E := E1 ∩ · · · ∩ EN. +(48) +Lemma 24 The probability of the clean event E (48) satisfies: +P(E) ≥ 1 − 2N +T . +Proof +Applying the Hoeffding bound Theorem 23 to the empirical mean ¯f(Ai) of m +rewards for action Ai and choosing ϵ = rad = +� +log(T)/2m gives +P( ¯Ei) = P +��� ¯f(Ai) − f(Ai) +�� ≥ rad +� +≤ 2exp +� +−2mrad2� += 2exp (−2m(log(T)/2m)) += 2exp (− log(T)) += 2 +T . +(49) +Then, we can bound the probability of clean events +P(E) = P(E1 ∩ · · · ∩ EN) += 1 − P( ¯E1 ∪ · · · ∪ ¯EN) +(De Morgan’s Law) +≥ 1 − +N +� +i=1 +P( ¯Ei) +(union bounds) +≥ 1 − 2N +T . +(using (49)) +C.3 Near Optimality of the final S (Exploitation Phase Action) +In Theorem 24, we showed that the clean event E will happen with high probability. When +the clean event E happens, we have | ¯f(A) − f(A)| ≤ rad for all evaluated action A. For an +online algorithm (with output S) using an (α, δ)-robust approximation as subroutine, we +have +f(S) ≥ αf(OPT) − δ · rad. +(50) +C.4 Final Regret +Now we are ready to show the regret of C-ETC (Theorem 3 in Section 4 of the main paper). +36 + +Case 1: clean event E happens +In the first case we analyse the expected regret under the condition that the clean event E +happens. In this section, all expectations will be conditioned on E, but to simplify notation +we will write E[·] instead of E[·|E] in some cases. +First we can break up the expected α-regret conditioned on E into two parts, one for +the first L exploration iterations, and the second for the exploitation iteration. Although +the number of actions taken per iteration and the number of iterations of the greedy is not +known a priori, we can upper bound the duration. Also recall that ft(At) is the random +reward for taking action At, which itself is random, depending on empirical means of actions +in earlier iterations. +E[R(T)|E] = αTf(OPT) − +T +� +t=1 +E[ft(At)] += αTf(OPT) − +T +� +t=1 +E[E[ft(At)|At]] +(law of total expectation) += αTf(OPT) − +T +� +t=1 +E[f(At)] +(f(·) defined as expected reward) += +T +� +t=1 +(αf(OPT) − E[f(At)]) +(rearranging) += +N +� +i=1 +m (αf(OPT) − E[f(Ai)]) +� +�� +� +Exploration phase ++ +T +� +t=TN+1 +(αf(OPT) − E[f(At)]) +� +�� +� +Exploitation phase += +N +� +i=1 +m (αf(OPT) − E[f(Ai)]) + +T +� +t=TN+1 +(αf(OPT) − E[f(S)]) . +(51) +Case 1 (clean event): Bounding exploration regret: +We will separately bound the +regret incurred from the exploration and exploitation. We begin with bounding regret from +exploration, +N +� +i=1 +m (αf(OPT) − E[f(Ai)]) +≤ +N +� +i=1 +m (α − 0) +(rewards are bounded in [0, 1]) +≤ Nm. +(52) +Case 1 (clean event): Bounding exploitation regret: +We next bound the regret +incurred during the exploitation iteration. Since the set S used during exploitation is a +random variable, we can take the expectation of (50) (conditioned on event E), to bound +37 + +the expected instantaneous regret for each time step of the exploitation iteration, +αf(OPT) − E[f(S)] ≤ δrad. +(53) +Using a loose bound for the duration of the exploitation iteration, T − TL + 1 < T, +T +� +t=TN+1 +(αf(OPT) − E[f(S)]) ≤ +T +� +t=TN+1 +δrad +(using (53)) +≤ Tδrad. +(54) +Case 1 (clean event): Bounding total regret: +Then the expected cumulative regret +(51) can be bounded as +E[R(T)|E] = +N +� +i=1 +m (αf(OPT) − E[f(Ai)]) + +T +� +t=TN+1 +(αf(OPT) − E[f(S)]) (copying (51)) +≤ Nm + Tδrad +(using (52), (54)) +Plugging in the formula for the confidence radius rad = +� +log(T)/2m, we have +E[R(T)|E] ≤ Nm + Tδ +� +log(T)/2m +We want to optimize m, the number of times each action is played. Denoting the regret +bound (55) as a function of m +g(m) = Nm + Tδ +� +log(T)/2m, +(55) +then +g′(m) = N − 1 +2Tδ +� +log(T)/2m−3/2. +(56) +Setting g′(m) = 0 and solving for m, +m∗ = δ2/3T 2/3 log(T)1/3 +2N2/3 +. +(57) +We next check the second derivative, +g′′(m) = 3 +4δT +� +log(T)/2m−5/2. +(58) +For positive values of m, g′′(m) > 0, thus g(m) reaches a minimum at (57). +Since m is the number of times actions are played, we (trivially) need m ≥ 1 and m to +be an integer. We choose +m† = +� +δ2/3T 2/3 log(T)1/3 +2N2/3 +� +. +(59) +38 + +Since from (58) we have that g′′(m) > 0 for positive m, g(m∗) ≤ g(m†). For T ≥ 2 +√ +2N +δ +, +we have m∗ ≥ 1. +Plugging (59) back in to (55), +E[R(T)|E] ≤ m†N + Tδ +� +log(T)/2m† +((55) with m† samples for each action) += ⌈m∗⌉N + Tδ +� +log(T)/2⌈m∗⌉ +≤ ⌈m∗⌉N + Tδ +� +log(T)/2m∗ +(Since ⌈m∗⌉ ≥ m∗) +≤ 2m∗N + Tδ +� +log(T)/2m∗ +(Since m∗ ≥ 1, ⌈m∗⌉ ≤ 2m∗) += 2δ2/3T 2/3 log(T)1/3 +2N2/3 +N ++ Tδ +� +log(T)/2 +� +δ2/3T 2/3 log(T)1/3 +2N2/3 +�−1/2 +(using (57)) += 3δ2/3N1/3T 2/3 log(T)1/3 +(60) += O +� +δ +2 +3 N +1 +3 T +2 +3 log(T) +1 +3 +� +. +In conclusion, the expected α-regret of C-ETC using an (α, δ)-robust approximation as +subroutine is upper bounded by (60) if the clean event E happens. +Case 2: clean event E does not happen +We next derive an upper bound for the expected α-regret for case that the event E does +not happen. By Theorem 24, +P( ¯E) = 1 − P(E) ≤ 2N +T . +Since the reward function ft(·) is upper bounded by 1, the expected α-regret incurred under +¯E for a horizon of T is at most T, +E[R(T)| ¯E] ≤ T. +(61) +Putting it all together +Combining Cases 1 and 2 we have, +E[R(T)] = E[R(T)|E] · P(E) + E[R(T)| ¯E] · P( ¯E) +(Law of total expectation) +≤ 3δ2/3N1/3T 2/3 log(T)1/3 · 1 + T · 2N +T +(using (60), Theorem 24, and (61)) += O +� +δ +2 +3 N +1 +3 T +2 +3 log(T) +1 +3 +� +. +This concludes the proof. +39 + +Algorithm 2 Online Greedy for Opaque Feedback Model (OGo) +Input: set of base arms Ω, horizon T, cost for each arm c(a), budget B +Initialize n ← |Ω|, cmin ← mina∈Ω{c(a)}, β ← +B +cmin , γ ← n1/3β +� +log(n) +T +�1/3 +, ϵ ← +� +β log(n) +γT +Initialize ω1 ← ones(β, n) +for t ∈ [1, · · · , T] do +St ← ∅ +l ← zeros(β, n) +// loss +Randomly sample a value ξ ∼ Uniform([0, 1]) +if ξ ≤ γ then +e ∼ Uniform({1, · · · , β}) +for i ∈ [1, · · · , e − 1] do +// For experts before e, exploit +Select an arm a with probability +ωt[i,a] +� ωt[i,:], re-sample if a ∈ St +St ← St ∪ {a} with probability cmin +c(a) ; St ← St−1 otherwise +end for +a ∼ Uniform({1, · · · , n}\St) +// For expert e, explore +St ← St ∪ {a} +Play action St, observe ft(St) +Update l[i, j] ← cminft(St) +c(a) +for all i = e and j ̸= a +// Feed cminft(St) +c(a) +back to expert +e associated with action a +Update ωt+1[i, j] ← ωt[i, j] exp(−ϵl[i, j]) for all pairs of i and j +else +// Exploitation with probability 1 − γ +for i ∈ [1, · · · , β] do +// For experts before e, exploit +Select arm a with probability +ωt[i,a] +� ωt[i,:], re-sample if a ∈ St +St ← St ∪ {a} with probability cmin +c(a) ; St ← St−1 otherwise +end for +Play action St, observe ft(St) +ωt+1[i, j] ← ωt[i, j] +// Since feeding back 0 to all expert-action payoffs, loss is 0, +no update +end if +end for +D. Implementation of Algorithm OGo +In this section we describe implementation details and parameter selection for OGo +algorithm Streeter and Golovin (2008). The choice of exploration probability is given by +the original paper:γ = n1/3β +� +log(n) +T +�1/3 +, where β = B/cmin. Note that in the original paper, +B is used instead of β, because they assume the minimum cost is 1. Here we generalize it +to arbitrary non-negative costs. ϵ is the learning rate for Randomized Weighted Majority +(WMR) expert algorithm Arora et al. (2012). It is chosen by setting the derivative of regret +40 + +upper bound to zero, which is ϵ = +� +log(n) +Te , where Te is the time spent on updating expert +e. Since it explores with probability γ, and there are β expert algorithms, we have Te ≈ γT +β . +Thus we pick ϵ = +� +β log(n) +γT +. In experiments, there are many cases the chosen γ is large or +even larger than 1, so we cap the probability of exploring γ by 1/2 to avoid exploring too +much. Note that unlike a hard budget in our setting, for OGo, it only requires the budget +to be satisfied in expectation, so in general we might choose sets over budget. Algorithm 2 +is the pseudo code for implementation details of OGo. +E. Comments on Lower bounds of Submodular CMAB +For the setting we explore in this paper, with stochastic (or even adversarial) knapsack- +constrained combinatorial MAB with submodular expected rewards and just bandit feed- +back, it remains an open question if ˜O(T 1/2) expected cumulative α-regret is possible (ig- +noring n and β). Both Streeter and Golovin (2008) and Niazadeh et al. (2021) analyze lower +bounds for the adversarial setting. However, Streeter and Golovin (2008) obtain bounds +for 1-regret (it is NP-hard in offline setting to obtain an approximation ratio better than +1 − 1/e). Niazadeh et al. (2021) obtain ˜Ω(T 2/3) lower bounds for the harder setting where +feedback is only available during “exploration” rounds chosen by the agent, who incurs an +associated penalty. +F. Dealing with Small Time Horizons in Experiments +In Section 6, we used N = ˜Kn as an upper bound on the number of function evaluations for +both C-ETC-K and C-ETC-Y, where n is the number of base arms and ˜K is an upper bound +of the cardinality of any feasible sets. When the time horizon T is small, it is possible that +the exploration phase will not finish due to the formula being optimized for m (the number +of plays for each action queried by A) uses a loose bound on the exploitation time. When +this is the case, we select the largest m (closest to the formula) for which we can guarantee +that exploration will finish. Recall that for C-ETC-Y and C-ETC-K, the number of oracle +calls can only be upper bounded in advance. +We first calculate m† using (59): +m† = +� +δ2/3T 2/3 log(T)1/3 +2 ˜K2/3n2/3 +� +. +Note that a (slightly tighter) upper bound on the number of subsets evaluated during the +exploration phase (with ˜K bounding the number of iterations of the greedy process) is +N ≤ n + (n − 1) + · · · + (n − ˜K + 1) += +� +n − +˜K +2 + 1 +2 +� +˜K. +We compare +� +n − ˜ +K +2 + 1 +2 +� +˜Km† with T. +41 + +• Case 1. If +� +n − ˜ +K +2 + 1 +2 +� +˜Km† < T, C-ETC can finish exploring. We select m = m†. +• Case 2. If +� +n − ˜ +K +2 + 1 +2 +� +˜Km† ≥ T, it is possible that the algorithm cannot finish +exploring. In this case, we will find a new m, so that the exploration can be guaranteed +to finish. We select the largest m (closest to m†) so that the exploration time is upper +bounded by T, +m = +T +� +n − ˜ +K +2 + 1 +2 +� +˜K +. +G. Basic Facts +Fact 1 For a monotonically non-decreasing submodular set function f defined over subsets +of Ω, we have for arbitrary subsets A, B ⊆ Ω, +f(B) − f(A) ≤ +� +j∈B\A +[f(A ∪ {j}) − f(A)] . +Fact 2 (Khuller et al., 1999) +For x1, · · · , xn ∈ R+ such that � xi = A, the function +[1 − �n +i=1(1 − xi +A )] achieves its minimum at x1 = x2 = · · · = xn = A/n. +Fact 3 For k ≥ 1, +1 − +� +1 − 1 +k +�k +≥ 1 − 1 +e. +H. Other Related Work for Adversarial CMAB with Knapsack +constraints +Streeter and Golovin (2008) propose and analyze an algorithm for adversarial CMAB with +submodular rewards, full-bandit feedback, and under a knapsack constraint (though only +in expectation, taken over randomness in the algorithm). We discuss this in more detail in +the supplemental material, here only highlighting a few key points. We also use this as a +baseline in our experiments in Section 7. The authors adapted a simpler greedy algorithm +than the one we adapt (Khuller et al., 1999), using an ϵ-greedy exploration type framework. +We provide evidence in our experiments that their algorithm requires large horizons to +learn. The offline algorithm they adapted achieves an approximation ratio (1 − 1/e) for +budgets that exactly match the cost used up by the greedy solution, but otherwise does not +achieve a constant approximation (Khuller et al., 1999). +In (Golovin et al., 2014), the authors propose an algorithm for adversarial setting with +submodular rewards when there is a matroid constraint (neither knapsack nor matroid +constraints are special cases of the other). +I. Related work on Stochastic Submodular CMAB with Semi-Bandit +Feedback +There are also a number of works that require additional “semi-bandit” feedback. +For +combinatorial MAB with submodular rewards, a common type of semi-bandit feedback are +42 + +marginal gains (Lin et al., 2015; Yue and Guestrin, 2011b; Yu et al., 2016; Takemori et al., +2020b), which enable the learner to take actions of maximal cardinality or budget, receive +a corresponding reward, and gain information not just on the set but individual elements. +For the full-bandit setting we consider, to greedily build a solution, we need to spend time +taking small cardinality actions to estimate their quality, incurring regret. +J. Experiments with Song Recommendation +We test our methods on the application of song recommendation on the Million Song Dataset +Bertin-Mahieux et al. (2011). In this problem, the agents aims to recommend a bundle of +songs to users such that they are liked by as many users as possible. +Data Set Description and Experiment Details +From the Million Song Dataset, we extract most popular 20 songs and 100 most active +users. As in Yue and Guestrin (2011a), we model the system as having a set of topics +(or genres) G with |G| = d and for each item e ∈ Ω, there is a feature vector x(e) := +(Pg(e))g∈G ∈ Rd that represents the information coverage on different genres. For each +genre g, we define the probabilistic coverage function fg(S) by 1 − � +e∈S (1 − Pg(e)) and +define the reward function f(S) = � +i wifi(S) with linear coefficients wi. The vector w := +[w1, . . . , wd] represents user preference on genres. In calculating Pg(e) and w, we use the +same formula for calculating ¯w(e, g) and θ∗ in Hiranandani et al. (2020). Like Takemori +et al. (2020a), we define the cost of a song by its length (in seconds). For each user, the +stochastic rewards of set S are sampled from a Bernoulli distribution with parameter f(S). +For the total reward, we take the average over all users. When making the plots, we use +statistics taken from 10 runs. +Results and Discussion +Figures 2a and 2b show average cumulative regret curves for C-ETC-K (in blue), C- +ETC-Y (in orange) and OGo (in green) for different horizon T values when the budget +constraint B is 500 and 800, respectively. Figures 2c and 2d are the instantaneous reward +plots over a single horizon T = 215, 443. Again, C-ETC significantly outperforms OGo for +all time horizons and budget considered. We again estimated the slopes for both methods +on log-log scale plots. Over the horizons tested, OGo’s cumulative regret (averaged over +ten runs) has a growth rate above 0.85. The growth rates of C-ETC-K for budgets 500 and +800 are 0.70 and 0.73, respectively. The growth rates of C-ETC-Y for budgets 500 and 800 +are 0.70 and 0.71, respectively. +43 + +(a) +(b) +(c) +(d) +Figure 2: Plots for song recommendation example. (a) and (b) are comparison results for +cumulative regret as a function of time horizon T. (c) and (d) are the moving average plot +with window size 100 of instantaneous reward as a function of t. The gray dashed lines in +(a) and (b) represent y = aT 2/3 for various values of a for visual reference. The gray dashed +lines in (c) and (d) represent expected rewards for the action chosen by an offline greedy +algorithm. +44 + +SR B=500 +Cumulative Regret +10° +4 +C-ETC-K +C-ETC-Y +103 +OGo +104 +105 +Horizon TSR B=800 +Cumulative Regret +10° +4 +C-ETC-K +C-ETC-Y +103 +3 +OGo +104 +105 +Horizon TSR B=500 +le-1l +Instantaneous Reward +6 +5 +4 +３ +C-ETC-K +C-ETC-Y +2 +OG° +0.0 +0.5 +1.0 +1.5 +2.0 +1e5 +Time-step tSR B=800 +1e-1 +Instantaneous Reward +6 +4 +C-ETC-K +C-ETC-Y +2 +OG° +0.0 +0.5 +1.0 +1.5 +2.0 +1e5 +Time-step t \ No newline at end of file diff --git a/WtFQT4oBgHgl3EQfcDb0/content/tmp_files/load_file.txt b/WtFQT4oBgHgl3EQfcDb0/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..165907769759228ddbb4118dead6837d27230c51 --- /dev/null +++ b/WtFQT4oBgHgl3EQfcDb0/content/tmp_files/load_file.txt @@ -0,0 +1,1177 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf,len=1176 +page_content='A Framework for Adapting Offline Algorithms to Solve Combinatorial Multi-Armed Bandit Problems with Bandit Feedback Guanyu Nie nieg@iastate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='edu Yididiya Y Nadew yididiya@iastate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='edu Yanhui Zhu yanhui@iastate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='edu Vaneet Aggarwal vaneet@purdue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='edu Christopher John Quinn cjquinn@iastate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='edu Editor: Abstract We investigate the problem of stochastic, combinatorial multi-armed bandits where the learner only has access to bandit feedback and the reward function can be non-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We provide a general framework for adapting discrete offline approximation algorithms into sublinear α-regret methods that only require bandit feedback, achieving O � T 2 3 log(T) 1 3 � expected cumulative α-regret dependence on the horizon T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' The framework only requires the offline algorithms to be robust to small errors in function evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' The adaptation procedure does not even require explicit knowledge of the offline approximation algorithm — the offline algorithm can be used as black box subroutine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' To demonstrate the utility of the proposed framework, the proposed framework is applied to multiple problems in submodular maximization, adapting approximation algo- rithms for cardinality and for knapsack constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' The new CMAB algorithms for knap- sack constraints outperform a full-bandit method developed for the adversarial setting in experiments with real-world data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Introduction Many real world sequential decision problems can be modeled using the framework of stochastic multi-armed bandits (MAB), such as scheduling, assignment problems, ad-campaigns, and product recommendations, among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' The decision maker sequentially selects ac- tions and receives stochastic rewards from an unknown distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' The goal of the decision maker is to maximize the expected cumulative reward over a (possibly unknown) time hori- zon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Actions result both in the immediate reward and, more importantly, information about that action’s reward distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Such problems result in a trade-off between trying actions the agent is uncertain of (exploring) or only taking the action that is empirically the best seen so far (exploiting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' In the classic MAB setting, the number of possible actions is small relative to the time horizon, meaning each action can be taken at least once, and there is no assumed relationship between the reward distributions of different arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' The combinatorial multi-armed bandit (CMAB) setting involves a large but structured action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' For example, in product recommendation problems, the decision maker may select a subset of products (base arms) 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='13326v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='LG] 30 Jan 2023 from among a large set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' There are several aspects that can affect the difficulty of these problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' First, MAB methods are typically compared against a learner with access to a value oracle of the reward function (an offline problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' For some problems, it is NP- hard for the baseline learner with value oracle access to optimize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' An example is if the expected/averaged reward function is submodular and actions are subsets constrained by cardinality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' At best, for these problems, approximation algorithms may exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Thus, unless the time horizon is large (exponentially long in the number of base arms, for instance), it would be more reasonable to compare the CMAB agent against the performance of the approximation algorithm for the related offline problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Likewise, one could apply state of the art methods for (unstructured) MAB problems treating each subset as a separate arm, and obtain ˜O(T 1 2 ) dependence on the horizon T for the subsequent regret bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' However, that dependence would only apply for exponentially large T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Feedback plays an important role in how challenging the problem is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' When the decision maker only observes a (numerical) reward for the action taken, that is known as bandit or full-bandit feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' When the decision maker observes additional information, such as contributions of each base arm in the action, that is semi-bandit feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Semi-bandit feedback greatly facilitates learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Suppose for instance that the reward function (on average) was monotone increasing over the inclusion lattice and there was a cardinality constraint of size k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' The agent would know from the start that no set of size smaller than k could be optimal (or could even be the near-optimal solution the baseline learning using a value oracle would find).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' However, there would be �n k � sets of size k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' For n = 100 and k = 10, the agent would need a horizon T > 1012 to try each cardinality k set even just once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' If the reward function belongs to a certain class, such as the class of submodular functions, then one approach would be to use a greedy procedure based on base arm values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' With semi-bandit feedback, the agent could on the one hand only take actions of cardinality k (putatively optimal actions), gain the subsequent rewards, and yet also observe samples of the base arms’ values to improve future actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Bandit feedback is much more challenging, as only the joint reward is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' In general, for non-linear reward functions, the individual values or marginal gains of base arms can only be loosely bounded if actions only consist of maximal subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Thus, to estimate values or marginal gains of base arms, the agent would need to deliberately spend time sampling actions (such as smaller sets) that are known to be sub-optimal in order to estimate their values to later better select actions of cardinality k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Standard MAB methods like UCB or TS based methods by design do not take actions known to be sub-optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Thus, while such strategies could be used when semi-bandit feedback is available, it is less clear whether they can be effectively used when only bandit feedback is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' There are important applications where semi-bandit feedback may not be available, such as in influence maximization and recommender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Influence maximization models the problem of identifying a low-cost subset (seed set) of nodes in a (known) social network that can influence the maximum number of nodes in a network (Nguyen and Zheng, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Leskovec et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=', 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Bian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Recent research has generalized the problem to online settings where the knowledge of the network and diffusion model is not required (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Perrault et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=', 2020a) but extra feedback is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' However, for many networks the user interactions and user accounts are private;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' only aggregate feedback 2 (such as the count of individuals using a coupon code or going to a website) might be visible to the decision maker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' In this work, we seek to address these challenges by proposing a general framework for adapting offline approximation algorithms into algorithms for stochastic CMAB problems when only bandit feedback is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We identify that a single condition related to the robustness of the approximation algorithm to erroneous function evaluations is sufficient to guarantee that a simple explore-then-commit (ETC) procedure accessing the approximation algorithm as a black box results in a sublinear α-regret CMAB algorithm despite having only bandit feedback available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' The approximation algorithm does not need to have any special structure (such as an iterative greedy design).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Importantly, no effort is needed on behalf of the user in mapping steps in the offline method into steps of the CMAB method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We demonstrate the utility of this framework by assessing the robustness of several approximation algorithms in the submodular optimization literature (three approximation algorithms designed for knapsack constraints and one designed for cardinality constraints) which immediately result in sublinear α-regret CMAB algorithms that only rely on bandit- feedback, the first such algorithms for CMAB problems with submodular rewards and knap- sack constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We also show that despite the simplicity and universal design of the adap- tation, the resulting CMAB algorithms work well on budgeted influence maximization and song recommendation problems using real world data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' The main contributions of this paper can be summarized as: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We provide a general framework for adapting discrete offline approximation algorithms into sublinear α-regret methods for stochastic CMAB problems where only bandit feedback is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' The framework only requires the offline algorithms to be robust to small errors in function evaluation, a property important in its own right for offline problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' The algorithms are not required to have a special structure — instead they are used as black boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Our procedure has minimal storage and time-complexity overhead, and achieves a regret bound with ˜O(T 2 3 ) dependence on the horizon T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We illustrate the utility of the proposed framework by assessing the robustness of several approximation algorithms for (offline) constrained submodular optimization, a class of re- ward functions lacking simplifying properties of linear or Lipschitz reward functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Specifi- cally, we prove the robustness of approximation algorithms given in Nemhauser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (1978);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Badanidiyuru and Vondr´ak (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Sviridenko (2004);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Khuller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (1999);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Yaroslavtsev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (2020) with cardinality or knapsack constraints, and use the general framework to give regret bounds for the stochastic CMAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' In particular, we note that this paper gives the first regret bounds for stochastic submodular CMAB with knapsack constraints under bandit feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We evaluate the performance of proposed framework through the stochastic submod- ular CMAB with knapsack constraints problem for two applications: Budgeted Influence Maximization, and Song Recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' The evaluation results demonstrate that the proposed approach significantly outperforms a full-bandit method for a related problem in the adversarial setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Related Work We now briefly discuss only the most closely related works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' See the supplementary material for more discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Adversarial CMAB The closest related works are on adversarial CMAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' In (Niazadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=', 2021), the authors propose a framework for transforming greedy α-approximation algorithms for offline problems to online methods in an adversarial bandit setting, for both semi-bandit (achieving �O(T 1/2) α−regret) and full-bandit feedback (achieving �O(T 2/3) α−regret).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Their framework requires the offline approximation algorithm to have an iter- ative greedy structure (unlike ours), satisfy a robustness property (like ours), and satisfy a property referred to as Blackwell reducibility (unlike ours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' In addition to these condi- tions, the adaptation depends on the number of subproblems (greedy iterations) which for some algorithms can be known ahead of time (such as with cardinality constraints) but for other algorithms can only be upper-bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (Our adaptation uses the offline algorithm as a black box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=') The authors check those conditions and explicitly adapt several offline approximation algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' In this paper, we consider an approach for converting offline approximation algorithm to online for stochastic CMAB, while requiring less assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We also note that (Niazadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=', 2021) do not consider submodular CMAB with knapsack constraints, and thus do not verify whether any approximation algorithms for the offline problem satisfy the required properties (of sub-problem structure or robustness or Blackwell reducibility) to be transformed, and this is an example we consider for our general framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Consequently, in our experiments for submodular CMAB with knapsack constraints in Section 7, we use the algorithm in (Streeter and Golovin, 2008) designed for a knapsack constraint (in expectation) as representative of methods for the adversarial setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Other related works for adversarial stochastic CMAB are described in Appendix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Stochastic Submodular CMAB with Full Bandit Feedback Recently, Nie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (2022) propose an algorithm for stochastic MAB with submodular rewards, when there is a cardinality constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Their algorithm is a specific adaptation of an offline greedy method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' In our work, we propose a general framework that employs the offline algorithm as a black box (and this result becomes a special case of our approach).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' While there are multiple results for semi-bandit feedback (see Appendix I), this paper considers full bandit feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Problem Statement We consider sequential, combinatorial decision-making problems over a finite time horizon T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Let Ω denote the ground set of base elements (arms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Let n = |Ω| denote the number of arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Let D ⊆ 2Ω denote the subset of feasible actions (subsets), for which we presume membership can be efficiently evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We will later consider applications with cardinality and knapsack constraints, though our methods are not limited to those.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We will use the terminologies subset and action interchangeably throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' At each time step t, the learner selects a feasible action At ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' After the subset At is selected, the learner receives reward ft(At).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We assume the reward ft is stochastic, bounded in [0, 1], and i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' conditioned on a given subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Define the expected reward function as f(A) = E[ft(A)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' 4 The goal of the learner is to maximize the cumulative reward �T t=1 ft(At).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' To measure the performance of the algorithm, one common metric is to compare the learner to an agent with access to a value oracle for f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' However, if optimizing f over D is NP-hard, such a comparison would not be meaningful unless the horizon is exponentially large in the problem parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' If there is a known approximation algorithm A with approximation ratio α ∈ (0, 1] for optimizing f over D, a more natural alternative is to evaluate the performance of a CMAB algorithm against what A could achieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Thus, we consider the the expected cumulative α-regret Rα,T , which is the difference between α times the cumulative reward of the optimal subset’s expected value and the average received reward, (we write RT when α is understood from context) E[RT ] = αTf(OPT) − E � T � t=1 ft(At) � , (1) where OPT is the optimal solution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=', OPT ∈ arg maxA∈D f(A) and the expectations are over both the random rewards and the sequence of actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Robustness of Offline Algorithms In this section, we introduce a criterion for an offline approximation algorithm’s sensitivity to (bounded) additive perturbations to function evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Investigating robustness of approximation algorithms in offline settings is valuable in its own right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Importantly, we will show that this property alone is sufficient to guarantee that the offline algorithm can be adapted to solve analogous combinatorial multi-armed bandit (CMAB) problems with just bandit feedback and yet achieve sub-linear regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Furthermore, the CMAB adaptation will not rely on any special structure of the algorithm design, instead employing it as a black box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Definition 1 ((α, δ)-Robust Approximation) An algorithm A is an (α, δ)-robust ap- proximation algorithm for the combinatorial optimization problem of maximizing a function f : D → R over a finite domain D ⊆ 2Ω if its output S∗ using a value oracle for ˆf satisfies the relation below with the optimal solution OPT under f, provided that for any ϵ > 0 that |f(S) − ˆf(S)| < ϵ for all S ∈ D, f(S∗) ≥ αf(OPT) − δϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Note that the perturbed ˆf is not required to be in the same class as f (linear, quadratic, submodular, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Thus, this definition is a stronger notion of robustness than one limited to ˆf in the same class that have bounded L∞ distance from f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' For (unstructured) k armed bandit problems, one can view the analogous offline algo- rithm with access to a value oracle for the elements as first evaluating each arm (D = {{1}, {2}, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' , {k}}), so N = k queries total, and then evaluating arg max over the k values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' That algorithm trivially is a (1, 2)-robust approximation algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Remark 2 In Niazadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (2021), there is a related definition of robustness for offline approximation algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' That definition and the subsequent offline-to-online adaptation 5 Algorithm 1 Combinatorial Explore-then-Commit Input: horizon T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' set of base elements Ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' an offline (α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' δ)-robust algorithm A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' and an upper-bound N on the number of A’s queries to the value oracle Initialize m ← � δ2/3T 2/3 log(T)1/3 2N2/3 � // Exploration Phase // while A queries the value of some A ⊆ Ω do For m times,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' play action A Calculate the empirical mean ¯f Return ¯f to A end while // Exploitation Phase // for remaining time do Play action S output by algorithm A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' end for procedure is restricted to approximation algorithms with an iterative greedy structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' The criterion Theorem 1 we consider does not require the approximation algorithm to have an iterative greedy structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' To illustrate the utility of our proposed framework, in Section 6 we will show that several approximation algorithms from the constrained submodular maximization literature are (α, δ)-robust, leading to new sublinear α-regret algorithms for related stochastic CMAB problems with submodular rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' C-ETC Algorithm: Offline to Stochastic In this section, we present our proposed algorithm for adapting offline approximation to algo- rithms for stochastic CMAB, Combinatorial Explore-Then-Commit (C-ETC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' The pseudo- code is shown in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' The algorithm takes an offline (α, δ) robust algorithm A with an upper bound N on the number of oracle queries by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' In the exploration phase, when the offline algorithm queries the value oracle for action A, C-ETC will play action A for m times, where m is a constant chosen to minimizing regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' C-ETC then computes the empirical mean ¯f of rewards for A and feeds ¯f back to the offline algorithm A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' In the exploitation phase, C-ETC keeps playing the solution S output from algorithm A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Thus, the CMAB procedure does not need A to have any special structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' No careful construc- tion is needed for the CMAB procedure beyond running A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' All that is needed is checking robustness (Theorem 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Also, there is no over-heard in terms of storage and per-round time complexities— C-ETC is as efficient as the offline algorithm A itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Now we analyze the α-regret for C-ETC (Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Theorem 3 For the sequential decision making problem defined in Section 2 and T ≥ 2 √ 2N δ , the expected cumulative α-regret of C-ETC using an (α, δ)-robust approximation al- 6 gorithm as subroutine is at most O � δ 2 3 N 1 3 T 2 3 log(T) 1 3 � , where N upper-bounds the number of value oracle queries made by the offline algorithm A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' The detailed proof is in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We highlight some key steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We show that with high probability, the empirical means of all actions taken during exploration phase will be within rad = � log T 2m of their corresponding statistical means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' As is common in proofs for ETC methods, we refer to this occurrence as the clean event E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Then, using an (α, δ)-robust approximation algorithm as subroutine will guarantee the quality of of the set S used in the exploitation phase of Algorithm 1: f(S) ≥ αf(OPT) − δ · rad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (2) We then break up the expected cumulative α-regret conditioned on the clean event E, E[R(T)|E] = N � i=1 m (αf(S∗) − E[f(St)]) � �� � exploration phase + T � t=TN+1 (αf(S∗) − E[f(S)]) � �� � exploitation phase .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (3) Using the fact that the reward is bounded between [0, 1], we have E[R(T)|E] ≤ Nm + Tδrad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Optimizing over m then results in E[R(T)|E] = O � δ 2 3 N 1 3 T 2 3 log(T) 1 3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We then show that because the clean event E happens with high probability, the expected cumulative regret E[R(T)] is dominated by E[R(T)|E], which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Lower bounds: For the general setting we explore in this paper, with stochastic (or even adversarial) combinatorial MAB and only bandit feedback, it is unknown whether ˜O(T 1/2) expected cumulative α-regret is possible (ignoring problem parameters like n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' For special cases, such as linear reward functions, ˜O(T 1/2) is known to be achievable even with bandit feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Even for the special case of submodular reward functions and a cardinality constraint, it remains an open question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Niazadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (2021) obtain ˜Ω(T 2/3) lower bounds for the harder setting where feedback is only available during “exploration” rounds chosen by the agent, who incurs an associated penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Remark 4 C-ETC uses knowledge of the horizon T to optimize the number m of samples per action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' When the time horizon T is not known, we can use geometric doubling trick to extend our result to an anytime algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We refer to the general detailed procedure in (Besson and Kaufmann, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' From Theorem 4 in (Besson and Kaufmann, 2018), we can show that the regret bound conserves the original T 2/3 log(T)1/3 dependence with only changes in constant factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' 7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Applications on Submodular Maximization In this section, we apply our general framework to stochastic CMAB problems with mono- tone submodular rewards where only bandit feedback is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' This application results in the first sublinear α-regret CMAB algorithms for knapsack constraints under bandit feed- back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We begin with a brief background, and analyze the robustness of offline approximation algorithms, and then obtain problem independent regret bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='1 Background and Definitions Denote the marginal gain f(e|A) = f(A ∪ e) − f(A) and the marginal density ρ(e|A) = f(A∪e)−f(A) c(e) for any subset A ⊆ Ω and element e ∈ Ω \\ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' A set function f : 2Ω → R defined on a finite ground set Ω is said to be submodular if it satisfies the diminishing return property: for all A ⊆ B ⊆ Ω, and e ∈ Ω \\ B, it holds that f(e|A) ≥ f(e|B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' A set function is said to be monotonically non-decreasing if f(A) ≤ f(B) for all A ⊆ B ⊆ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Our aim is to find a set S such that f(S) is maximized subject to some constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' For knapsack constraints, we assume that the cost function c : Ω → R>0 is known and linear, so the cost of a subset is be the sum of the costs of individual items: c(A) = � v∈A c(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' To simplify the presentation, we avoid the cases of trivially large budgets B > � v∈Ω c(v) and assume all items have non-trivial costs 0 < c(v) ≤ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' A cardinality constraint is a special case with unit costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' In the following, we consider both types of those constraints: cardinality and knapsack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Maximizing a monotone submodular set function under a k-cardinality constraint is NP- hard even with a value oracle Nemhauser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' The best achievable approximation ratio with a polynomial time algorithm is 1−1/e Nemhauser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (1978) using O(nk) oracle calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' In Badanidiyuru and Vondr´ak (2014), 1−1/e−ϵ′ is achieved within O( n ϵ′ log n ϵ′ ) time, where ϵ′ is a user selected parameter to balance accuracy and time complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Maximizing a monotone submodular set function under a knapsack constraint is conse- quently also NP-hard Khuller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' The best achievable approximation ratio with a polynomial time algorithm is 1 − 1/e (Sviridenko, 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Khuller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=', 1999), but that requires O(n5) function evaluations, making it prohibitive for many applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' There are other offline algorithms that achieve worse approximation ratios but are much more ef- ficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We adapt a 1 2 approximation algorithm (Yaroslavtsev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=', 2020) and a 1 2(1 − 1/e) approximation algorithm (Khuller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=', 1999), both of which use O(n2) function evalua- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' There is another algorithm proposed recently in Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (2022), but since it queries infeasible sets, we do not consider it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='2 Offline Approximation Algorithms – Robustness For an overview of offline approximation algorithms for submodular optimization, please refer Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We next state our results on (α, δ)-robustness of the offline algorithms considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' The assumption of complete/noiseless access to a value oracle is often a strong assumption for real world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Thus, even for offline applications, it is worthwhile knowing how robust an algorithm is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' So the following results are relevant even in the offline setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' For the CMAB setting we consider, robustness is also a sufficient property to 8 guarantee a no-regret adaptation of the offline algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Detailed proofs are included in Appendix B in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Theorem 5 (Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='3 of Nie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (2022)) Greedy in Nemhauser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (1978) is a (1 − 1 e, 2k)-robust approximation algorithm for submodular maximization under a k- cardinality constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Theorem 6 ThresholdGreedy Badanidiyuru and Vondr´ak (2014) is a (1− 1 e −ϵ′, 2(2− ϵ′)k)-robust approximation algorithm for submodular maximization under a k-cardinality constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Theorem 7 PartialEnumeration Sviridenko (2004);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Khuller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (1999) is a (1− 1 e, 4+ 2 ˜K + 2β)-robust approximation algorithm for submodular maximization under a knapsack constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Theorem 8 Greedy+Max Yaroslavtsev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (2020) is a ( 1 2, 1 2 + ˜K + 2β)-robust approx- imation algorithm for submodular maximization problem under a knapsack constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Theorem 9 Greedy+ Khuller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (1999) is a ( 1 2(1− 1 e), 2+ ˜K+β)-robust approximation algorithm for submodular maximization problem under a knapsack constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Remark 10 For the offline setting, Greedy+Max is superior to Greedy+, as it achieves a better α approximation ratio with the same calls to the value oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' However, their (α, δ) pairs are incomparable, as for β > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='5 (with β = 1 corresponding to a cardinality con- straint), Greedy+ has a smaller δ (thus more robust) which affects exploration time in their adaptations and in turn affects their regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' To illustrate the robustness analysis, we highlight some key steps for the proof of Theo- rem 8 for Greedy+Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Let o1 ∈ arg maxe:e∈OPT c(e) denote the most expensive element in OPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Inspired by the proof techniques in (Yaroslavtsev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=', 2020), we consider the last item added by the greedy solution (based on noisy evaluation) before the cost of this solution exceeds B −c(o1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Let Gi denote the set selected by Greedy that has cardinality i and denote the constituent elements as Gi = {g1, · · · , gi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Denote Gℓ as the largest greedy sequence that consumes less than B−c(o1) of the budget B, so c(Gℓ) ≤ B−c(o1) < c(Gℓ+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Let Si denote the augmented set at i-th iteration and S denote the final output of the algo- rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Denote ˆf(e|S) := ˆf(S ∪ e) − ˆf(S) and ˆρ(e|S) := ˆf(S∪e)− ˆf(S) c(e) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We prove the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Lemma 11 (Greedy+Max inequality) For i ∈ {0, 1, · · · , ℓ}, the following inequality holds: ˆf(Gi ∪ o1)+ max{0, ˆρ(gi+1|Gi)}(B − c(o1)) ≥ f(OPT) − (2 ˜K − 1)ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' For i = ℓ, Theorem 11 tells us that there can be two cases: ˆf(Gℓ ∪ o1) ≥ 1 2f(OPT) − � ˜K − 1 2 + γ � ϵ, or 9 ˆρ(gℓ+1|Gℓ)(B − c(o1)) ≥ 1 2f(OPT) − � ˜K − 1 2 − γ � ϵ, where γ will be selected later to minimize the additive error δ coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' If ˆf(Gℓ ∪ o1) ≥ 1 2f(OPT) − � ˜K − 1 2 + γ � ϵ, then denote aℓ = arg maxe∈Ω\\Gℓ ˆf(e|Gℓ), which is the element selected to augment Gℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We have ˆf(Gℓ ∪ aℓ) ≥ ˆf(Gℓ ∪ o1) ≥ 1 2f(OPT) − � ˜K − 1 2 + γ � ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (4) Then the final output of the algorithm S will satisfy f(S) ≥ ˆf(S) − ϵ ≥ ˆf(Gℓ ∪ aℓ) − ϵ ≥ 1 2f(OPT) − � ˜K + 1 2 + γ � ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (using (4)) If ˆρ(gℓ+1|Gℓ)(B − c(o1)) ≥ 1 2f(OPT) − ( ˜K − 1 2 − γ)ϵ, rearranging we have ˆρ(gℓ+1|Gℓ) ≥ f(OPT) 2(B − c(o1)) − ( ˜K − 1 2 − γ)ϵ B − c(o1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (5) Moreover, ˆf(Gℓ) = l−1 � j=0 ˆρ(gj+1|Gj)c(gj+1) ≥ l−1 � j=0 ˆρ(gℓ+1|Gj)c(gj+1) (6) ≥ l−1 � j=0 � ρ(gℓ+1|Gj) − 2ϵ c(gℓ+1) � c(gj+1) ≥ l−1 � j=0 � ρ(gℓ+1|Gℓ) − 2ϵ c(gℓ+1) � c(gj+1) (7) = � ρ(gℓ+1|Gℓ) − 2ϵ c(gℓ+1) � c(Gℓ) ≥ � ˆρ(gℓ+1|Gℓ) − 4ϵ c(gℓ+1) � c(Gℓ) ≥ ˆρ(gℓ+1|Gℓ)c(Gℓ) − 4βϵ, (8) 10 where (6) follows from the greedy selection rule, the (7) follows from submodularity of f, and (8) follows from the definition of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We then have ˆf(Gℓ+1) = ˆf(Gℓ) + c(gℓ+1)ˆρ(gℓ+1|Gℓ) ≥ � ˆρ(gℓ+1|Gℓ)c(Gℓ) − 4βϵ � + c(gℓ+1)ˆρ(gℓ+1|Gℓ) (9) = ˆρ(gℓ+1|Gℓ)c(Gℓ+1) − 4βϵ ≥ 1 2f(OPT) − ( ˜K − 1 2 − γ)ϵ B − c(o1) c(Gℓ+1) − 4βϵ (10) ≥ 1 2f(OPT) − ( ˜K − 1 2 − γ)ϵ − 4βϵ (11) = 1 2f(OPT) − � ˜K − 1 2 − γ + 4β � ϵ, (12) where (9) follows from (8), (10) follows from (5), and (11) follows from the chosen ℓ satisfies c(Gℓ+1) > B − c(o1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Thus, the final output of the algorithm S will satisfy f(S) ≥ ˆf(S) − ϵ ≥ ˆf(Gℓ+1) − ϵ ≥ 1 2f(OPT) − � ˜K + 1 2 − γ + 4β � ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Finally, combining both cases and selecting γ = 2β completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='3 CMAB algorithms for Submodular Rewards with Knapsack Constraints Now that we have analyzed the robustness of several offline algorithms, we can invoke Theorem 3 to bound the expected cumulative α regret for stochastic CMAB adaptations that rely only on bandit feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We name the adapted algorithms as C-ETC-N, C-ETC- B for cardinality constraint, C-ETC-S C-ETC-K and C-ETC-Y for knapsack constraint, respectively, based on which offline algorithm it is adapted from (using the first author’s last name);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' which are in order Nemhauser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (1978);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Badanidiyuru and Vondr´ak (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Sviridenko (2004);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Khuller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (1999);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Yaroslavtsev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' PartialEnumeration was first proposed and analyzed by Khuller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (1999) for maximum coverage problems and then analyzed by Sviridenko (2004) for monotone submodular functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' To distinguish CMAB adaptations of Greedy+ and C-ETC-K, both proposed in Khuller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (1999), we use C-ETC-S for the adaption of PartialEnumeration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' The following corollaries hold immediately: Corollary 12 For an online submodular maximization under a cardinality constraint, the expected cumulative (1 − 1/e)-regret of C-ETC-N is at most O � kn 1 3 T 2 3 log(T) 1 3 � for T ≥ √ 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Remark 13 This result improves upon the result from Nie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (2022) by a factor of k 1 3 despite our use of a generic framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' 11 Corollary 14 For an online submodular maximization under a cardinality constraint, the expected cumulative (1−1/e−ϵ′)-regret of C-ETC-B is at most O � k 2 3 n 1 3 (ϵ′) 1 3 (log n ϵ′ ) 1 3 T 2 3 log(T) 1 3 � for T ≥ √ 2n (2−ϵ′)ϵ′k log n ϵ′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Corollary 15 For an online submodular maximization under a knapsack constraint, the expected cumulative (1 − 1/e)-regret of C-ETC-S is at most O � β 2 3 ˜K 1 3 n 4 3 T 2 3 log(T) 1 3 � for T ≥ √ 2 ˜ Kn4 2+ ˜ K+β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Corollary 16 For an online submodular maximization under a knapsack constraint, the expected cumulative 1 2-regret of C-ETC-Y is at most O � β 2 3 ˜K 1 3 n 1 3 T 2 3 log(T) 1 3 � for T ≥ 2 √ 2 ˜ Kn 1 2 + ˜ K+2β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Corollary 17 For an online submodular maximization under a knapsack constraint, the expected cumulative 1 2(1 − 1 e)-regret of C-ETC-K is at most O � β 2 3 ˜K 1 3 n 1 3 T 2 3 log(T) 1 3 � for T ≥ 2 √ 2 ˜ Kn 2+ ˜ K+β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Storage and Per-Round Time Complexities: C-ETC-Y and C-ETC-K have low storage complexity and per-round time-complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' During exploitation, only the indices of at most ˜K base arms are needed in memory and does not need any computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' During exploration, they just need to update the empirical mean for the current action at time t, which can be done in O(1) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' It additionally stores the highest empirical density so far in the current iteration of the greedy routine and its associated base arm (C-ETC-K needs to store one more arm and C-ETC-Y an additional O( ˜K) storage is needed to store the augmented set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Thus, C-ETC-Y and C-ETC-K have O( ˜K) storage complexity and O(1) per-round time complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' For comparison, the algorithm proposed by Streeter and Golovin (2008) for an averaged knapsack constraint in the adversarial setting uses O(n ˜K) storage complexity and O(n) per-round time complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Some comments on lower bound are given in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Experiments In this section, we conduct experiments on real world data with a Budgeted Influence Maxi- mization (BIM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We also conduct experiments on Song Recommendation (SR) in Appendix J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Both of these are applications of stochastic CMAB with submodular rewards under a knapsack constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' There are three adaptions we considered in Section 6 for knapsack constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Since the time complexity for PartialEnumeration is much larger than the other two offline algorithms we consider, it will use at least T ≈ 108 for C-ETC-S to finish exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' For this reason, we do not consider C-ETC-S in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' To our knowledge, our work is the first to consider these applications with only bandit feedback available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Baseline: The only other algorithm designed for combinatorial MAB with general sub- modular rewards, under a knapsack constraint, and using full-bandit feedback is Online Greedy with opaque feedback model (OGo) proposed by Streeter and Golovin (2008) 12 (a) (b) (c) (d) Figure 1: Plots for budgeted influence maximization (BIM) example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (a) and (b) are comparison results for cumulative regret as a function of time horizon T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (c) and (d) are the moving average plot with window size 100 of instantaneous reward as a function of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' The gray dashed lines in (a) and (b) represent y = aT 2/3 for various values of a for visual reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' The gray dashed lines in (c) and (d) represent expected rewards for the action chosen by an offline greedy algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' for the adversarial setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' However, OGo only satisfies the knapsack constraint in expecta- tion, while our algorithms C-ETC-K ands C-ETC-Y satisfies a strict constraint (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' every action At must be under budget).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' See Appendix D for more details about OGo and its implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' In Section 6, we used N = ˜Kn as an upper bound on the number of function evaluations for both C-ETC-K and C-ETC-Y, where n is the number of base arms and ˜K is an upper bound of the cardinality of any feasible set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' When the time horizon T is small, it is possible that the exploration phase will not finish due to the formula being optimized for m (the number of plays for each action queried by A) uses a loose bound on the exploitation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' When this is the case, we select the largest m (closest to the formula) for which we can guarantee that exploration will finish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' For details, see Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' 13 BIM B=6 Cumulative Regret 10 4 C-ETC-K C-ETC-Y 103 3 OGo 104 105 Horizon TBIM B=8 Cumulative Regret 10 4 C-ETC-K C-ETC-Y 103 3 OGo 104 105 Horizon TBIM B=6 1e-1 Instantaneous Reward 3 2 C-ETC-K C-ETC-Y OGo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='00 1e5 Time-step tBIM B=8 1e-1 Instantaneous Reward 3 2 C-ETC-K C-ETC-Y OGo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='00 Time-step t 1e5We first conduct experiments for the application of budgeted influence maximization (BIM) on a portion of the Facebook network graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' BIM models the problem of identifying a low-cost subset (seed set) of nodes in a (known) social network that can influence the maximum number of nodes in a network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' While there are prior works proposing algorithms for budgeted online influence maximization problems, the state of the art (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=', Perrault et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (2020b)) presumes knowledge of the diffusion model (such as independent cascade) and, more importantly, extensive semi-bandit feedback on individual diffusions, such as which specific nodes became active or along which edges successful infections occurred, in order to estimate diffusion parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' For social networks with user privacy, this information is not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Data Set Description and Experiment Details: The Facebook network dataset was introduced in Leskovec and Mcauley (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' To facilitate running multiple experiments for different horizons, we used the community detection method proposed by Blondel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (2008) to detect a community with 354 nodes and 2853 edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We further changed the network to be directed by replacing every undirected edge by two directed edge with opposite directions, yielding a directed network with 5706 edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' The diffusion process is simulated using the independent cascade model (Kempe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=', 2003), where in each discrete step, an active node (that was inactive at the previous time step) independently attempts to infect each of its inactive neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Following existing work of Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (2015, 2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Bian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (2020), we set the probability of each edge (u, v) as 1/din(v), where din(v) is the in-degree of node v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Moreover, we consider a user u is more influential if the user has more out-degrees, dout(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' In our experiment, we only consider influential users to spend our budget more efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We pick the users with out-degrees that are above 95th percentile (18 users).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Denote this set as I, then for a user u ∈ I, the cost is defined as c(u) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='01dout(u) + 1, similar to (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' For each time horizon that was used, we ran each method ten times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' For this set of experiments, instead of cumulative 1 2-regret, which requires knowing OPT, we compare the cumulative rewards achieved by C-ETC and OGo against Tf(Sgrd), where Sgrd is the solution returned by the offline 1 2-approximation algorithm proposed by Yaroslavtsev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Tf(Sgrd) ≥ 1 2Tf(OPT), so Tf(Sgrd) is a more challenging reference value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Results and Discussion: Figures 1a and 1b show average cumulative regret curves for C-ETC-K (in blue), C-ETC-Y (in orange) and OGo (in green) for different horizon T values when the budget constraint B is 6 and 8, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' For B = 8, the turning point is T = 21544.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Standard errors of means are presented as error bars, but might be too small to be noticed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Figures 1c and 1d are the instantaneous reward plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' The peaks at the very beginning of exploration phase correspond to the time step that the single person with highest influence is sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' C-ETC significantly outperforms OGo for all time horizons and budget considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' To evaluate the gap between the empirical performance and the theoretical guarantee, we estimated the slope for both methods on log-log scale plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Over the horizons tested, OGo’s cumulative regret (averaged over ten runs) has a growth rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' The growth rates of C-ETC-K for budgets 6 and 8 are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='76 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='68, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' The growth rates of C-ETC-Y for budgets 6 and 8 are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='75 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='69, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' The slopes are close to the 2/3 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='67 theoretical guarantee, and notably, the performance for larger B is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' 14 References Sanjeev Arora, Elad Hazan, and Satyen Kale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' The multiplicative weights update method: a meta-algorithm and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=', 8:121–164, 2012.' metadata={'source': 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+page_content=' Shawe-Taylor, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Zemel, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Bartlett, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Pereira, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Weinberger, editors, Advances in Neural Information Processing Systems, volume 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Curran Associates, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=', 2011a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' URL https://proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='neurips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='cc/paper/2011/ file/33ebd5b07dc7e407752fe773eed20635-Paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Yisong Yue and Carlos Guestrin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Linear submodular bandits and their application to diversified retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 24, 2011b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' 17 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Offline Approximation Algorithms – Overview We give a brief overview of the offline approximation algorithms which we will analyze (α, δ) robustness for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' For a k-cardinality constraint, the greedy algorithm Greedy proposed in Nemhauser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (1978) starts from an empty set G ← ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Then it repeatedly add the element with highest marginal gain f(e|G) until the cardinality |G| reaches k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' ThresholdGreedy, pro- posed in Badanidiyuru and Vondr´ak (2014), considers a sequence of decreasing thresholds: {τ = d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' τ ≥ ϵ′ n d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' τ ← (1−ϵ′)τ} where d = maxe∈Ω f(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Then starting from empty set G = ∅, the algorithm includes any element e /∈ G such that f(e|G) ≥ τ whenever the cardinality is smaller than k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' The algorithm then repeats using a lower threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Badanidiyuru and Vondr´ak (2014) showed that ThresholdGreedy can achieve 1 − 1/e − ϵ′ approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' For a knapsack constraint, several algorithms run the following greedy subroutine, which we refer to as Greedy (cardinality is a special case of this routine with budget k and unit cost, so we keep the same name without confusion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Start with empty set G ← ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Repeatedly add the element e with the highest marginal density ρ(e|G) that fits into the budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Let Gi denote the set selected by Greedy that has cardinality i and denote the constituent elements as Gi = {g1, · · · , gi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Let L denote the cardinality of the final greedy set (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' when no more elements remain that are under budget), so GL is output by Greedy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Note that L can only be bounded ahead of time—there could be maximal subsets (to which no other elements could be added without violating the budget) of different cardinalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Greedy can have an unbounded approximation ratio Khuller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (1999) for knapsack constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Khuller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (1999) proposed Greedy+, which outputs the better of the best individual element a∗ ∈ arg maxe∈Ω f(e) and the output of Greedy, arg maxS∈{GL,a∗} f(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Khuller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (1999) proved that Greedy+ achieves a 1 2(1− 1 e) approximation ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Then, Sviridenko (2004);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Khuller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (1999) proposed PartialEnumeration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' It first enumerate all sets with cardinality up to three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' For each enumerated triplets, it build the rest of the solution set greedily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Then it outputs the set with largest value among all evaluated sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' They showed that PartialEnumeration can achieve 1 − 1/e approximation ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Greedy+Max generalizes Greedy+ by augmenting each set {Gi}L i=1 in the nested se- quence produced by Greedy with another element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' For 0 ≤ i ≤ L − 1, define G′ i ← Gi ∪ arg maxe∈Ω:c(Gi)+c(e)≤B f(Gi ∪ e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' By construction, G′ 0 = {a∗}, the best individual element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' For i = L, G′ L ← GL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Greedy+Max then outputs the best set in the augmented sequence, arg maxS∈{G′ 0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=',G′ L} f(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Yaroslavtsev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (2020) proposed Greedy+Max and proved it achieves an approximation ratio of 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' A bound on the number of value oracle calls will be important in adapting offline meth- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Denote β := B/cmin and ˜K := min{n, β} as an upper bound of the number of items in any feasible set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We note here that while PartialEnumeration uses O( ˜Kn4) function evaluations, both Greedy+Max and Greedy+ use O( ˜Kn) oracle calls, same as Greedy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We use N = ˜Kn in the analysis for Greedy+Max and Greedy+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Proof for Robustness of Offline Algorithms In this section, we prove the (α, δ) robustness of algorithms considered in Section 6 of the main paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' 18 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='1 Notation We first review notations used in the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Recall that we are only able to evaluate the surrogate function ˆf such that | ˆf(S) − f(S)| ≤ ϵ for any feasible set S and some ϵ > 0, we further denote ˆf(e|S) = ˆf(S ∪ e) − ˆf(S) and ˆρ(e|S) = ˆf(S∪e)− ˆf(S) c(e) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Let Gi denote the set selected by basic Greedy (based on surrogate function ˆf) as described in Section 3 up until ith item and Gi = {g1, · · · , gi} in the order of each item is selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Without loss of generality, define G0 = ∅ and f(G0) = ˆf(G0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Denote cmin = mine∈Ω c(e) be the item with lowest individual cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Let β = B/cmin and ˜K = min{n, β} being an upper bound of the number of items in any feasible set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Since all selected actions should be feasible, for ease of notation, we omit denoting that condition throughout the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' For example, we write arg maxe∈Ω\\A f(e|A) to simplify the notation of arg maxe:e∈Ω\\A and A∪e∈D f(e|A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Let S be the set returned by modified algorithms in corresponding context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='2 Robustness of Offline Methods for Submodular Maximization under Cardinality Constraint B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='1 Greedy We consider the original greedy algorithm Greedy proposed in Nemhauser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (1978), which gives a (1 − 1 e)-approximation guarantee for submodular maximization under a k- cardinality constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' To restate Theorem 5 in the main paper, Greedy is a (1 − 1 e, 2k)- robust approximation algorithm for submodular maximization under a k-cardinality con- straint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' The result follows from Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='3 of Nie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (2022), part of the regret analysis for a CMAB adaptation of Greedy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='2 ThresholdGreedy We then consider the threshold greedy algorithm ThresholdGreedy proposed in Badani- diyuru and Vondr´ak (2014), which gives a (1− 1 e −ϵ′)-approximation guarantee for submod- ular maximization under a k-cardinality constraint, where ϵ′ is a user specified parameter to balance accuracy and run time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Restating Theorem 6 in the main paper, Thresh- oldGreedy is a (1 − 1 e − ϵ′, 2(2 − ϵ′)k)-robust approximation algorithm for submodular maximization under a k-cardinality constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Proof From the assumption of the surrogate function ˆf we know f(e|S) − 2ϵ ≤ ˆf(e|S) ≤ f(e|S) + 2ϵ for any e ∈ Ω \\ S and S ⊆ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Now assume the the next chosen element is a and the current partial solution is S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' On one hand, we have ˆf(a|S) ≥ w =⇒ f(a|S) ≥ w − 2ϵ, (13) on the other hand, for every e ∈ OPT \\ S, ˆf(e|S) ≤ w 1 − ϵ′ =⇒ f(e|S) ≤ w 1 − ϵ′ + 2ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (14) Combining and manipulating (13) and (14) we have for any e ∈ OPT \\ S: f(a|S) + 2ϵ ≥ (f(e|S) − 2ϵ)(1 − ϵ′) =⇒ f(a|S) ≥ (1 − ϵ′)f(e|S) − 2(2 − ϵ′)ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (15) 19 Taking an average over all e ∈ OPT \\ S, f(a|S) ≥ 1 − ϵ′ |OPT \\ S| � e∈OPT\\S f(e|S) − 2(2 − ϵ′)ϵ ≥ 1 − ϵ′ k � e∈OPT\\S f(e|S) − 2(2 − ϵ′)ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (16) Now consider after i ∈ [k − 1] steps, we get a partial solution Si = {a1, · · · , ai}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' By (16), we have f(ai+1|Si) ≥ 1 − ϵ′ k � e∈OPT\\S f(e|Si) − 2(2 − ϵ′)ϵ ≥ 1 − ϵ′ k f(OPT|Si) − 2(2 − ϵ′)ϵ (submodularity) ≥ 1 − ϵ′ k (f(OPT) − f(Si)) − 2(2 − ϵ′)ϵ, (monotonicity) and hence for i ∈ [k − 1], f(Si+1) − f(Si) = f(ai+1|Si) ≥ 1 − ϵ′ k (f(OPT) − f(Si)) − 2(2 − ϵ′)ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (17) Using (17) as induction hypothesis, we then prove by induction (omitted) that for i ∈ [k−1], f(Si+1) ≥ � 1 − � 1 − 1 − ϵ′ k �i+1� f(OPT) − 2(i + 1)(2 − ϵ′)ϵ, and plugging in i = k − 1 we get f(Sk) ≥ � 1 − � 1 − 1 − ϵ′ k �k� f(OPT) − 2k(2 − ϵ′)ϵ ≥ (1 − e−(1−ϵ′))f(OPT) − 2k(2 − ϵ′)ϵ ≥ (1 − 1/e − ϵ′)f(OPT) − 2k(2 − ϵ′)ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We finish the proof by observing that Sk is the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='3 Proof for Robustness of Greedy+Max In this section, we give a detailed proof for Theorem 8 in Section 6 of the main paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Recall the statement is that Greedy+Max is a ( 1 2, 1 2 + ˜K + 2β)-robust approximation algorithm for submodular maximization problem under a knapsack constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Let o1 ∈ arg maxe:e∈OPT c(e) denote the most expensive element in OPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' During the ith iteration of the greedy process, having previously selected the set Gi−1 with i − 1 elements, 20 it will select the element gi with highest marginal density (based on surrogate function ˆf) among feasible elements, gi = arg max e: e∈Ω\\Gi−1 ˆρ(e|Gi−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (18) Inspired by the proof techniques in Yaroslavtsev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (2020), we consider the last item added by the greedy solution (based the surrogate function ˆf) before the cost of this solution exceeds B − c(o1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Denote Gℓ as the largest greedy sequence that consumes less than B − c(o1) budgets, c(Gℓ) ≤ B − c(o1) < c(Gℓ+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Let ai denote the element selected to augment with the greedy solution Gi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=', ai = arg maxe∈Ω\\Gi ˆf(e|Gi), and Si denote the augmented set at i-th iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Before proving the theorem, we show Theorem 11 in Section 6 of the main paper, that for i ∈ {0, 1, · · · , ℓ}, the following inequality holds: ˆf(Gi ∪ o1) + max{0, ˆρ(gi+1|Gi)}(B − c(o1)) ≥ f(OPT) − (2 ˜K − 1)ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Proof Recall that from the definition of ˆf, we have | ˆf(S) − f(S)| ≤ ϵ for any evaluated set S and some ϵ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Consequently, we have for any i ∈ {0, 1, · · · , ℓ}, | ˆf(Gi) − f(Gi)| ≤ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (19) Now we evaluate the set Gi ∪ o1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Case 1: If o1 has already been added, o1 ∈ Gi, then | ˆf(Gi ∪ o1) − f(Gi ∪ o1)| = | ˆf(Gi) − f(Gi)| ≤ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Case 2: If o1 /∈ Gi, then ˆf(Gi ∪ o1) is evaluated in iteration i + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' This iteration i + 1 does exist1 because for any i ∈ {0, 1, · · · , ℓ}, we only used less than B − c(o1) budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' For the remaining budget, at least o1 can still fit into the budget so Gi ∪ o1 will be evaluated in iteration i + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' In this case, we still have | ˆf(Gi ∪ o1) − f(Gi ∪ o1)| ≤ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Combining these two cases, we have | ˆf(Gi ∪ o1) − f(Gi ∪ o1)| ≤ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (20) Also, for any evaluated action in iteration i + 1, namely the actions {Gi ∪ e|e ∈ Ω \\ Gi and c(e) + c(Gi) ≤ B}, we have ρ(e|Gi) = f(Gi ∪ e) − f(Gi) c(e) ≤ ˆf(Gi ∪ e) − ˆf(Gi) c(e) + 2ϵ c(e) = ˆρ(e|Gi) + 2ϵ c(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (21) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' For (α, δ) robustness alone, this point is not necessary due to the assumption of |f(S)− ˆf(S)| ≤ ϵ for all S ⊆ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' For the regret bound proof of our proposed C-ETC method in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='4, the “clean event” (corresponding to concentration of empirical mean of set values around their statistical means) will only imply concentration for those actions taken and thus for which empirical estimates exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' 21 Then we have f(OPT) ≤ f(Gi ∪ OPT) (Monotonicity of f) ≤ f(Gi ∪ o1) + f(OPT \\ (Gi ∪ o1)|Gi ∪ o1) ≤ f(Gi ∪ o1) + � e∈OPT\\(Gi∪o1) f(e|Gi ∪ o1) (Submodularity of f) ≤ ˆf(Gi ∪ o1) + ϵ + � e∈OPT\\(Gi∪o1) c(e)ρ(e|Gi ∪ o1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (22) where (22) uses (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Since we picked iteration i such that c(Gi) ≤ B −c(o1), then all items in OPT\\(Gi ∪o1) still fit, as o1 is the largest item in OPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Since the greedy algorithm always selects the item with the largest marginal density with respect to the surrogate function ˆf, gi = arg maxe∈Ω\\Gi ˆρ(e|Gi), thus we have ˆρ(gi+1|Gi) = max e∈Ω\\Gi ˆρ(e|Gi) ≥ max e∈Ω\\(Gi∪o1) ˆρ(e|Gi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (23) Hence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' continuing with (22),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='f(OPT) ≤ ˆf(Gi ∪ o1) + ϵ + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='e∈OPT\\(Gi∪o1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='c(e)ρ(e|Gi ∪ o1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='≤ ˆf(Gi ∪ o1) + ϵ + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='e∈OPT\\(Gi∪o1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='c(e)ρ(e|Gi) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='(Submodularity) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='≤ ˆf(Gi ∪ o1) + ϵ + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='e∈OPT\\(Gi∪o1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='c(e) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='ˆρ(e|Gi) + 2ϵ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='c(e) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='(using (21)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='≤ ˆf(Gi ∪ o1) + ϵ + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='e∈OPT\\(Gi∪o1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='c(e)ˆρ(e|Gi) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='+ 2ϵ|OPT \\ (Gi ∪ o1)| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='≤ ˆf(Gi ∪ o1) + ϵ + ˆρ(gi+1|Gi) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='e∈OPT\\(Gi∪o1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='c(e) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='+ 2ϵ|OPT \\ (Gi ∪ o1)| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='(Using (23)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='≤ ˆf(Gi ∪ o1) + ϵ + ˆρ(gi+1|Gi)c(OPT \\ (Gi ∪ o1)) + 2ϵ|OPT \\ (Gi ∪ o1)| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='≤ ˆf(Gi ∪ o1) + ϵ + max{0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' ˆρ(gi+1|Gi)}c(OPT \\ (Gi ∪ o1)) + 2ϵ|OPT \\ (Gi ∪ o1)| ≤ ˆf(Gi ∪ o1) + ϵ + max{0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' ˆρ(gi+1|Gi)}(gi+1|Gi)(B − c(o1)) + 2ϵ|OPT \\ (Gi ∪ o1)| ≤ ˆf(Gi ∪ o1) + max{0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' ˆρ(gi+1|Gi)}(gi+1|Gi)(B − c(o1)) + (2 ˜K − 1)ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Rearranging terms gives the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Now we are ready to prove Theorem 8 (robustness of Greedy+Max algorithm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Applying Theorem 11 (Greedy+Max inequality) for i = ℓ, and recalling that ℓ is chosen as the index of the last greedy set such that c(Gℓ) ≤ B − c(o1) < c(Gℓ+1), ˆf(Gℓ ∪ o1) + max{0, ˆρ(gℓ+1|Gℓ)}(B − c(o1)) ≥ f(OPT) − (2 ˜K − 1)ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (24) 22 From (24), we will next argue at least one of the terms in the left hand side must be large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We will consider cases for the two terms being large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' To minimize the worst-case additive error term from the cases, we will split the cases into whether ˆf(Gℓ ∪ o1) is larger than or equal to 1 2f(OPT) − ( ˜K − 1 2 + γ)ϵ, or max{0, ˆρ(gℓ+1|Gℓ}(B − c(o1)) is larger than or equal to 1 2f(OPT) − ( ˜K − 1 2 − γ)ϵ, where γ will be selected later to minimize the additive error δ coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Case 1: If ˆf(Gℓ ∪ o1) ≥ 1 2f(OPT) − ( ˜K − 1 2 + γ)ϵ, recall that aℓ as the element selected to augment with the greedy solution Gℓ, aℓ = arg maxe∈Ω\\Gℓ ˆf(e|Gℓ), then ˆf(Gℓ ∪ aℓ) ≥ ˆf(Gℓ ∪ o1) ≥ 1 2f(OPT) − � ˜K − 1 2 + γ � ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (25) The set S that the algorithm selects in the end will be the set with the highest mean (based on surrogate function ˆf) among all those evaluated (both sets in the greedy process and their augmentations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Also, its observed value ˆf(Sℓ) is at most ϵ above f(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Thus f(S) ≥ ˆf(S) − ϵ ≥ ˆf(Gℓ ∪ aℓ) − ϵ ≥ 1 2f(OPT) − � ˜K + 1 2 + γ � ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (using (25)) Case 2(a): If max{0, ˆρ(gℓ+1|Gℓ)}(B−c(o1)) ≥ 1 2f(OPT)−( ˜K− 1 2−γ)ϵ and ˆρ(gℓ+1|Gℓ) > 0, rearranging we have ˆρ(gℓ+1|Gℓ) ≥ f(OPT) 2(B − c(o1)) − ( ˜K − 1 2 − γ)ϵ B − c(o1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (26) 23 Then, ˆf(Gℓ) = ˆf(Gℓ) − ˆf(Gℓ−1) + ˆf(Gℓ−1) + · · · − ˆf(G1) + ˆf(G1) − ˆf(G0) (telescoping sum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' G0 = ∅, ˆf(G0) := 0) = l−1 � j=1 ˆf(gj+1|Gj) (Definition of ˆf(·|·)) = l−1 � j=0 ˆρ(gj+1|Gj)c(gj+1) (Definition of ˆρ(·|·)) ≥ l−1 � j=0 ˆρ(gℓ+1|Gj)c(gj+1) (greedy choice of gj+1) ≥ l−1 � j=0 � ρ(gℓ+1|Gj) − 2ϵ c(gℓ+1) � c(gj+1) ≥ l−1 � j=0 � ρ(gℓ+1|Gℓ) − 2ϵ c(gℓ+1) � c(gj+1) (submodularity of f) = � ρ(gℓ+1|Gℓ) − 2ϵ c(gℓ+1) � c(Gℓ) (simplifying) ≥ � ˆρ(gℓ+1|Gℓ) − 4ϵ c(gℓ+1) � c(Gℓ) ≥ ˆρ(gℓ+1|Gℓ)c(Gℓ) − 4βϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (27) Recalling that ℓ is chosen as the index of the last greedy set that has a remaining budget as big as the cost of the heaviest element in OPT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' c(Gℓ) ≤ B − c(o1) < c(Gℓ+1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' ˆf(Gℓ+1) = ˆf(Gℓ ∪ gℓ+1) = ˆf(Gℓ) + c(gℓ+1)ˆρ(gℓ+1|Gℓ) ≥ � ˆρ(gℓ+1|Gℓ)c(Gℓ) − 4βϵ � + c(gℓ+1)ˆρ(gℓ+1|Gℓ) (from (27)) = ˆρ(gℓ+1|Gℓ)c(Gℓ+1) − 4βϵ (simplifying) ≥ 1 2f(OPT) − ( ˜K − 1 2 − γ)ϵ B − c(o1) c(Gℓ+1) − 4βϵ (case 2 condition) ≥ 1 2f(OPT) − ( ˜K − 1 2 − γ)ϵ − 4βϵ (ℓ chosen so that c(Gℓ+1) > B − c(o1)) = 1 2f(OPT) − � ˜K − 1 2 − γ + 4β � ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (28) The set S that the algorithm selects at the end of the exploitation phase will be the set with the highest empirical mean among all those explored (both sets in the greedy process 24 and augmented sets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Thus its empirical mean is at most ϵ above f(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' f(S) ≥ ˆf(S) − ϵ ≥ ˆf(Gℓ+1) − ϵ ≥ 1 2f(OPT) − � ˜K + 1 2 − γ + 4β � ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (using (28)) Case 2(b): If max{0, ˆρ(gℓ+1|Gℓ)}(B−c(o1)) ≥ 1 2f(OPT)−( ˜K− 1 2−γ)ϵ and ˆρ(gℓ+1|Gℓ) ≤ 0, then the set S that the algorithm selects at the end satisfies f(S) ≥ 0 ≥ 1 2f(OPT) − ( ˜K − 1 2 − γ)ϵ (Case 2(b) condition) ≥ 1 2f(OPT) − ( ˜K − 1 2 − γ + 4β)ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Thus, combining cases 1 and 2, and selecting γ = 2β, the additive 1 2-approximation error we get by the modified Greedy+Max algorithm is at most (1 2 + ˜K + 2β)ϵ, which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='4 Proof for Robustness of Greedy+ In this section, we prove Theorem 9 in Section 6 of the main paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' The following state- ments, Lemmas 18,19 and 21, and their proofs are adapted from the proof of 1 2(1 − 1 e) approximation ratio in the offline setting Khuller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (1999) using a value oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Krause and Guestrin (2005) adapted the proof of Khuller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (1999) to an offline setting where the greedy process relies on an exact oracle to evaluate individual element values and to compare the best individual element to the set output by the greedy process, but use an inexact value oracle (within ϵ of the correct value) to evaluate marginal densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' The main differences arise from (i) the algorithms of Khuller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (1999);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Krause and Guestrin (2005) evaluate densities before checking for feasibility,2 leading to different definitions of the augmented greedy sequence, necessitating us to use more care to show analogous properties, (ii) exact value oracles for best individual elements and for selecting OPT are used in Khuller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (1999);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Krause and Guestrin (2005), simplifying work to conclude the final bound for the approximation ratio α = 1 2(1− 1 e) and leading to a different δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Recall that Theorem 9 in Section 6 of the main paper states that Greedy+ is a ( 1 2(1 − 1 e), 2+ ˜K +β)-robust approximation algorithm for submodular maximization problem under a knapsack constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We define Gi and gi the same as previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Recall that the greedy process (using a surrogate ˆf) produces a nested sequence of subsets ∅ = G0 ⊂ G1 ⊂ · · · ⊂ GL, where L denotes the cardinality of the set final output of the greedy process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' For the proof, we describe the greedy process as running for L + 1 iterations, though on the final iteration no elements are added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' As noted in Footnote 1, concentration of estimates (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' the surrogate ˆf) used by C-ETC in the bandit setting will only be for evaluated subsets, which by restriction will all be feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' 25 For any action Gi−1 ∪a evaluated in iteration i of the greedy process, its marginal gains are upper bounded by that of the best subset based on surrogate function ˆf, f(Gi−1 ∪ a) − f(Gi−1) − 2ϵ c(a) ≤ ˆf(Gi−1 ∪ a) − ˆf(Gi−1) c(a) ≤ ˆf(Gi−1 ∪ gi) − ˆf(Gi−1) c(gi) (gi selected by greedy rule based on ˆf) ≤ f(Gi−1 ∪ gi) − f(Gi−1) + 2ϵ c(gi) = f(Gi) − f(Gi−1) + 2ϵ c(gi) , (29) where (29) just uses the definition of Gi ← Gi−1 ∪ gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We will use (29) to lower bound the true marginal gains (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' in terms of f) achieved for each iteration of the greedy process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Let ℓ ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' , L + 1} denote the first iteration for which there was an element a′ ∈ Ω\\Gℓ−1 whose cost exceeds the remaining budget (c(a′)+c(Gℓ−1) > B) (thus subset Gℓ−1∪a′ was not sampled), yet whose marginal density was higher than the marginal density of the chosen element gℓ up to ±2ϵ normalized by the cost, specifically, for ℓ ≤ L, f(Gℓ−1 ∪ a′) − f(Gℓ−1) − 2ϵ c(a′) > f(Gℓ−1 ∪ aℓ) − f(Gℓ−1) + 2ϵ c(ar) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (30) If there is no such iteration ℓ < L+1, then for ℓ = L+1, we take the element a′ maximizing the term on the left hand side of (30), a′ = arg max a∈Ω\\Gℓ−1 f(Gℓ−1 ∪ a) − f(Gℓ−1) − 2ϵ c(a) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (31) Likewise, if there is more than one element satisfying (30) for some (earliest) iteration r, then we also take the maximizer (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We define an “augmented” greedy sequence of length ℓ which matches the greedy se- quence up to the set of cardinality ℓ, where the element a′ is selected despite violating the budget, { �G0 = G0 = ∅, �G1 = G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' , �Gℓ−1 = Gℓ−1, �Gℓ = Gℓ−1 ∪ {a′}} (32) and correspondingly enumerate the elements of �Gℓ in the order they were selected, {�g1 = g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' , �gℓ−1 = gℓ−1, �gℓ = g′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (33) We first prove the following lemma, bounding the marginal gains of the augmented greedy sequence { �G0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' , �Gℓ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Lemma 18 For all i ∈ {1, 2, · · · , ℓ}, the following inequality holds: f( �Gi) − f( �Gi−1) ≥ c(�gi) B � f(OPT) − f( �Gi−1) � − 2 � 1 + ˜Kc(�gi) B � ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' 26 Proof Set any i ∈ {1, 2, · · · , ℓ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Let {v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' · · · , vk} = OPT \\ �Gi−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Note that by construction (32), we have �Gi−1 = Gi−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' The difference f(OPT) − f( �Gi−1) can be bounded by the marginal gains of elements in the set difference, f(OPT) − f( �Gi−1) ≤ k � j=1 � f( �Gi−1 ∪ vj) − f( �Gi−1) � (Fact 1) = k � j=1 � f( �Gi−1 ∪ vj) − f( �Gi−1) − 2ϵ + 2ϵ � = k � j=1 c(vj)f( �Gi−1 ∪ vj) − f( �Gi−1) − 2ϵ c(vj) + 2kϵ ≤ k � j=1 c(vj)f( �Gi−1 ∪ �gi) − f( �Gi−1) + 2ϵ c(�gi) + 2kϵ (34) = k � j=1 c(vj)f( �Gi) − f( �Gi−1) + 2ϵ c(�gi) + 2kϵ (35) where (34) holds by following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We consider four cases, depending on whether or not ˆf(Gi−1∪ vj) was evaluated during the iteration i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Case 1 ( ˆf(Gi−1 ∪ vj) was evaluated and i < ℓ): At iteration i (necessarily i ≤ L since no subsets were evaluated in iteration L + 1) with current greedy set Gi−1, adding the element vj to the current greedy set was feasible, c(vj) ≤ B − c(Gi−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Then Greedy+ would have evaluated ˆf(Gi−1 ∪ vj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Since vj was not selected, the chosen element gi = Gi\\Gi−1 must have had a higher surrogate density ˆf(Gi−1∪vj) > ˆf(Gi−1 ∪ gi), so for i < ℓ, for which �gi = gi by construction (33), (29) implies (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Case 2 ( ˆf(Gi−1 ∪ vj) was evaluated and i = ℓ): By the reasoning in the previous case,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' for the item aℓ chosen at iteration ℓ by the greedy process (due to feasibility and having the highest surrogate density),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' we still have the bound (29) on true values,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' which coupled with our specific construction of �gℓ (30) means f( �Gℓ−1 ∪ vj) − f( �Gℓ−1) − 2ϵ c(vj) ≤ f( �Gℓ−1 ∪ ar) − f( �Gℓ−1) + 2ϵ c(ar) (by (29)) < f( �Gℓ−1 ∪ �gr) − f( �Gℓ−1) − 2ϵ c(�gr) (by construction (30)) < f( �Gℓ−1 ∪ �gr) − f( �Gℓ−1) + 2ϵ c(�gr) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Case 3 ( ˆf(Gi−1 ∪ vj) was not evaluated and i < ℓ): At iteration i < ℓ ≤ L + 1 with the current greedy set Gi−1, adding the element vj to the current greedy set was 27 not feasible, c(vj) > B −c(Gi−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' By construction of the augmented greedy sequence, only at iteration ℓ was there an infeasible element whose surrogate marginal density satisfied the inequality (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Thus, for iterations i < ℓ, Gi−1 = �Gi−1 and Gi = �Gi, so (34) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Case 4 ( ˆf(Gi−1 ∪ vj) was not evaluated and i = ℓ): For iteration i = ℓ, with current greedy set Gi−1, the augmented greedy sequence construction implies (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Namely, with i = ℓ, f( �Gℓ−1 ∪ vj) − f( �Gℓ−1) − 2ϵ c(vj) < f( �Gℓ−1 ∪ �gr) − f( �Gℓ−1) − 2ϵ c(�gr) (by (31)) < f( �Gℓ−1 ∪ �gr) − f( �Gℓ−1) + 2ϵ c(�gr) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' menaing (34) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We now continue lower bounding f(OPT) − f( �Gi−1), f(OPT) − f( �Gi−1) ≤ � � k � j=1 c(vj)f( �Gi) − f( �Gi−1) + 2ϵ c(�gi) � � + 2kϵ (copying (35)) = � � k � j=1 c(vj) � � f( �Gi) − f( �Gi−1) + 2ϵ c(�gi) + 2kϵ ≤ B f( �Gi) − f( �Gi−1) + 2ϵ c(�gi) + 2kϵ (OPT is feasible, so �k j=1 c(vj) ≤ B) ≤ B c(�gi) � f( �Gi) − f( �Gi−1) � + 2 � B c(�gi) + ˜K � ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (rearranging;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' k ≤ ˜K) Multiplying both sides by c(�gi) B and rearranging finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We unravel the recurrence in Theorem 18 to lower bound f( �Gi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Lemma 19 For all i ∈ {1, 2, · · · , ℓ}, f( �Gi) ≥ � �1 − i� j=1 (1 − c(�gj) B ) � � f(OPT) − 2(β + ˜K)ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Remark 20 The steps to unravel the recurrence to obtain the first term (coefficient of f(OPT)) is the same as the proof for the analogous result in the offline setting Khuller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' The second term (with ϵ) is due to working with marginal densities of a 28 surrogate function ˆf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' The basic steps for working with that second term is the same as Krause and Guestrin (2005), though we use a looser bound β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' in Krause and Guestrin (2005) we think there may be a mistake in applying the induction step (with “c(Xi)” fixed for different i in the proof), though they were loosely bounded with β later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Proof The proof will follow by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We first show the base case i = 1 using Theorem 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' f( �G1) = f( �G1) − f( �G0) (f is normalized;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' �G0 = ∅) ≥ c(�g1) B � f(OPT) − f( �G0) � − 2 � 1 + ˜Kc(�g1) B � ϵ (using Theorem 18) = � 1 − � 1 − c(�g1) B �� f(OPT) − 2 � 1 + ˜Kc(�g1) B � ϵ (36) where (36) follows from rearranging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' For the second term in (36), using that 1 + ˜Kc(�g1) B ≤ B c(�g1) � 1 + ˜Kc(�g1) B � (since B c(�g1) ≥ 1) = B c(�g1) + ˜K ≤ B cmin + ˜K = β + ˜K, (37) then f( �G1) ≥ � 1 − � 1 − c(�g1) B �� f(OPT) − 2 � 1 + ˜Kc(�g1) B � ϵ (copying (36)) ≥ � 1 − � 1 − c(�g1) B �� f(OPT) − 2(β + ˜K)ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (using (37)) This completes the base case of i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' 29 We next consider i > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Unraveling the recurrence shown in Theorem 18,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='f( �Gi) = f( �Gi) − f( �Gi−1) + f( �Gi−1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='≥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='c(�gi) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='f(OPT) − f( �Gi−1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='− 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='1 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='˜Kc(�gi) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='ϵ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='+ f( �Gi−1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='(using Theorem 18) ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='� f(OPT) − 2(β + ˜K)ϵ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='(induction step) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='�1 − (1 − c(�gi) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='B ) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='1 − c(�gi) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='�1 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='i−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='j=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='(1 − c(�gj) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=') ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='� f(OPT) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='− 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='1 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='˜Kc(�gi) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='1 − c(�gi) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='(β + ˜K) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='ϵ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='(rearranging) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='�1 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='i� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='j=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='(1 − c(�gj) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=') ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='� f(OPT) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='− 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='1 + β − β c(�gi) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='+ ˜K ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (38) For the second term in (38), using that β c(�gi) B = B cmin c(�gi) B (def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' of β) = c(�gi) cmin ≥ 1, (39) then −2 � 1 + β − β c(�gi) B + ˜K � ϵ = −2 � β + ˜K � ϵ + 2 � β c(�gi) B − 1 � ϵ (rearranging) ≥ −2 � β + ˜K � ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (using (39)) 30 Applying this to (38) completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' The inequality in Theorem 19 for the augmented greedy set of cardinality ℓ can be further simplified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We will use the following observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Lemma 21 The following inequality holds: f( �Gℓ) ≥ (1 − 1 e)f(OPT) − 2(β + ˜K)ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Proof Applying i = ℓ to Theorem 19 and bounding the coefficient for f(OPT), f( �Gℓ) ≥ � �1 − ℓ� j=1 (1 − c(�gj) B ) � � f(OPT) − 2(β + ˜K)ϵ ≥ � �1 − ℓ� j=1 (1 − c(�gj) c( �Gℓ) ) � � f(OPT) − 2(β + ˜K)ϵ (by construction, c( �Gℓ) > B) ≥ � �1 − ℓ� j=1 (1 − c( �Gℓ)/ℓ c( �Gℓ) ) � � f(OPT) − 2(β + ˜K)ϵ (using Fact 2) = � 1 − (1 − 1 ℓ )ℓ � f(OPT) − 2(β + ˜K)ϵ (simplifying) ≥ � 1 − 1 e � f(OPT) − 2(β + ˜K)ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (using Fact 3) Using the aforementioned lemmas, we are now ready to complete the proof for Theorem 3 (robustness of Greedy+ algorithm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We will bound the value of set GL using the results on the augmented greedy set (32) of cardinality ℓ, and in turn bound the value of the set S, the final output of Greedy+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Recall that Greedy+ chooses the set S to be either the best individual element (based on ˆf) a∗ ← arg maxe∈Ω ˆf(e) or the output of the greedy process GL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Let aOPT = arg maxe∈Ω f(e) denote the element with the highest value under f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Then f(a∗) ≥ ˆf(a∗) − ϵ ≥ ˆf(aOPT) − ϵ (by definition of a∗) ≥ f(aOPT) − 2ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (40) By construction (32), �Gℓ includes one more element a′ than �Gℓ−1 (and a′ maximizes (31)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' By submodularity, the marginal gain of a′ is bounded by f(a′) and in turn by the 31 best individual element based on surrogate function ˆf, f( �Gℓ−1) + f(aOPT) ≥ f( �Gℓ−1) + f(a′) (by definition of aOPT) ≥ f( �Gℓ−1) + � f( �Gℓ−1 ∪ a′) − f( �Gℓ−1) � (by submodularity) = f( �Gℓ−1 ∪ a′) = f( �Gℓ) (by construction (32)) ≥ (1 − 1 e)f(OPT) − 2(β + ˜K)ϵ, (41) where (41) follows from Theorem 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Also by construction (32), the greedy and augmented greedy processes match up to and including the set of cardinality ℓ − 1, so f(GL) ≥ f(Gℓ−1) (monotonicity) = f( �Gℓ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (By construction (32)) Thus, f(GL) + f(aOPT) ≥ f( �Gℓ−1) + f(aOPT) ≥ (1 − 1 e)f(OPT) − 2(β + ˜K)ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (using (41)) At least one of f(GL) and f(aOPT) is at least half of the value of the right hand side, max{f(GL), f(aOPT)} ≥ 1 2(1 − 1 e)f(OPT) − (β + ˜K)ϵ (42) Thus, for the chosen set S f(S) ≥ ˆf(S) − ϵ = max{ ˆf(GL), ˆf(a∗)} − ϵ ≥ max{ ˆf(GL), ˆf(aOPT)} − ϵ (a∗ is the element with largest ˆf value) ≥ max{f(GL) − ϵ, f(aOPT) − ϵ} − ϵ (element-wise dominance) = max{f(GL), f(aOPT)} − 2ϵ ≥ 1 2(1 − 1 e)f(OPT) − (β + ˜K)ϵ − 2ϵ (from (42)) = 1 2(1 − 1 e)f(OPT) − (2 + β + ˜K)ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='5 Proof for Robustness of PartialEnumeration Now we analyze the PartialEnumeration algorithm for submodular maximization under a knapsack constraint proposed in Sviridenko (2004);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Khuller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Recall that Theorem 7 in Section 6 of the main paper states PartialEnumeration is a (1 − 1 e, 4 + 32 2 ˜K + 2β)-robust approximation algorithm for submodular maximization under a knapsack constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Proof Assume |OPT| > 3, otherwise the algorithm finds a (1, 2)-robust approximation, so it is also a (1− 1 e, 2( ˜K+β))-robust approximation for non-trivial cases where ˜K ≥ 1 and β ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Enumerate the elements of the optimal solution as OPT = {Y1, · · · , Ym}, corresponding to the order they would be selected by the simple greedy algorithm (iteratively selecting the element with the largest marginal gain, not the largest marginal density) Yi+1 = arg max Y ∈OPT f({Y1, · · · , Yi, Y }) − f({Y1, · · · , Yi}), (43) and let R = {Y1, Y2, Y3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Consider the iteration where the algorithm considers R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Define the function f′(A) = f(A ∪ R) − f(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (44) f′ is a non-decreasing submodular set function with f′(∅) = 0, and the optimal solution (with budget B − c(R)) is OPT \\ R since for any set S with cost c(S) ≤ B − c(R), f′(OPT \\ R) = f(OPT ∪ R) − f(R) (def of f′) = f(OPT) − f(R) (R ⊆ OPT by construction) ≥ f(S ∪ R) − f(R) = f′(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Hence we can apply Greedy+ algorithm to f′ (based on noisy evaluations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Let gℓ be the first element from OPT \\ R which could not be added due to budget constraints, and let A = {g1, · · · , gℓ−1} be first ℓ−1 elements selected by Greedy+ algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Let G = A∪R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Using Theorem 21, we get f′(A ∪ gℓ) ≥ (1 − 1 e)f′(OPT \\ R) − 2(β′ + ˜K′)ϵ, where β′ = B−c(R) c′ min , ˜K′ = min{n − 3, β′} and c′ min = mine∈Ω\\R c(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Simple calculation can show that β′ ≤ β and ˜K′ ≤ ˜K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Thus, f′(A ∪ gℓ) ≥ (1 − 1 e)f′(OPT \\ R) − 2(β + ˜K)ϵ, From the definition of f′, we have f(G) = f′(A) + f(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Let ∆ = f′(A ∪ gℓ) − f′(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We have f′(A) + ∆ ≥ (1 − 1 e)f′(OPT \\ R) − 2(β + ˜K)ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (45) Further observe that elements in OPT are ordered that for all 1 ≤ i ≤ 3, f({Y1, · · · , Yi}) − f({Y1, · · · , Yi−1}) ≥f({Y1, · · · , Yi−1, gℓ}) − f({Y1, · · · , Yi−1}) (ordering rule) ≥f(R ∪ A ∪ gℓ) − f(R ∪ A) ({Y1, · · · , Yi−1} ⊆ R when 1 ≤ i ≤ 3 and submodularity) =f(R ∪ A ∪ gℓ) − f(R) − (f(R ∪ A) − f(R)) =f′(A ∪ gℓ) − f′(A) =∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' 33 By telescoping sum, f(R) ≥ 3∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Now we get f(G) = f(R) + f′(A) ≥ f(R) + (1 − 1 e)f′(OPT \\ R) − 2(β + ˜K)ϵ − ∆ ≥ f(R) + (1 − 1 e)f′(OPT \\ R) − 2(β + ˜K)ϵ − f(R)/3 ≥ (1 − 1 3)f(R) + (1 − 1 e)f′(OPT \\ R) − 2(β + ˜K)ϵ ≥ (1 − 1 e) � f′(OPT \\ R) + f(R) � − 2(β + ˜K)ϵ (e ≤ 3) = (1 − 1 e)f(OPT) − 2(β + ˜K)ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (definition of f′) The output of the algorithm is not necessarily G because the values of the evaluated triplets are based on surrogate function ˆf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Denote O as the output of the algorithm and denote G′ as the best evaluated set (with respect to ˆf) with size ℓ + 2 (same as G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We must have that ˆf(G′) ≥ ˆf(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Also denote the final set (until violating budget) continuing G′ as G′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We have, f(O) ≥ ˆf(O) − ϵ ≥ ˆf(G′′) − ϵ (selection rule of the algorithm) ≥ f(G′′) − 2ϵ ≥ f(G′) − 2ϵ (G′ ⊆ G′′ and monotonicity of f) ≥ ˆf(G′) − 3ϵ ≥ ˆf(G) − 3ϵ ≥ f(G) − 4ϵ ≥ (1 − 1 e)f(OPT) − (4 + 2β + 2 ˜K)ϵ, finishing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Proof for Regret of C-ETC In this section, we prove Theorem 3 in Section 4 of the main paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We restate the theorem: For the sequential decision making problem defined in Section 2 and T ≥ 2 √ 2N δ , the expected cumulative α-regret of C-ETC using an (α, δ)-robust approximation algorithm as subroutine is at most O � δ 2 3 N 1 3 T 2 3 log(T) 1 3 � , where N upper-bounds the number of value oracle queries made by the offline algorithm A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='1 Overview and Notations We will separate the proof into two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' The first case is for when the clean event E happens, which we will show in Theorem 24 happens with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Under the 34 clean event, using the fact that the offline algorithm is an (α, δ)-robust approximation, C- ETC’s chosen set S for the exploitation phase will nonetheless be near-optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' The second case is when the complementary event happens, which occurs with low probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' The proof structure analyzing a high-probability “clean event” where empirical estimates are sufficiently concentrated around their means is analogous to that for the unstructured non-combinatorial setting (see for instance, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='2 in (Slivkins, 2019)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' However, un- like the ETC procedure for non-combinatorial MAB problems, C-ETC makes sequences of decisions during exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Furthermore, the combinatorial action space, non-linearity of the reward function, and lack of extra feedback (like marginal gains) make the problem challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Even in the special setting of deterministic rewards, the standard MAB prob- lem becomes trivial (finding the largest of n base arms) while the problem we considered are NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Recap that for any (feasible) action A, ft(A) denotes a (random) reward at time t for the agent taking that action, f(A) denotes the expected value for action A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Let ¯ft(A) denote the empirical mean of rewards received from playing action A up to and including time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' In the following, we will drop the subscript t from the empirical mean, writing ¯f(A) when it is clear from context that action A has been played m times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Also, we use Ai, i ∈ {1, · · · , N} denotes the i-th action the algorithm samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We further denote Ti, i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' , N} as the time step when the sampling of the i-th action has been determined, or Ai has been played m times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' For notation consistency, we also denote T0 = 0 and TN+1 = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='2 Probability of the Clean Event Now we define events that are important in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Recall that for each action A being explored, the m rewards are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' with mean f(A) and bounded in [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Thus, we can bound the deviation of the (unbiased) empirical mean ¯f(Ai) from the expected value f(Ai) for each action played.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Specifically, we can use a two-sided Hoeffding bound for bounded variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Remark 22 For convenience, we assume the reward function bounded in [0, 1], but the result can be generalized to the case where the deviation of the true reward and the expected reward has a light tailed distribution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=', sub-Gaussian).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Lemma 23 (Hoeffding’s inequality) Let X1, · · · , Xn be independent random variables bounded in the interval [0, 1], and let ¯X denote their empirical mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Then we have for any ϵ > 0, P ��� ¯X − E[ ¯X] �� ≥ ϵ � ≤ 2exp � −2nϵ2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (46) By C-ETC, each sampled action will be played the same number of times, denoted by m, so we consider bounding the probabilities of equal-sized confidence radii rad := � log(T)/2m for all the actions played during exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We next analyze the probability of the event that the empirical means of all actions played during exploration are concentrated around their statistical means within a radius rad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Denote the corresponding events for each action played having empirical means con- centrated around their respective statistical means as Ei, Ei := � { �� ¯f(Ai) − f(Ai) �� < rad}, i ∈ {1, · · · , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (47) 35 Define the clean event E to be the event that the empirical means of all actions played in the exploration phase are within rad of their corresponding statistical means: E := E1 ∩ · · · ∩ EN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (48) Lemma 24 The probability of the clean event E (48) satisfies: P(E) ≥ 1 − 2N T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Proof Applying the Hoeffding bound Theorem 23 to the empirical mean ¯f(Ai) of m rewards for action Ai and choosing ϵ = rad = � log(T)/2m gives P( ¯Ei) = P ��� ¯f(Ai) − f(Ai) �� ≥ rad � ≤ 2exp � −2mrad2� = 2exp (−2m(log(T)/2m)) = 2exp (− log(T)) = 2 T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (49) Then, we can bound the probability of clean events P(E) = P(E1 ∩ · · · ∩ EN) = 1 − P( ¯E1 ∪ · · · ∪ ¯EN) (De Morgan’s Law) ≥ 1 − N � i=1 P( ¯Ei) (union bounds) ≥ 1 − 2N T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (using (49)) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='3 Near Optimality of the final S (Exploitation Phase Action) In Theorem 24, we showed that the clean event E will happen with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' When the clean event E happens, we have | ¯f(A) − f(A)| ≤ rad for all evaluated action A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' For an online algorithm (with output S) using an (α, δ)-robust approximation as subroutine, we have f(S) ≥ αf(OPT) − δ · rad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (50) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='4 Final Regret Now we are ready to show the regret of C-ETC (Theorem 3 in Section 4 of the main paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' 36 Case 1: clean event E happens In the first case we analyse the expected regret under the condition that the clean event E happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' In this section, all expectations will be conditioned on E, but to simplify notation we will write E[·] instead of E[·|E] in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' First we can break up the expected α-regret conditioned on E into two parts, one for the first L exploration iterations, and the second for the exploitation iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Although the number of actions taken per iteration and the number of iterations of the greedy is not known a priori, we can upper bound the duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Also recall that ft(At) is the random reward for taking action At, which itself is random, depending on empirical means of actions in earlier iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' E[R(T)|E] = αTf(OPT) − T � t=1 E[ft(At)] = αTf(OPT) − T � t=1 E[E[ft(At)|At]] (law of total expectation) = αTf(OPT) − T � t=1 E[f(At)] (f(·) defined as expected reward) = T � t=1 (αf(OPT) − E[f(At)]) (rearranging) = N � i=1 m (αf(OPT) − E[f(Ai)]) � �� � Exploration phase + T � t=TN+1 (αf(OPT) − E[f(At)]) � �� � Exploitation phase = N � i=1 m (αf(OPT) − E[f(Ai)]) + T � t=TN+1 (αf(OPT) − E[f(S)]) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (51) Case 1 (clean event): Bounding exploration regret: We will separately bound the regret incurred from the exploration and exploitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We begin with bounding regret from exploration, N � i=1 m (αf(OPT) − E[f(Ai)]) ≤ N � i=1 m (α − 0) (rewards are bounded in [0, 1]) ≤ Nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (52) Case 1 (clean event): Bounding exploitation regret: We next bound the regret incurred during the exploitation iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Since the set S used during exploitation is a random variable, we can take the expectation of (50) (conditioned on event E), to bound 37 the expected instantaneous regret for each time step of the exploitation iteration, αf(OPT) − E[f(S)] ≤ δrad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (53) Using a loose bound for the duration of the exploitation iteration, T − TL + 1 < T, T � t=TN+1 (αf(OPT) − E[f(S)]) ≤ T � t=TN+1 δrad (using (53)) ≤ Tδrad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (54) Case 1 (clean event): Bounding total regret: Then the expected cumulative regret (51) can be bounded as E[R(T)|E] = N � i=1 m (αf(OPT) − E[f(Ai)]) + T � t=TN+1 (αf(OPT) − E[f(S)]) (copying (51)) ≤ Nm + Tδrad (using (52), (54)) Plugging in the formula for the confidence radius rad = � log(T)/2m, we have E[R(T)|E] ≤ Nm + Tδ � log(T)/2m We want to optimize m, the number of times each action is played.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Denoting the regret bound (55) as a function of m g(m) = Nm + Tδ � log(T)/2m, (55) then g′(m) = N − 1 2Tδ � log(T)/2m−3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (56) Setting g′(m) = 0 and solving for m, m∗ = δ2/3T 2/3 log(T)1/3 2N2/3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (57) We next check the second derivative, g′′(m) = 3 4δT � log(T)/2m−5/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (58) For positive values of m, g′′(m) > 0, thus g(m) reaches a minimum at (57).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Since m is the number of times actions are played, we (trivially) need m ≥ 1 and m to be an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We choose m† = � δ2/3T 2/3 log(T)1/3 2N2/3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (59) 38 Since from (58) we have that g′′(m) > 0 for positive m, g(m∗) ≤ g(m†).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' For T ≥ 2 √ 2N δ , we have m∗ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Plugging (59) back in to (55), E[R(T)|E] ≤ m†N + Tδ � log(T)/2m† ((55) with m† samples for each action) = ⌈m∗⌉N + Tδ � log(T)/2⌈m∗⌉ ≤ ⌈m∗⌉N + Tδ � log(T)/2m∗ (Since ⌈m∗⌉ ≥ m∗) ≤ 2m∗N + Tδ � log(T)/2m∗ (Since m∗ ≥ 1, ⌈m∗⌉ ≤ 2m∗) = 2δ2/3T 2/3 log(T)1/3 2N2/3 N + Tδ � log(T)/2 � δ2/3T 2/3 log(T)1/3 2N2/3 �−1/2 (using (57)) = 3δ2/3N1/3T 2/3 log(T)1/3 (60) = O � δ 2 3 N 1 3 T 2 3 log(T) 1 3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' In conclusion, the expected α-regret of C-ETC using an (α, δ)-robust approximation as subroutine is upper bounded by (60) if the clean event E happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Case 2: clean event E does not happen We next derive an upper bound for the expected α-regret for case that the event E does not happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' By Theorem 24, P( ¯E) = 1 − P(E) ≤ 2N T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Since the reward function ft(·) is upper bounded by 1, the expected α-regret incurred under ¯E for a horizon of T is at most T, E[R(T)| ¯E] ≤ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (61) Putting it all together Combining Cases 1 and 2 we have, E[R(T)] = E[R(T)|E] · P(E) + E[R(T)| ¯E] · P( ¯E) (Law of total expectation) ≤ 3δ2/3N1/3T 2/3 log(T)1/3 · 1 + T · 2N T (using (60), Theorem 24, and (61)) = O � δ 2 3 N 1 3 T 2 3 log(T) 1 3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' 39 Algorithm 2 Online Greedy for Opaque Feedback Model (OGo) Input: set of base arms Ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' horizon T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' cost for each arm c(a),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' budget B Initialize n ← |Ω|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' cmin ← mina∈Ω{c(a)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' β ← B cmin ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' γ ← n1/3β � log(n) T �1/3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' ϵ ← � β log(n) γT Initialize ω1 ← ones(β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' n) for t ∈ [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' T] do St ← ∅ l ← zeros(β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' n) // loss Randomly sample a value ξ ∼ Uniform([0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' 1]) if ξ ≤ γ then e ∼ Uniform({1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' β}) for i ∈ [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' e − 1] do // For experts before e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' exploit Select an arm a with probability ωt[i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='a] � ωt[i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=':],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' re-sample if a ∈ St St ← St ∪ {a} with probability cmin c(a) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' St ← St−1 otherwise end for a ∼ Uniform({1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' n}\\St) // For expert e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' explore St ← St ∪ {a} Play action St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' observe ft(St) Update l[i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' j] ← cminft(St) c(a) for all i = e and j ̸= a // Feed cminft(St) c(a) back to expert e associated with action a Update ωt+1[i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' j] ← ωt[i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' j] exp(−ϵl[i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' j]) for all pairs of i and j else // Exploitation with probability 1 − γ for i ∈ [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' β] do // For experts before e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' exploit Select arm a with probability ωt[i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='a] � ωt[i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=':],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' re-sample if a ∈ St St ← St ∪ {a} with probability cmin c(a) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' St ← St−1 otherwise end for Play action St, observe ft(St) ωt+1[i, j] ← ωt[i, j] // Since feeding back 0 to all expert-action payoffs, loss is 0, no update end if end for D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Implementation of Algorithm OGo In this section we describe implementation details and parameter selection for OGo algorithm Streeter and Golovin (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' The choice of exploration probability is given by the original paper:γ = n1/3β � log(n) T �1/3 , where β = B/cmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Note that in the original paper, B is used instead of β, because they assume the minimum cost is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Here we generalize it to arbitrary non-negative costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' ϵ is the learning rate for Randomized Weighted Majority (WMR) expert algorithm Arora et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' It is chosen by setting the derivative of regret 40 upper bound to zero, which is ϵ = � log(n) Te , where Te is the time spent on updating expert e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Since it explores with probability γ, and there are β expert algorithms, we have Te ≈ γT β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Thus we pick ϵ = � β log(n) γT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' In experiments, there are many cases the chosen γ is large or even larger than 1, so we cap the probability of exploring γ by 1/2 to avoid exploring too much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Note that unlike a hard budget in our setting, for OGo, it only requires the budget to be satisfied in expectation, so in general we might choose sets over budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Algorithm 2 is the pseudo code for implementation details of OGo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Comments on Lower bounds of Submodular CMAB For the setting we explore in this paper, with stochastic (or even adversarial) knapsack- constrained combinatorial MAB with submodular expected rewards and just bandit feed- back, it remains an open question if ˜O(T 1/2) expected cumulative α-regret is possible (ig- noring n and β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Both Streeter and Golovin (2008) and Niazadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (2021) analyze lower bounds for the adversarial setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' However, Streeter and Golovin (2008) obtain bounds for 1-regret (it is NP-hard in offline setting to obtain an approximation ratio better than 1 − 1/e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Niazadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (2021) obtain ˜Ω(T 2/3) lower bounds for the harder setting where feedback is only available during “exploration” rounds chosen by the agent, who incurs an associated penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Dealing with Small Time Horizons in Experiments In Section 6, we used N = ˜Kn as an upper bound on the number of function evaluations for both C-ETC-K and C-ETC-Y, where n is the number of base arms and ˜K is an upper bound of the cardinality of any feasible sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' When the time horizon T is small, it is possible that the exploration phase will not finish due to the formula being optimized for m (the number of plays for each action queried by A) uses a loose bound on the exploitation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' When this is the case, we select the largest m (closest to the formula) for which we can guarantee that exploration will finish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Recall that for C-ETC-Y and C-ETC-K, the number of oracle calls can only be upper bounded in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We first calculate m† using (59): m† = � δ2/3T 2/3 log(T)1/3 2 ˜K2/3n2/3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Note that a (slightly tighter) upper bound on the number of subsets evaluated during the exploration phase (with ˜K bounding the number of iterations of the greedy process) is N ≤ n + (n − 1) + · · · + (n − ˜K + 1) = � n − ˜K 2 + 1 2 � ˜K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We compare � n − ˜ K 2 + 1 2 � ˜Km† with T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' 41 Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' If � n − ˜ K 2 + 1 2 � ˜Km† < T, C-ETC can finish exploring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We select m = m†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' If � n − ˜ K 2 + 1 2 � ˜Km† ≥ T, it is possible that the algorithm cannot finish exploring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' In this case, we will find a new m, so that the exploration can be guaranteed to finish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We select the largest m (closest to m†) so that the exploration time is upper bounded by T, m = T � n − ˜ K 2 + 1 2 � ˜K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Basic Facts Fact 1 For a monotonically non-decreasing submodular set function f defined over subsets of Ω, we have for arbitrary subsets A, B ⊆ Ω, f(B) − f(A) ≤ � j∈B\\A [f(A ∪ {j}) − f(A)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Fact 2 (Khuller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=', 1999) For x1, · · · , xn ∈ R+ such that � xi = A, the function [1 − �n i=1(1 − xi A )] achieves its minimum at x1 = x2 = · · · = xn = A/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Fact 3 For k ≥ 1, 1 − � 1 − 1 k �k ≥ 1 − 1 e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Other Related Work for Adversarial CMAB with Knapsack constraints Streeter and Golovin (2008) propose and analyze an algorithm for adversarial CMAB with submodular rewards, full-bandit feedback, and under a knapsack constraint (though only in expectation, taken over randomness in the algorithm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We discuss this in more detail in the supplemental material, here only highlighting a few key points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We also use this as a baseline in our experiments in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' The authors adapted a simpler greedy algorithm than the one we adapt (Khuller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=', 1999), using an ϵ-greedy exploration type framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We provide evidence in our experiments that their algorithm requires large horizons to learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' The offline algorithm they adapted achieves an approximation ratio (1 − 1/e) for budgets that exactly match the cost used up by the greedy solution, but otherwise does not achieve a constant approximation (Khuller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=', 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' In (Golovin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=', 2014), the authors propose an algorithm for adversarial setting with submodular rewards when there is a matroid constraint (neither knapsack nor matroid constraints are special cases of the other).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Related work on Stochastic Submodular CMAB with Semi-Bandit Feedback There are also a number of works that require additional “semi-bandit” feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' For combinatorial MAB with submodular rewards, a common type of semi-bandit feedback are 42 marginal gains (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Yue and Guestrin, 2011b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Takemori et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=', 2020b), which enable the learner to take actions of maximal cardinality or budget, receive a corresponding reward, and gain information not just on the set but individual elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' For the full-bandit setting we consider, to greedily build a solution, we need to spend time taking small cardinality actions to estimate their quality, incurring regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Experiments with Song Recommendation We test our methods on the application of song recommendation on the Million Song Dataset Bertin-Mahieux et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' In this problem, the agents aims to recommend a bundle of songs to users such that they are liked by as many users as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Data Set Description and Experiment Details From the Million Song Dataset, we extract most popular 20 songs and 100 most active users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' As in Yue and Guestrin (2011a), we model the system as having a set of topics (or genres) G with |G| = d and for each item e ∈ Ω, there is a feature vector x(e) := (Pg(e))g∈G ∈ Rd that represents the information coverage on different genres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' For each genre g, we define the probabilistic coverage function fg(S) by 1 − � e∈S (1 − Pg(e)) and define the reward function f(S) = � i wifi(S) with linear coefficients wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' The vector w := [w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' , wd] represents user preference on genres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' In calculating Pg(e) and w, we use the same formula for calculating ¯w(e, g) and θ∗ in Hiranandani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Like Takemori et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (2020a), we define the cost of a song by its length (in seconds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' For each user, the stochastic rewards of set S are sampled from a Bernoulli distribution with parameter f(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' For the total reward, we take the average over all users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' When making the plots, we use statistics taken from 10 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Results and Discussion Figures 2a and 2b show average cumulative regret curves for C-ETC-K (in blue), C- ETC-Y (in orange) and OGo (in green) for different horizon T values when the budget constraint B is 500 and 800, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Figures 2c and 2d are the instantaneous reward plots over a single horizon T = 215, 443.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Again, C-ETC significantly outperforms OGo for all time horizons and budget considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' We again estimated the slopes for both methods on log-log scale plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' Over the horizons tested, OGo’s cumulative regret (averaged over ten runs) has a growth rate above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' The growth rates of C-ETC-K for budgets 500 and 800 are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='70 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='73, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' The growth rates of C-ETC-Y for budgets 500 and 800 are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='70 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='71, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' 43 (a) (b) (c) (d) Figure 2: Plots for song recommendation example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (a) and (b) are comparison results for cumulative regret as a function of time horizon T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' (c) and (d) are the moving average plot with window size 100 of instantaneous reward as a function of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' The gray dashed lines in (a) and (b) represent y = aT 2/3 for various values of a for visual reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' The gray dashed lines in (c) and (d) represent expected rewards for the action chosen by an offline greedy algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content=' 44 SR B=500 Cumulative Regret 10° 4 C-ETC-K C-ETC-Y 103 OGo 104 105 Horizon TSR B=800 Cumulative Regret 10° 4 C-ETC-K C-ETC-Y 103 3 OGo 104 105 Horizon TSR B=500 le-1l Instantaneous Reward 6 5 4 ３ C-ETC-K C-ETC-Y 2 OG° 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='0 1e5 Time-step tSR B=800 1e-1 Instantaneous Reward 6 4 C-ETC-K C-ETC-Y 2 OG° 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} +page_content='0 1e5 Time-step t' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFQT4oBgHgl3EQfcDb0/content/2301.13326v1.pdf'} diff --git a/_NAzT4oBgHgl3EQfS_tm/content/tmp_files/2301.01241v1.pdf.txt b/_NAzT4oBgHgl3EQfS_tm/content/tmp_files/2301.01241v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..14a34fa0de9de2c35d34c72d39bfcfddff368c6e --- /dev/null +++ b/_NAzT4oBgHgl3EQfS_tm/content/tmp_files/2301.01241v1.pdf.txt @@ -0,0 +1,1055 @@ +1 +Developing and deploying deep learning models in brain MRI: +a review +Kunal Aggarwal1.2, Marina Manso Jimeno3,4, Keerthi Sravan Ravi3,4, Gilberto Gonzalez5, Sairam +Geethanath1 + +1 Accessible MR Laboratory, Biomedical Engineering, and Imaging Institute, Dept. of Diagnostic, +Molecular and Interventional Radiology, Mount Sinai Hospital, New York City, New York +2 Department of Electrical and Computer Engineering, Technical University Munich, Munich, +Germany +3 Department of Biomedical Engineering, Columbia University in the City of New York, New York +City, New York, USA +4 Columbia University Magnetic Resonance Research Center, Columbia University in the City of +New York, New York City, New York, USA +5 Division of Neuroradiology, Department of Radiology, Massachusetts General Hospital, +Boston, Massachusetts + + +Word Count: 5952 + + + + + + + + +2 +Abstract +Magnetic Resonance Imaging (MRI) of the brain has benefited from deep learning (DL) to alleviate +the burden on radiologists and MR technologists, and improve throughput. The easy accessibility +of DL tools have resulted in the rapid increase of DL models and subsequent peer-reviewed +publications. However, the rate of deployment in clinical settings is low. Therefore, this review +attempts to bring together the ideas from data collection to deployment into the clinic building on +the guidelines and principles that accreditation agencies have espoused. We introduce the need +for and the role of DL to deliver accessible MRI. This is followed by a brief review of DL examples +in the context of neuropathologies. Based on these studies and others, we collate the +prerequisites to develop and deploy DL models for brain MRI. We then delve into the guiding +principles to practice good machine learning practices in the context of neuroimaging with a focus +on explainability. A checklist based on the FDA's good machine learning practices is provided as +a summary of these guidelines. Finally, we review the current challenges and future opportunities +in DL for brain MRI. + +Keywords: Accessible MRI, Neuroimaging, GMLPs, Explainable AI, FDA, Deep Learning, +Deployment, Brain + + + + +3 +Abbreviations: +CAD: Computer Aided Detection +CAM: Class Activation Mapping +CNN: Convolutional Neural Network +DAGMNet: Dual attention gate network +DICOM: Digital Imaging and Communications in Medicine +DSC: DICE Similarity Coefficient +FL: Federated Learning +GDPR: General Data Protection Regulation +GMLPs: Good Machine Learning Practices +Grad-CAM: Gradient-weighted Class Activation Mapping +HR: High-Resolution +IRB: Institutional Review Board +LR: Low-Resolution +MCI: Mild Cognitive Impairment +OECD: Organisation for Economic Co-operation and Development +OOD: Out-of-Distribution +PPML: Privacy Protecting Machine Learning +PSNR: Peak Signal-to-Noise Ratio +RF: Radio Frequency +SSIM: Structural Similarity Index Measure +XAI: Explainable Artificial Intelligence + + + +4 +Introduction +As of 20181-2, the density of MR scanners was least in geographies with the highest populations +such as in sub-Saharan Africa and the Indian subcontinent. The severe lack of neurosurgeons +and skilled human resources required to operate, use, and interpret data from MR scanners is a +major barrier to accessing this life-saving technology2. In contrast, contemporary radiology +departments in the Organisation for Economic Co-operation and Development (OECD) countries +require a close interplay of a team with diverse expertise: radiologists specializing in different +anatomical sites, MR technologists, medical physicists, and radiologic nurses supported by the +vendor’s service and application engineers. This MR inaccessibility and the resulting disparity +necessitates the development and deployment of automated methods to augment existing local +expertise. Recently, there has been the development of autonomous MRI methods to automate +protocolling3-4, identify artifacts5-6, and reconstruct images from accelerated scans7. These +advances are expected to assist local MR technicians, accelerate acquisitions, improve +throughput by assisting or replacing manual data processing steps, and reduce waiting times for +radiology reporting. In addition to positively impacting MR accessibility, these automation methods +facilitate MRI-based big data studies such as the Human Connectome Project8, UK Biobank9, and +Rhineland study10, among others. This is due to the high volume and velocity associated with +such studies. + +These automation methods leverage the recent resurgence of artificial intelligence techniques in +general and supervised learning methods in particular. A deep cascade of neural networks - +termed deep learning (DL) models - is trained with a set of input features on one side and the +resulting outcomes (labels) on the other side, using existing apriori annotated data11. These +trained DL models are then validated against an unseen but similar data set to fine-tune +“hyperparameters” of the DL model. Subsequently, the tuned DL model is used to perform + +5 +inference on a test dataset to evaluate its performance by comparing it with a human-evaluated +outcome. Finally, once the classification-tasked model performs satisfactorily with respect to +false-positive (FP), false-negative (FN), true-positive (TP), and true-negative (TN) prediction +metrics (captured in a “confusion matrix”), then it is considered for a deployment study and down- +stream accreditation processes11. + +In this review, we discuss studies that have developed and deployed deep learning models tasked +with multi-task classification. We then present an analysis of the related literature to provide +critical steps involved in data collection and curation required to set up deep learning studies. We +then highlight the importance of good machine learning practices and the explainability of the DL +results by illustrating examples and tools. A suggestive set of steps to enable mounting a +successful development and deployment of MRI DL models will be then discussed based on +recent literature. Finally, a brief overview of the challenges and opportunities related to MRI-based +DL studies is presented. + +MRI DL studies of the brain +Easy and direct availability of vast amounts of MRI data from publicly available repositories such +as HCP8, UK Biobank9, among others, as well as accessible tools to build12 and optimize DL +models13 have significantly accelerated the application of DL methods to address challenges in +MRI. These studies have resulted in a substantial body of peer-reviewed literature (Figure 1), with +most of them sharing an open-source implementation of their models with sample data. This +review focuses on MRI DL studies that meet the following criteria: (i) published in the last five +years; (ii) Methods mostly focusing on classification and not regression tasks; (iii) studies +incorporating explainable AI components; (iv) works demonstrating deployment of the DL models +and preferably across multiple vendors and sites. These criteria were used to focus our review on + +6 +specific developments. Notably, the generic tools developed by mathematicians and computer +scientists in the DL community for explainable AI such as GradCAM14 and other methods15-16 have +focused more on classification tasks compared to regression tasks (Figure 1). The quantification +of the outcomes of a classifier network is relatively straightforward compared to a regression +model. The outcomes are directly compared to the “true annotated labels'' and hence result in a +binary or a multi-class decision. These can easily be binned into TP, FP, TN, and FN and +evaluated for sensitivity and specificity. In contrast, regression models accomplishing tasks such +as DL denoising and synthesizing images are quantified using peak signal-to-noise ratio (PSNR) +and structural similarity index measure (SSIM). These metrics have a continuous value and are +hard to set thresholds for acceptance. In addition, regression models are typically explained using +activation maps that are subsequently interpreted by the authors rather than community-wide +tools independent of the developed methods. The interested reader is referred to 7 for a detailed +review of machine learning-based image reconstruction methods that leverage regression +models. + +Example DL MRI solutions for neuroimaging +Brain tumors: Nalepa et al. have demonstrated a fully automated pipeline for DCE-MRI analysis +of brain tumors called Sens.AI DCE17. In particular, they have substituted manual segmentation +of brain tumors in T2 FLAIR images with deep-learning-based methods to demonstrate improved +reproducibility. They complement this with a real-time image processing algorithm to determine +the vascular input region. Finally, they include a new cubic model of the vascular function for PK +modeling. They have validated their package with the BraTs dataset for DICE coefficient and area +under the curve as well as by twelve readers from two institutions. These results show good to +excellent agreement between the gold standard BraTS dataset and Sens.AI DCE with a total +execution time of approximately 3 minutes. Another study focused on the automated identification +and classification of brain tumor MRI data classified into glioma, meningioma, and pituitary tumors + +7 +with an accuracy of 93.7% and a DICE similarity coefficient (DSC) of 95.8%18. A subsequent task +of classifying gliomas into high or low grade had an accuracy of 96.5% and a DSC of 94.3%. +These models were based on the GoogleNet variant architectures to efficiently combine local +features. The identification and classification tasks were accomplished in less than 3 minutes. +The method compared well with other state-of-the-art methods with respect to accuracy. Although +the study did not explicitly focus on explainable AI methods such as Grad-CAM for interpretation, +the authors performed an ablation study to demonstrate the effect of the locally chosen features. +Brain extraction is a key step in multiple neuroimaging pre-processing pipelines and in complying +with privacy laws. This task becomes more challenging in the presence of pathology such as +diffuse gliomas as most analytical and deep learning methods focus on healthy brain extraction. +Thakur et al have addressed this gap by developing and testing their models for brain extraction +in the presence of diffuse glioma, in a multi-institutional manner19. The authors considered +multiparametric MRI data from private and public repositories acquired with different acquisition +protocols to train a “modality-agnostic” tool that does not require retraining. The work +demonstrated a similar or better accuracy compared to other brain extraction models that worked +only on healthy brain extractions. The interested reader is pointed to these reviews for further +reading on deep learning methods for brain tumor imaging20, classification21, and segmentation22. + +Stroke: The need for automated segmentation and classification of images, especially in +emergency room settings in this time-critical pathology is well understood. In this direction, Liu et +al developed a deep learning model that detected and segmented abnormalities in acute ischemic +stroke23. The work included several steps of pre-processing the data, such as skull stripping and +DWI intensity normalization, among others. The study compared T-score and the modified c-fuzzy +methods for lesion segmentation. In addition, the authors implemented the 3D Dual attention gate +network (DAGMNet) as a supervised learning method to delineate the lesions. The developed +model performs better than the unsupervised generic tools and is faster, publicly available, and + +8 +easy to deploy. They tested their algorithm on a hold-out data set of 280 MRIs and quantified the +improved performance using DICE scores, precision, sensitivity, subject detection rate, DICE +scores for the lesion volumes and lesion DWI contrast. Another study on stroke detection +performed by Zhang et al demonstrated an accuracy of 89.77% over 300 ischemic stroke +patients24. The authors evaluated three network architectures with labels drawn by experienced +radiologists from two hospitals. Their models predicted a bounding box covering the lesions. +Statistical analysis performed on the location, size, and shape correlated well with the radiologists’ +labeling. This implementation aids in rapid localization and preliminary characterization of the +lesion. The authors have committed to making the data available once a thousand patient data +are collected. An important task in imaging stroke is to grade the severity of the ischemia. A multi- +class classification solution for detecting the severity has been developed by Acharya et al.25 The +authors extracted higher-order features from MR images, such as bispectrum entropy and its +phase, followed by support vector machines to classify the severity of the stroke into LACS, PACS +and TACS. The algorithm demonstrated high levels of accuracy without the need for any manual +intervention to augment the neuroradiologist. + +Alzheimer’s disease: Supervised learning models have demonstrated the utility of automation +in the imaging of Dementia. A specific challenge in this area is the classification and staging of +the progression in mild cognitive impairment (MCI) patients. Kwak et al. have developed a deep +learning model based on brain atrophy patterns and associated these changes with differences +in amyloid burden, cognition, and metabolism26. This model is used to classify AD patients from +cognitively normal subjects. A secondary classification helps identify the trajectory of cognitive +decline in individuals with MCI. These results were validated with cognitive tests, fluid biomarkers, +and PET uptake data with good agreement. The approach is expected to benefit from integrating +cognitive and neurobiological features to capture the heterogeneity of MCI. Another approach +involved the development and testing of whole-brain 3D convolutional neural networks to detect + +9 +AD27. This model did not include any patient-specific information to allow the generalization of the +algorithm. The implementation included four steps of brain extraction, normalization, 3D CNN +followed by domain adaptation. A key feature of this study is the authors' focus on accomplishing +accountability. The method outperformed other state-of-the-art methods in the CAD Dementia +challenge test, along with explainable AI visualizations to aid the interpretation of classification +results. Ahmed et al. also used a 3D approach but collected an ensemble of 2D patches in three +orientations to train an ROI-based neural network to stage AD using MR images as per NIA +labels28. This approach was demonstrated on the GARD and ADNI datasets. This method was +compared to other state-of-the-art methods, and the performance was similar or better. The +landmarks delineated by the algorithm to indicate AD correlated well with known neuroanatomical +areas. However, the model could not accurately predict asymptomatic AD based on the high rates +of false positives. + +Prerequisites for DL-based neuro-MRI +Data from routine MRI studies result in high volume, velocity, and variety: characteristics of big +data29–35. For training DL models, high volume and velocity are favorable factors: more data is +better than lesser; high velocity of data requires automated processing methods. However, the +variety in MRI data due to a large number of available acquisition parameters, reconstruction +methods, receive coil configurations, post-processing steps requires attention to fine details +before collating data for annotation and subsequent training. In line with the classification +suggested by Wald36, the sources of these variabilities need to be binned as emanating from the +MR system characteristics, subject-induced variations, and pathology-specific factors. Typically, +the goal of the DL models discussed in this work is to glean the subtle changes in +pathophysiological states based on the MR images. Therefore, standardization of parameters +getting affected by the system and subject priors is critical to ensure that the DL models focus + +10 +their attention on the pathology being investigated. The removal or reduction of the confounding +variables is therefore essential in building these DL models. This exercise is facilitated by the use +of explainable AI tools. Therefore, two critical requirements to understand an MRI-based DL study +are standardization of procedures and methods and explainable AI tools. A detailed discussion +on these prerequisites is performed below. +DL models require large amounts of data to achieve high accuracy levels37–39. Unlike models +trained on natural images, large datasets are challenging to achieve for medical imaging +applications37–39. Training a DL model of medical images entails data collection, curation, and +annotation. Additionally, data augmentation might be necessary in cases when the data are +insufficient or to strengthen the generalization ability of the model39–41. The steps of data collection +and annotation are the most expensive and time-consuming, and patient privacy policies often +restrict the use and sharing of images37,39,42,43. These factors significantly impact the models’ +performance in clinical settings, limiting their ultimate deployment40,42. The purpose of this section +is to review strategies and recommendations (Figure 2) at the data level for maximizing the +likelihood of successful deployment after training based on previously published work. +Data collection, including patient selection, imaging protocol, sequences, and scan parameters, +is determined based on the intended use of the model and its targeted application. Cohorts can +be prospective or retrospective, depending on whether the data are acquired for the study or +retrieved from a public or private repository44. An initial step in the process is the approval by an +Institutional Review Board (IRB) or a similar board. Additionally, participants provide informed +consent about the use of their personal data, which is typically de-identified during data curation. +Privacy Protecting Machine Learning (PPML) is a niche area of research that aims at maximizing +the confidentiality of patient data while optimizing its use on data-driven models45. In this field, +Federated Learning (FL) alleviates the shortage of data problems by allowing training models on +large-scale, multi-center data without data sharing. FL feasibility in MRI has been explored by + +11 +Sarma et al. and Sheller et al. for brain tumor and prostate segmentation tasks46,47. When dealing +with multi-institutional data, protocol harmonization can avoid model bias that may arise from +differences in image contrast, intensity, or noise distribution. In MRI, these differences may stem +from multiple sources, including variations in system manufacturer, field strength, Radio +Frequency (RF) coils, patient positioning, acquisition sequence and scan time, and even pre- +processing and reconstruction pipelines. +Publicly available databases are the results of extensive research projects and contain large +amounts of data that can be leveraged for training. These datasets are typically acquired using +the same protocol and scanner or using highly harmonized protocols. Patient inclusion and +exclusion criteria in these research cohorts are strict, and the datasets are well-curated and +usually undergo multi-step post-processing pipelines and standardization operations. While the +reproducibility of models trained on publicly available datasets is easier to assess, the data lack +the heterogeneity characteristic of clinical data observed during deployment. Martensson et al.43 +systematically studied the performance variability of a DL model trained on different combinations +of training sets, including publicly available datasets and more heterogeneous Out-of-Distribution +(OOD) datasets. They observed that performance drops when models trained on homogeneous +data are applied to clinical data. However, inference on clinical data showed a better agreement +level with a radiologist reading if clinical cohorts were present in the training data. +A benefit of local data acquisition for training is having access to raw data. Most public or private +imaging repositories contain data in the Digital Imaging and Communications in Medicine +(DICOM) format. The raw data undergo several pre-processing steps, including coil-combination, +filtering, artifact correction, and phase removal before storage, stripping the images of features +that DL models in the process could recognize. Raw data might be favored for certain DL tasks, +data augmentation techniques, or for data synthesis via forward modeling. This is exemplified by +the increasing usage of the fastMRI dataset48, the only publicly-available dataset of raw MR knee + +12 +and brain data. It has become a benchmark for the validation and reproducibility assessment of +DL-based image reconstruction algorithms. Additionally, models for the detection or correction of +k-space-occurring artifacts such as motion49–51 and Gibbs ringing5,52 typically leverage raw data +for the simulation of artifact-corrupted images. +Data collection is followed by data curation. This step is performed to standardize and improve +dataset quality for subsequent deep neural network training42. The data used for the model +development entails a trade-off between distribution heterogeneity and representation bias. It +should represent varying patient populations and anatomy disparities while avoiding biasing +network representation. Successfully deployed models are typically developed with data acquired +using the same imaging protocol and the same system as the site targeted for deployment53,54. +Failure mode analysis of the prospective evaluation of an automatic kidney segmentation model +after deployment revealed segmentation errors arising from common clinical scenarios such as a +fluid-filled stomach and a distended bladder54. These cases are typically excluded during cohort +building or data curation and reduce the model’s tolerance to data variations. Poor generalizability +to unseen domains is one of the major challenges to successfully deploying DL models in the +clinic55. +For fully and semi-supervised learning tasks, data labeling or annotation is typically the most time- +consuming step of an AI project. It may require localizing, delineating, or segmenting lesions or +organs of interest or labeling or annotating characteristics of the data. This step is performed +manually by an experienced reader via visual inspection. When human observations or clinicians' +expertise is required to annotate the data, multiple readers and ideally with variable levels of +experience, are preferred to estimate inter-reader variability and compare it to the model’s +performance. + +13 +Finally, data augmentation is the process of generating additional versions of the initial data to +enlarge the training set and improve the model's robustness, thereby avoiding overfitting56. +Typical techniques include translation, flipping, rotation, and cropping. Depending on the +application, other approaches may be useful, such as random k-space oversampling for model- +based reconstruction techniques. Data augmentation can also be leveraged to reduce the gap +between the training data and prospective clinical data, for example, by simulating noise and +motion artifacts in the images. A recent study57 for accelerated MR reconstruction demonstrated +that with the introduction of these commonly-occurring artifacts using MR physics-driven data +augmentation techniques, model performance on both in-distribution and OOD data increases +compared to state-of-the-art image-based data augmentation methods. + +Good Machine Learning Practices for MRI +AI's novelty and current regulatory paradigms are not well adapted to strike a balance between +patient safety and promoting the expansion of this new industry58. For assessing commercially +accessible algorithms to guarantee their dependability and safety, defining best practices is an +area of active research59,60, significant regulatory problems need to be resolved to move clinical +AI toward becoming safe and robust61. Wu et. al. reviewed the FDA database for parameters that +were used to evaluate the AI algorithms of products. The parameters they found were - (i) number +of patients and sites used in the evaluation, (ii) prospective and retrospective collection of data, +and (iii) whether the performance was stratified by disease subtypes or not. Based on the FDA +summary, the study revealed that 126 out of 130 AI devices conducted solely retrospective +investigations at their submission. The influence of the AI decision tool on clinical practice must +be fully characterized, though, and this is crucial since human-computer interaction might differ +significantly from a model's intended purpose. For instance, most computer-aided detection + +14 +(CAD) diagnostic tools are meant to serve as decision-support aids rather than primary diagnostic +instruments61. Instead of independently diagnosing, staging, or triaging pathology, CAD is meant +to identify, mark, highlight, or otherwise draw attention to imaging characteristics62. The FDA +suggests regulating AI software based on function rather than technical components or intended +use, which is different from the case for most pharmaceutical items, gadgets, and foods58. +Therefore, FDA advocates ten guiding principles for medical device development known as 10 +Good Machine Learning Practices (GMLPs) that take into consideration the prerequisites +discussed above. According to the first and second principle, all expertise related to the product +development should work together from the development phase until integration into the clinical +workflow. This includes neuroradiologists, neuroimaging scientists, MR technicians, and data +scientists implementing good software engineering and security practices. The third principle +states the importance of metadata in developing DL models and connects to the concept of data +security from the second principle. The fourth and fifth principles mention the importance of +datasets. The training and testing datasets should be independent and the reference datasets +should have the same characteristics as of the patients in the former datasets. According to the +sixth principle, the intended use of a model must be clearly defined along with its risks and +performance limitations on different datasets. This relates to principle number three in a sense +that metadata defines the scope of the model being used on the specific patients. Principle seven +states the involvement of humans and the fact that human intervention cannot be avoided at any +stage of development or deployment. This principle focuses more on AI in the loop rather than +human in the loop. Eighth and ninth principle centers on the user and states that the model should +be easy to understand for the end user and must list all the possible precautions in order to avoid +harm to the patient. Principle ten mentions that updates in the model are a mandatory part of the +DL deployment and must be considered frequently59. A checklist based on these GMLPs is +provided as a summary specifically designed for experts working in brain MRI (Table 1). + +15 +Explainable AI +Although deep learning techniques produce outcomes, they do not explain how those results were +obtained. One cannot just analyze the deep neural network to understand how that choice was +made. As a result, deep learning models are sometimes referred to as "Black Boxes"63. Medical +professionals believe these "black boxes" may be prejudiced in some way, which might have +negative effects when used in practical applications63. Additionally, laws like the General Data +Protection Regulation (GDPR, Article 15) of the European Union specify that patients have the +right to request an explanation for how a given diagnosis was reached if the standard deep +learning models cannot64. Therefore, Explainable Artificial Intelligence (XAI) techniques recently +developed with the primary objective of visualizing and interpreting the results of machine learning +(ML) and deep learning (DL) networks represent a potential remedy to close this gap between +high performance and deep-level understanding65 (Figure 3). They have been utilized in a variety +of applications, including the categorization of ECGs66 and the visualization of feature maps at +various Convolutional Neural Network (CNN) layers67. +Velden et al. categorized XAI approaches into three groups based on three criteria: (i) model- +based vs post-hoc; (ii) model-specific against model-agnostic; and (iii) global versus local. These +categories are visual, textual, and example based. The most prevalent type of XAI in medical +imaging, out of these three categories, is the visual explanation. These approaches, sometimes +referred to as saliency mapping, employ a backpropagation methodology to highlight the key +elements of a picture for a certain model’s decision by emphasizing the pixels that had the +greatest influence on the results of the investigation64. Class activation mapping (CAM), a +technique used in the backpropagation methodology, was introduced by Zhou et al. in 2016. They +used global average pooling on the last convolutional feature maps to substitute the fully +connected layers at the conclusion of a CNN. It is a weighted linear sum of the visual patterns +that were observed and recorded by the filters at various spatial positions68. + +16 +Gradient-weighted class activation mapping is a generic strategy that includes CAM as one of its +specialized methods (Grad-CAM). Grad-CAM can operate with any CNN, but CAM needs global +average pooling in particular64. Grad-CAM delivers the ROI on an input image that has the +greatest influence on class prediction. Grad-CAM allows us to track the spatial attention changes +that occur between network layers, or more precisely, what each network layer focuses on in each +input image. To do this, the output gradient with respect to each neuron in the network is +calculated to ascertain its relative significance69. +Grad-CAM has been widely used to describe deep learning models. Jimeno et. al. used it to +identify and classify wrap-around and Gibbs ringing artifacts5. It was used by Windisch et al. in +2020 to identify brain MRI regions that caused the classifier to determine the existence of a +malignancy70. A model’s prediction of the fetus's brain age may also be explained using Grad- +CAM, according to a 2020 publication by Liao et al., which will help avoid congenital +malformations71. It was also employed by Natekar et al. in 2020 to describe the brain tumor +segmentation network69. +Occlusion Sensitivity technique is another XAI tool64. The input MR image is disturbed by a small +perturbation, and the categorization choice is changed and examined. In order to quantify the +variation in the output prediction, it covers a piece of the input picture with a black patch. After +moving the patch across the whole picture, it is simple to determine which parts of the brain are +responsible for the categorization choice in question by looking at this variation65. This approach +was utilized by Bordin et al. to identify relationships between White matter hyperintensities and +the anatomical areas that are most important for the categorization of Alzheimer's disease. In +conclusion, these XAI approaches present a potentially important addition that may eventually +boost radiologist's confidence in the usage of AI models. + + +17 +Challenges and opportunities +DL research has recently witnessed accelerating adoption in the field of MRI (Figure 1) impacting +image acquisition, reconstruction, processing, and radiological reporting tasks. +Image acquisition: Currently, DL for MRI acquisition can be classified into two broad categories: +(i) automatically generating MR pulse sequences for a target contrast or signal-to-noise ratio. In +this approach, once a vendor hardware of interest has been identified, imposing appropriate +constraints on the cost function (for example, slew rate) will facilitate easy implementation of the +optimized pulse sequence on the chosen hardware4. Second is the acceleration of existing +vendor-defined protocols, potentially relying on post-acquisition methods to recover SNR72–74. +This approach is inherently limited to a particular protocol and vendor. The emergence of physics- +informed DL methods will allow researchers to develop models that are privy to the underlying +physical phenomena, potentially resulting in improved interpretability since the outputs can be +evaluated using existing task-specific knowledge75–79. Performing automated and intelligent slice +planning for localizers is also an active area of research80,81. +Image reconstruction and processing: Based on the work by Chaudhari et al.40, applications +of DL to image reconstruction and processing are classified into model-free image synthesis, +model-based image reconstruction, and classification and segmentation. Model-free image +synthesis pertains to the mapping of input images to output images. Examples are image super- +resolution, denoising, artifact reduction or removal, and synthesis of missing contrasts. Image +super-resolution enables the acquisition of multiple low-resolution images, which can be upscaled +using DL models82–87. Compiling a training dataset for this task is not straightforward since it is +not trivial to acquire paired low-resolution (LR) and high-resolution (HR) data. Apart from logistical +challenges, image registration is a primary concern. It is therefore convenient to acquire HR +images and subsequently perform retrospective downsampling to generate LR images. However, +this does not faithfully replicate MRI encoding, and hence does not accurately represent real- + +18 +world LR data. In some other cases, HR data is not readily available. One workaround is to +leverage a self-supervised learning framework to synthesise low-resolution images from high- +resolution data, thereby mitigating the requirement of image registration88. Image denoising +models improve SNR post-acquisition72,74. Two common approaches to achieve image denoising +are to either directly synthesise the denoised image, or to synthesise the residual from which the +final denoised image can be obtained. In the first approach, the models are trained on pairs of +noisy/clean images to optimize for image quality whilst avoiding blurring artifacts and retaining +the anatomical structures present in the original image89–92. An alternative method is to obtain the +final denoised image from the difference of the original input image and the predicted residual93,94. +Next, artifact reduction or removal models improve image quality by partially or completely +correcting MR image artifacts that might otherwise interfere with diagnosis or reduce image +quality52,95–97. Finally, contrast-synthesis models enable performing a limited MR exam whilst still +obtaining the same diagnostic information as from a comprehensive MR exam, by generating the +missing contrasts98,99. They can also enable performing contrast-enhanced MR examinations with +reduced dosages of the exogenous contrast agents100,101. Model-based image reconstruction +involves transforming undersampled data into fully-sampled reconstructed images102–105. One +primary challenge associated with image synthesis and reconstruction is hallucination40. This +relates to the addition of features that are not present in the input image. Since the model’s +representations are learned implicitly, hallucinations typically tend to reflect the characteristics of +the training dataset. The challenge of distinguishing true image signals from hallucinated signals +is exacerbated in the task of contrast-synthesis. Existing explainable AI approaches applicable to +other tasks are not amenable to image synthesis tasks. Consequently, mitigation strategies to +avoid hallucinations are an active area of research in the broader DL community. For image +reconstruction, embedding data-consistency steps into the reconstruction process is a viable +strategy to mitigate hallucinations. + +19 +Radiological reporting: The typical workflow of a radiologist involves identifying, localizing, and +characterizing the pathology of interest. This is labour-intensive, and recent DL implementations +have attempted to alleviate this burden on the radiologist106. Examples range from predicting +diagnosis from input images, to generating a text-based radiological report from input images. +The superior performance of DL methods on identification and classification tasks lends itself to +the automated detection of findings from acquired images. Furthermore, several works have also +demonstrated a potential for automated interpretation of findings107. Finally, assisting clinical +decision support systems could improve quality of care108,109. However, the attribution of clinical +decisions that were assisted by DL systems is an unresolved problem. Along with other ethical +and legal challenges such as those related to data sharing and bias (refer to sections on +prerequisites and GMLP), these bottlenecks need to be addressed prior to a potential deployment +in a real-world scenario. Wang et al. discuss the entire workflow of medical imaging: from +tomographic raw data/features to reconstructed images and then extracted diagnostic +features/readings7. +Figure 4 briefly captures the broader challenges and opportunities associated with employing DL +in medical imaging. In general, DL methods present other challenges and opportunities apart from +the application-specific ones discussed above. First, the current state-of-the-art DL qualifies as +narrow intelligence since it lacks global context108. This results in severe performance degradation +when tackling out of distribution data (OOD). Furthermore, it is not trivial to identify whether +unseen data is OOD110. This problem is exacerbated in diagnostic healthcare imaging because +the generated data is heterogeneous, noisy, and incomplete. This can be attributed to the +differences in vendor hardware and software, and the plethora of component configurations111. +Second, the lack of interpretability of DL models does not allow clinical users to develop trust in +the models’ predictions, resulting in stymied adoption and deployment in healthcare. Third, +training DL models to achieve the level of robustness necessary to handle this variety requires an +ImageNet-like breakthrough in the medical imaging community at large112. Recent works such as + +20 +RadImageNet112 are encouraging, and can potentially facilitate such advancements. However, +with ever-growing scales of data collection, the closely coupled and critical task of data curation +grows in complexity, at least for supervised learning frameworks. This relates to the fifth +challenge, which involves ensuring bias-free data curation. The performance of any DL model is +directly dependent on the quality of the data it was trained on. To avoid any biases in the output +which could potentially compound in downstream analyses, the training dataset has to be free of +all confounding factors. Training DL models on large-scale datasets requires prohibitively +expensive hardware setups to provide the required compute, coupled with extremely long training +durations. Consequently, this time-, cost-, and resource-intensive workflow raises the +development barrier thereby mostly limiting research efforts to well-funded organizations and +institutions. However, the recently increasing availability of commercial cloud solutions by +Amazon, Microsoft, Google, etc., unlocks cost-effective compute that is globally accessible. In +addition to the pre-existing heterogeneity of the data, acquisition methods are constantly evolving, +introducing another dimension of variability to the data. This will require deployed DL models to +be capable of online training to avoid incorrect or irrelevant predictions, or potential misdiagnoses +in downstream analyses when encountering OOD data. Any development workflow lag, +regardless of the duration, will result in incorrect treatment planning until updated models are +deployed. On the other hand, disengaging the models until newer versions are available will result +in workflow interruptions and throughput degradation. Lastly, the ethical and legal uncertainties +involved critically need to be resolved prior to any potential deployments. Different countries +enforce different medical data custody laws, necessitating region-specific modifications to the DL +tool and the data pipeline to ensure compliance. Most importantly, the ownership of a DL-assisted +clinical decision is an open question. Despite these challenges, the ability to automate tasks such +as image interpretation and diagnosis will alleviate the immense burden on healthcare providers, +allowing them to focus on other important tasks whilst improving the quality of their work lives. +Providing more accurate and timely diagnosis, reduced costs, increased efficiency, and tailored + +21 +treatments to individual patients based on their specific characteristics and needs all result in +improved patient outcomes. These are strong motivators to strategize immediate or near-future +adoption of existing DL methods. Directing research efforts to explore opportunities and +simultaneously addressing existing issues will aid in the wider adoption and improved realization +of DL’s potential. Along with addressing weaknesses and leveraging strengths, incorporating the +GMLP principles (Section 4) across the development lifecycle of DL-assisted medical applications +will aid in maximizing safety, efficiency, and quality during clinical deployment. +Conclusion +Our literature review indicates an increase in DL models for brain MRI tasks related to the +acquisition, reconstruction, image analysis, and reporting in the last five years across +neuropathologies such as tumors, stroke, and Alzheimer’s disease. These studies were +summarized as a suggestive DL pipeline for brain MRI studies. Importantly, the proportion of +studies that adhere to GMLP principles and contain XAI components are significantly low. This +DL neuro-MRI GMLP checklist in this review is motivated by this gap and emanates from the ten- +point guidelines espoused by the accreditation agencies for these principles tailored to brain MRI. +Finally, our assessment of the opportunities and challenges in DL studies on brain MRI indicates +that the inclusion of the GMLPs significantly reduces the challenges associated with cost, and +lack of interpretability, bias in the training data among others (Figure 4). Overcoming these +challenges will unlock the potential to improve multiple aspects of neuroimaging using MRI +through the successful deployment of accreditation agency-approved DL models. +References: +[1] World Health Organization. Global atlas of medical devices. Published online 2022. +https://www.who.int/publications/i/item/9789240062207 +[2] Geethanath S, Vaughan JT Jr. Accessible magnetic resonance imaging: A review. J Magn +Reson Imaging. 2019;49(7):e65-e77. + +22 +[3] Ravi KS, Geethanath S. 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We used the PubMed +database with the following keywords - regression MRI deep learning, classification MRI deep +learning, and MRI deep learning. + + + + +Figure 2: Steps for creating a dataset for the development of a deep learning model. + + +Studydesign +DataQA +Data +Database +andIRB +and +privacyand +creation +curation +security +and sharing +ApprovalData +Data +Data +acquisition +annotation +augmentation +orretrievallearningMRl +publications +2018 +2019 +2020 +2021 +2022 +Year of Publication Publications +Abstracts +containing +1000- +'classification" +Deep +# +500-Deep learning MRl publications +2000 +Abstracts +containing32 + + +Figure 3: Mountain represents the number of publications getting reduced as we focus our +scope. At the base of the mountain, we have a number of publications on AI in MRI. As we +move up the mountain, we specialize more into accreditation agency approved XAI models in +MRI following GMLPs. We can see that a small fraction of all publications actually make it to the +top of the mountain, i.e., follow all the requirements of the accreditation agency. Many of the +publications follow black box approach which doesn’t explain the decision making methodology +of their models and thus end up nowhere. + + + +CAM +Artificial Intelligence in MRI +(6486publications)imaging-392 +Guided Backpropagation +LRP +Black BoxApproach-Doesn'tanswer +Occlusion Sensitivity +why, what and how +GradCAMAccreditation agency approved XAl models in MRI following Good +MachineLearningPractices(GMLPs)-35 +Accreditation agency approved XAl models in MRl-166 +Accreditation agency approved XAl models in medical33 + + +Figure 4: Challenges and opportunities of employing deep learning (DL) in neuroimaging. +Developing DL methods poses several challenges, such as (clockwise from top) ensuring bias- +free data curation and annotation, difficulty in assembling datasets reflective of real-world +heterogeneity, black-box models stymieing development of trust in predictions, region-specific +ethical and legal requirements, and managing the massive compute requirements to train and +deploy models. However, unrealized opportunities such as (clockwise from top) more timely and +accurate clinical decisions, augmenting available human expertise to alleviate the burden on +skilled personnel, increased throughput and the associated reduction in operating costs are +strong motivators to work toward a successful deployment. + + + +andradiologists +Improved +Lack oftrust inmodels'predictions +throughput +dueto lack of interpretabilitydecisions +$$$ +$$$ +Ethical& legal +Growing compute +Lackof +Reduced costs +Augment expertise +requirements +requirementsto +robustness due +from increased +to alleviate burden +train and deploy +tovariegate data +efficiency +on MRtechniciansCHALLENGES +OPPORTUNITIES +Bias-free data +curation and +Improved +annotation +clinical34 + +Checklist of GMLPs for brain MRI +1. +Are neuroradiologists, neuroimaging scientists, MR technician and data scientist +working together throughout the whole life cycle of the product? + +2. +Is the patient's personal information anonymous in the brain MR images? + +3. +Is the metadata being filled for each patient scan with proper details of all +parameters? + +4. +Does training and testing MR datasets contain different scans? There shouldn’t be +any common scan in both datasets. + +5. +Does reference MR dataset for validation of model have completely unique scans +with same parameters as training and testing dataset? + +6. +Are you using the model for segmenting brain structures from the specific contrast +for which it has been trained for? If so, don’t use it for other contrasts. + +7. +Is the output of the model accepted and readable by the neuroradiologist? + +8. +Has the model been tested in the neuroradiology department under the supervision +of an expert neuroradiologist before deployment? + +9. +Are the precautions and potential dangers of using the model explicitly +mentioned? + +10. +Is the model being updated frequently for incorporating the changes in the new +scans that may occur naturally? + + +Table 1: This checklist represents the 10 guiding principles framed by the FDA for medical +device development known as Good Machine Learning Practices (GMLPs). This is a simpler +version of those principles rephrased specifically for experts working in brain MRI, such as +neuroradiologists, neuroimaging scientists and associated data scientists. In order to get +approved by the FDA, the AI model developed for MRI must fulfill all these questions. + + diff --git a/_NAzT4oBgHgl3EQfS_tm/content/tmp_files/load_file.txt b/_NAzT4oBgHgl3EQfS_tm/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cb54134dd8c507ea872c7af85df02df7db4bb998 --- /dev/null +++ b/_NAzT4oBgHgl3EQfS_tm/content/tmp_files/load_file.txt @@ -0,0 +1,1040 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf,len=1039 +page_content='1 Developing and deploying deep learning models in brain MRI: a review Kunal Aggarwal1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content='2, Marina Manso Jimeno3,4, Keerthi Sravan Ravi3,4, Gilberto Gonzalez5, Sairam Geethanath1 1 Accessible MR Laboratory, Biomedical Engineering, and Imaging Institute, Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' of Diagnostic,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Molecular and Interventional Radiology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Mount Sinai Hospital,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' New York City,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' New York 2 Department of Electrical and Computer Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Technical University Munich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Munich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Germany 3 Department of Biomedical Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Columbia University in the City of New York,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' New York City,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' New York,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' USA 4 Columbia University Magnetic Resonance Research Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Columbia University in the City of New York,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' New York City,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' New York,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' USA 5 Division of Neuroradiology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Department of Radiology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Massachusetts General Hospital,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Boston,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Massachusetts Word Count: 5952 2 Abstract Magnetic Resonance Imaging (MRI) of the brain has benefited from deep learning (DL) to alleviate the burden on radiologists and MR technologists,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' and improve throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' The easy accessibility of DL tools have resulted in the rapid increase of DL models and subsequent peer-reviewed publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' However, the rate of deployment in clinical settings is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Therefore, this review attempts to bring together the ideas from data collection to deployment into the clinic building on the guidelines and principles that accreditation agencies have espoused.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' We introduce the need for and the role of DL to deliver accessible MRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' This is followed by a brief review of DL examples in the context of neuropathologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Based on these studies and others, we collate the prerequisites to develop and deploy DL models for brain MRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' We then delve into the guiding principles to practice good machine learning practices in the context of neuroimaging with a focus on explainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=" A checklist based on the FDA's good machine learning practices is provided as a summary of these guidelines." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Finally, we review the current challenges and future opportunities in DL for brain MRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Keywords: Accessible MRI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Neuroimaging,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' GMLPs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Explainable AI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' FDA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Deep Learning,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Deployment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Brain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content='Abbreviations: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content='CAD: Computer Aided Detection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content='CAM: Class Activation Mapping ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content='CNN: Convolutional Neural Network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content='DAGMNet: Dual attention gate network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content='DICOM: Digital Imaging and Communications in Medicine ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content='DSC: DICE Similarity Coefficient ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content='FL: Federated Learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content='GDPR: General Data Protection Regulation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content='GMLPs: Good Machine Learning Practices ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content='Grad-CAM: Gradient-weighted Class Activation Mapping ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content='HR: High-Resolution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content='IRB: Institutional Review Board ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content='LR: Low-Resolution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content='MCI: Mild Cognitive Impairment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content='OECD: Organisation for Economic Co-operation and Development ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content='OOD: Out-of-Distribution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content='PPML: Privacy Protecting Machine Learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content='PSNR: Peak Signal-to-Noise Ratio ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content='RF: Radio Frequency ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content='SSIM: Structural Similarity Index Measure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content='XAI: Explainable Artificial Intelligence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content='Introduction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content='As of 20181-2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' the density of MR scanners was least in geographies with the highest populations such as in sub-Saharan Africa and the Indian subcontinent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' The severe lack of neurosurgeons and skilled human resources required to operate, use, and interpret data from MR scanners is a major barrier to accessing this life-saving technology2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' In contrast, contemporary radiology departments in the Organisation for Economic Co-operation and Development (OECD) countries require a close interplay of a team with diverse expertise: radiologists specializing in different anatomical sites, MR technologists, medical physicists, and radiologic nurses supported by the vendor’s service and application engineers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' This MR inaccessibility and the resulting disparity necessitates the development and deployment of automated methods to augment existing local expertise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Recently, there has been the development of autonomous MRI methods to automate protocolling3-4, identify artifacts5-6, and reconstruct images from accelerated scans7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' These advances are expected to assist local MR technicians, accelerate acquisitions, improve throughput by assisting or replacing manual data processing steps, and reduce waiting times for radiology reporting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' In addition to positively impacting MR accessibility, these automation methods facilitate MRI-based big data studies such as the Human Connectome Project8, UK Biobank9, and Rhineland study10, among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' This is due to the high volume and velocity associated with such studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' These automation methods leverage the recent resurgence of artificial intelligence techniques in general and supervised learning methods in particular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' A deep cascade of neural networks - termed deep learning (DL) models - is trained with a set of input features on one side and the resulting outcomes (labels) on the other side, using existing apriori annotated data11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' These trained DL models are then validated against an unseen but similar data set to fine-tune “hyperparameters” of the DL model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Subsequently, the tuned DL model is used to perform 5 inference on a test dataset to evaluate its performance by comparing it with a human-evaluated outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Finally, once the classification-tasked model performs satisfactorily with respect to false-positive (FP), false-negative (FN), true-positive (TP), and true-negative (TN) prediction metrics (captured in a “confusion matrix”), then it is considered for a deployment study and down- stream accreditation processes11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' In this review, we discuss studies that have developed and deployed deep learning models tasked with multi-task classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' We then present an analysis of the related literature to provide critical steps involved in data collection and curation required to set up deep learning studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' We then highlight the importance of good machine learning practices and the explainability of the DL results by illustrating examples and tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' A suggestive set of steps to enable mounting a successful development and deployment of MRI DL models will be then discussed based on recent literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Finally, a brief overview of the challenges and opportunities related to MRI-based DL studies is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' MRI DL studies of the brain Easy and direct availability of vast amounts of MRI data from publicly available repositories such as HCP8, UK Biobank9, among others, as well as accessible tools to build12 and optimize DL models13 have significantly accelerated the application of DL methods to address challenges in MRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' These studies have resulted in a substantial body of peer-reviewed literature (Figure 1), with most of them sharing an open-source implementation of their models with sample data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' This review focuses on MRI DL studies that meet the following criteria: (i) published in the last five years;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' (ii) Methods mostly focusing on classification and not regression tasks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' (iii) studies incorporating explainable AI components;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' (iv) works demonstrating deployment of the DL models and preferably across multiple vendors and sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' These criteria were used to focus our review on 6 specific developments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Notably, the generic tools developed by mathematicians and computer scientists in the DL community for explainable AI such as GradCAM14 and other methods15-16 have focused more on classification tasks compared to regression tasks (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' The quantification of the outcomes of a classifier network is relatively straightforward compared to a regression model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=" The outcomes are directly compared to the “true annotated labels'' and hence result in a binary or a multi-class decision." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' These can easily be binned into TP, FP, TN, and FN and evaluated for sensitivity and specificity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' In contrast, regression models accomplishing tasks such as DL denoising and synthesizing images are quantified using peak signal-to-noise ratio (PSNR) and structural similarity index measure (SSIM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' These metrics have a continuous value and are hard to set thresholds for acceptance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' In addition, regression models are typically explained using activation maps that are subsequently interpreted by the authors rather than community-wide tools independent of the developed methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' The interested reader is referred to 7 for a detailed review of machine learning-based image reconstruction methods that leverage regression models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Example DL MRI solutions for neuroimaging Brain tumors: Nalepa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' have demonstrated a fully automated pipeline for DCE-MRI analysis of brain tumors called Sens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content='AI DCE17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' In particular, they have substituted manual segmentation of brain tumors in T2 FLAIR images with deep-learning-based methods to demonstrate improved reproducibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' They complement this with a real-time image processing algorithm to determine the vascular input region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Finally, they include a new cubic model of the vascular function for PK modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' They have validated their package with the BraTs dataset for DICE coefficient and area under the curve as well as by twelve readers from two institutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' These results show good to excellent agreement between the gold standard BraTS dataset and Sens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content='AI DCE with a total execution time of approximately 3 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Another study focused on the automated identification and classification of brain tumor MRI data classified into glioma, meningioma, and pituitary tumors 7 with an accuracy of 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content='7% and a DICE similarity coefficient (DSC) of 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content='8%18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' A subsequent task of classifying gliomas into high or low grade had an accuracy of 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content='5% and a DSC of 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content='3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' These models were based on the GoogleNet variant architectures to efficiently combine local features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' The identification and classification tasks were accomplished in less than 3 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' The method compared well with other state-of-the-art methods with respect to accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Although the study did not explicitly focus on explainable AI methods such as Grad-CAM for interpretation, the authors performed an ablation study to demonstrate the effect of the locally chosen features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Brain extraction is a key step in multiple neuroimaging pre-processing pipelines and in complying with privacy laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' This task becomes more challenging in the presence of pathology such as diffuse gliomas as most analytical and deep learning methods focus on healthy brain extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Thakur et al have addressed this gap by developing and testing their models for brain extraction in the presence of diffuse glioma, in a multi-institutional manner19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' The authors considered multiparametric MRI data from private and public repositories acquired with different acquisition protocols to train a “modality-agnostic” tool that does not require retraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' The work demonstrated a similar or better accuracy compared to other brain extraction models that worked only on healthy brain extractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' The interested reader is pointed to these reviews for further reading on deep learning methods for brain tumor imaging20, classification21, and segmentation22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Stroke: The need for automated segmentation and classification of images, especially in emergency room settings in this time-critical pathology is well understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' In this direction, Liu et al developed a deep learning model that detected and segmented abnormalities in acute ischemic stroke23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' The work included several steps of pre-processing the data, such as skull stripping and DWI intensity normalization, among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' The study compared T-score and the modified c-fuzzy methods for lesion segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' In addition, the authors implemented the 3D Dual attention gate network (DAGMNet) as a supervised learning method to delineate the lesions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' The developed model performs better than the unsupervised generic tools and is faster, publicly available, and 8 easy to deploy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' They tested their algorithm on a hold-out data set of 280 MRIs and quantified the improved performance using DICE scores, precision, sensitivity, subject detection rate, DICE scores for the lesion volumes and lesion DWI contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Another study on stroke detection performed by Zhang et al demonstrated an accuracy of 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content='77% over 300 ischemic stroke patients24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' The authors evaluated three network architectures with labels drawn by experienced radiologists from two hospitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Their models predicted a bounding box covering the lesions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Statistical analysis performed on the location, size, and shape correlated well with the radiologists’ labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' This implementation aids in rapid localization and preliminary characterization of the lesion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' The authors have committed to making the data available once a thousand patient data are collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' An important task in imaging stroke is to grade the severity of the ischemia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' A multi- class classification solution for detecting the severity has been developed by Acharya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content='25 The authors extracted higher-order features from MR images, such as bispectrum entropy and its phase, followed by support vector machines to classify the severity of the stroke into LACS, PACS and TACS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' The algorithm demonstrated high levels of accuracy without the need for any manual intervention to augment the neuroradiologist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Alzheimer’s disease: Supervised learning models have demonstrated the utility of automation in the imaging of Dementia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' A specific challenge in this area is the classification and staging of the progression in mild cognitive impairment (MCI) patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Kwak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' have developed a deep learning model based on brain atrophy patterns and associated these changes with differences in amyloid burden, cognition, and metabolism26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' This model is used to classify AD patients from cognitively normal subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' A secondary classification helps identify the trajectory of cognitive decline in individuals with MCI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' These results were validated with cognitive tests, fluid biomarkers, and PET uptake data with good agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' The approach is expected to benefit from integrating cognitive and neurobiological features to capture the heterogeneity of MCI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Another approach involved the development and testing of whole-brain 3D convolutional neural networks to detect 9 AD27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' This model did not include any patient-specific information to allow the generalization of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' The implementation included four steps of brain extraction, normalization, 3D CNN followed by domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=" A key feature of this study is the authors' focus on accomplishing accountability." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' The method outperformed other state-of-the-art methods in the CAD Dementia challenge test, along with explainable AI visualizations to aid the interpretation of classification results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Ahmed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' also used a 3D approach but collected an ensemble of 2D patches in three orientations to train an ROI-based neural network to stage AD using MR images as per NIA labels28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' This approach was demonstrated on the GARD and ADNI datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' This method was compared to other state-of-the-art methods, and the performance was similar or better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' The landmarks delineated by the algorithm to indicate AD correlated well with known neuroanatomical areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' However, the model could not accurately predict asymptomatic AD based on the high rates of false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Prerequisites for DL-based neuro-MRI Data from routine MRI studies result in high volume, velocity, and variety: characteristics of big data29–35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' For training DL models, high volume and velocity are favorable factors: more data is better than lesser;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' high velocity of data requires automated processing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' However, the variety in MRI data due to a large number of available acquisition parameters, reconstruction methods, receive coil configurations, post-processing steps requires attention to fine details before collating data for annotation and subsequent training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' In line with the classification suggested by Wald36, the sources of these variabilities need to be binned as emanating from the MR system characteristics, subject-induced variations, and pathology-specific factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Typically, the goal of the DL models discussed in this work is to glean the subtle changes in pathophysiological states based on the MR images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Therefore, standardization of parameters getting affected by the system and subject priors is critical to ensure that the DL models focus 10 their attention on the pathology being investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' The removal or reduction of the confounding variables is therefore essential in building these DL models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' This exercise is facilitated by the use of explainable AI tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Therefore, two critical requirements to understand an MRI-based DL study are standardization of procedures and methods and explainable AI tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' A detailed discussion on these prerequisites is performed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' DL models require large amounts of data to achieve high accuracy levels37–39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Unlike models trained on natural images, large datasets are challenging to achieve for medical imaging applications37–39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Training a DL model of medical images entails data collection, curation, and annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Additionally, data augmentation might be necessary in cases when the data are insufficient or to strengthen the generalization ability of the model39–41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' The steps of data collection and annotation are the most expensive and time-consuming, and patient privacy policies often restrict the use and sharing of images37,39,42,43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' These factors significantly impact the models’ performance in clinical settings, limiting their ultimate deployment40,42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' The purpose of this section is to review strategies and recommendations (Figure 2) at the data level for maximizing the likelihood of successful deployment after training based on previously published work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Data collection, including patient selection, imaging protocol, sequences, and scan parameters, is determined based on the intended use of the model and its targeted application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Cohorts can be prospective or retrospective, depending on whether the data are acquired for the study or retrieved from a public or private repository44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' An initial step in the process is the approval by an Institutional Review Board (IRB) or a similar board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Additionally, participants provide informed consent about the use of their personal data, which is typically de-identified during data curation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Privacy Protecting Machine Learning (PPML) is a niche area of research that aims at maximizing the confidentiality of patient data while optimizing its use on data-driven models45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' In this field, Federated Learning (FL) alleviates the shortage of data problems by allowing training models on large-scale, multi-center data without data sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' FL feasibility in MRI has been explored by 11 Sarma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' and Sheller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' for brain tumor and prostate segmentation tasks46,47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' When dealing with multi-institutional data, protocol harmonization can avoid model bias that may arise from differences in image contrast, intensity, or noise distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' In MRI, these differences may stem from multiple sources, including variations in system manufacturer, field strength, Radio Frequency (RF) coils, patient positioning, acquisition sequence and scan time, and even pre- processing and reconstruction pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Publicly available databases are the results of extensive research projects and contain large amounts of data that can be leveraged for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' These datasets are typically acquired using the same protocol and scanner or using highly harmonized protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Patient inclusion and exclusion criteria in these research cohorts are strict, and the datasets are well-curated and usually undergo multi-step post-processing pipelines and standardization operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' While the reproducibility of models trained on publicly available datasets is easier to assess, the data lack the heterogeneity characteristic of clinical data observed during deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Martensson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content='43 systematically studied the performance variability of a DL model trained on different combinations of training sets, including publicly available datasets and more heterogeneous Out-of-Distribution (OOD) datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' They observed that performance drops when models trained on homogeneous data are applied to clinical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' However, inference on clinical data showed a better agreement level with a radiologist reading if clinical cohorts were present in the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' A benefit of local data acquisition for training is having access to raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Most public or private imaging repositories contain data in the Digital Imaging and Communications in Medicine (DICOM) format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' The raw data undergo several pre-processing steps, including coil-combination, filtering, artifact correction, and phase removal before storage, stripping the images of features that DL models in the process could recognize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Raw data might be favored for certain DL tasks, data augmentation techniques, or for data synthesis via forward modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' This is exemplified by the increasing usage of the fastMRI dataset48, the only publicly-available dataset of raw MR knee 12 and brain data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' It has become a benchmark for the validation and reproducibility assessment of DL-based image reconstruction algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Additionally, models for the detection or correction of k-space-occurring artifacts such as motion49–51 and Gibbs ringing5,52 typically leverage raw data for the simulation of artifact-corrupted images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Data collection is followed by data curation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' This step is performed to standardize and improve dataset quality for subsequent deep neural network training42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' The data used for the model development entails a trade-off between distribution heterogeneity and representation bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' It should represent varying patient populations and anatomy disparities while avoiding biasing network representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Successfully deployed models are typically developed with data acquired using the same imaging protocol and the same system as the site targeted for deployment53,54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Failure mode analysis of the prospective evaluation of an automatic kidney segmentation model after deployment revealed segmentation errors arising from common clinical scenarios such as a fluid-filled stomach and a distended bladder54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' These cases are typically excluded during cohort building or data curation and reduce the model’s tolerance to data variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Poor generalizability to unseen domains is one of the major challenges to successfully deploying DL models in the clinic55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' For fully and semi-supervised learning tasks, data labeling or annotation is typically the most time- consuming step of an AI project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' It may require localizing, delineating, or segmenting lesions or organs of interest or labeling or annotating characteristics of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' This step is performed manually by an experienced reader via visual inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=" When human observations or clinicians' expertise is required to annotate the data, multiple readers and ideally with variable levels of experience, are preferred to estimate inter-reader variability and compare it to the model’s performance." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=" 13 Finally, data augmentation is the process of generating additional versions of the initial data to enlarge the training set and improve the model's robustness, thereby avoiding overfitting56." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Typical techniques include translation, flipping, rotation, and cropping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Depending on the application, other approaches may be useful, such as random k-space oversampling for model- based reconstruction techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Data augmentation can also be leveraged to reduce the gap between the training data and prospective clinical data, for example, by simulating noise and motion artifacts in the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' A recent study57 for accelerated MR reconstruction demonstrated that with the introduction of these commonly-occurring artifacts using MR physics-driven data augmentation techniques, model performance on both in-distribution and OOD data increases compared to state-of-the-art image-based data augmentation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=" Good Machine Learning Practices for MRI AI's novelty and current regulatory paradigms are not well adapted to strike a balance between patient safety and promoting the expansion of this new industry58." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' For assessing commercially accessible algorithms to guarantee their dependability and safety, defining best practices is an area of active research59,60, significant regulatory problems need to be resolved to move clinical AI toward becoming safe and robust61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Wu et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' reviewed the FDA database for parameters that were used to evaluate the AI algorithms of products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' The parameters they found were - (i) number of patients and sites used in the evaluation, (ii) prospective and retrospective collection of data, and (iii) whether the performance was stratified by disease subtypes or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Based on the FDA summary, the study revealed that 126 out of 130 AI devices conducted solely retrospective investigations at their submission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=" The influence of the AI decision tool on clinical practice must be fully characterized, though, and this is crucial since human-computer interaction might differ significantly from a model's intended purpose." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' For instance, most computer-aided detection 14 (CAD) diagnostic tools are meant to serve as decision-support aids rather than primary diagnostic instruments61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Instead of independently diagnosing, staging, or triaging pathology, CAD is meant to identify, mark, highlight, or otherwise draw attention to imaging characteristics62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' The FDA suggests regulating AI software based on function rather than technical components or intended use, which is different from the case for most pharmaceutical items, gadgets, and foods58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Therefore, FDA advocates ten guiding principles for medical device development known as 10 Good Machine Learning Practices (GMLPs) that take into consideration the prerequisites discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' According to the first and second principle, all expertise related to the product development should work together from the development phase until integration into the clinical workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' This includes neuroradiologists, neuroimaging scientists, MR technicians, and data scientists implementing good software engineering and security practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' The third principle states the importance of metadata in developing DL models and connects to the concept of data security from the second principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' The fourth and fifth principles mention the importance of datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' The training and testing datasets should be independent and the reference datasets should have the same characteristics as of the patients in the former datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' According to the sixth principle, the intended use of a model must be clearly defined along with its risks and performance limitations on different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' This relates to principle number three in a sense that metadata defines the scope of the model being used on the specific patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Principle seven states the involvement of humans and the fact that human intervention cannot be avoided at any stage of development or deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' This principle focuses more on AI in the loop rather than human in the loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Eighth and ninth principle centers on the user and states that the model should be easy to understand for the end user and must list all the possible precautions in order to avoid harm to the patient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Principle ten mentions that updates in the model are a mandatory part of the DL deployment and must be considered frequently59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' A checklist based on these GMLPs is provided as a summary specifically designed for experts working in brain MRI (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' 15 Explainable AI Although deep learning techniques produce outcomes, they do not explain how those results were obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' One cannot just analyze the deep neural network to understand how that choice was made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' As a result, deep learning models are sometimes referred to as "Black Boxes"63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Medical professionals believe these "black boxes" may be prejudiced in some way, which might have negative effects when used in practical applications63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Additionally, laws like the General Data Protection Regulation (GDPR, Article 15) of the European Union specify that patients have the right to request an explanation for how a given diagnosis was reached if the standard deep learning models cannot64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Therefore, Explainable Artificial Intelligence (XAI) techniques recently developed with the primary objective of visualizing and interpreting the results of machine learning (ML) and deep learning (DL) networks represent a potential remedy to close this gap between high performance and deep-level understanding65 (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' They have been utilized in a variety of applications, including the categorization of ECGs66 and the visualization of feature maps at various Convolutional Neural Network (CNN) layers67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Velden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' categorized XAI approaches into three groups based on three criteria: (i) model- based vs post-hoc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' (ii) model-specific against model-agnostic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' and (iii) global versus local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' These categories are visual, textual, and example based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' The most prevalent type of XAI in medical imaging, out of these three categories, is the visual explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' These approaches, sometimes referred to as saliency mapping, employ a backpropagation methodology to highlight the key elements of a picture for a certain model’s decision by emphasizing the pixels that had the greatest influence on the results of the investigation64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Class activation mapping (CAM), a technique used in the backpropagation methodology, was introduced by Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' in 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' They used global average pooling on the last convolutional feature maps to substitute the fully connected layers at the conclusion of a CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' It is a weighted linear sum of the visual patterns that were observed and recorded by the filters at various spatial positions68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' 16 Gradient-weighted class activation mapping is a generic strategy that includes CAM as one of its specialized methods (Grad-CAM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Grad-CAM can operate with any CNN, but CAM needs global average pooling in particular64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Grad-CAM delivers the ROI on an input image that has the greatest influence on class prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Grad-CAM allows us to track the spatial attention changes that occur between network layers, or more precisely, what each network layer focuses on in each input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' To do this, the output gradient with respect to each neuron in the network is calculated to ascertain its relative significance69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Grad-CAM has been widely used to describe deep learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Jimeno et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' used it to identify and classify wrap-around and Gibbs ringing artifacts5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' It was used by Windisch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' in 2020 to identify brain MRI regions that caused the classifier to determine the existence of a malignancy70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=" A model’s prediction of the fetus's brain age may also be explained using Grad- CAM, according to a 2020 publication by Liao et al." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=', which will help avoid congenital malformations71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' It was also employed by Natekar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' in 2020 to describe the brain tumor segmentation network69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Occlusion Sensitivity technique is another XAI tool64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' The input MR image is disturbed by a small perturbation, and the categorization choice is changed and examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' In order to quantify the variation in the output prediction, it covers a piece of the input picture with a black patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' After moving the patch across the whole picture, it is simple to determine which parts of the brain are responsible for the categorization choice in question by looking at this variation65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' This approach was utilized by Bordin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=" to identify relationships between White matter hyperintensities and the anatomical areas that are most important for the categorization of Alzheimer's disease." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=" In conclusion, these XAI approaches present a potentially important addition that may eventually boost radiologist's confidence in the usage of AI models." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' 17 Challenges and opportunities DL research has recently witnessed accelerating adoption in the field of MRI (Figure 1) impacting image acquisition, reconstruction, processing, and radiological reporting tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Image acquisition: Currently, DL for MRI acquisition can be classified into two broad categories: (i) automatically generating MR pulse sequences for a target contrast or signal-to-noise ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' In this approach, once a vendor hardware of interest has been identified, imposing appropriate constraints on the cost function (for example, slew rate) will facilitate easy implementation of the optimized pulse sequence on the chosen hardware4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Second is the acceleration of existing vendor-defined protocols, potentially relying on post-acquisition methods to recover SNR72–74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' This approach is inherently limited to a particular protocol and vendor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' The emergence of physics- informed DL methods will allow researchers to develop models that are privy to the underlying physical phenomena, potentially resulting in improved interpretability since the outputs can be evaluated using existing task-specific knowledge75–79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Performing automated and intelligent slice planning for localizers is also an active area of research80,81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Image reconstruction and processing: Based on the work by Chaudhari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content='40, applications of DL to image reconstruction and processing are classified into model-free image synthesis, model-based image reconstruction, and classification and segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Model-free image synthesis pertains to the mapping of input images to output images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Examples are image super- resolution, denoising, artifact reduction or removal, and synthesis of missing contrasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Image super-resolution enables the acquisition of multiple low-resolution images, which can be upscaled using DL models82–87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Compiling a training dataset for this task is not straightforward since it is not trivial to acquire paired low-resolution (LR) and high-resolution (HR) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Apart from logistical challenges, image registration is a primary concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' It is therefore convenient to acquire HR images and subsequently perform retrospective downsampling to generate LR images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' However, this does not faithfully replicate MRI encoding, and hence does not accurately represent real- 18 world LR data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' In some other cases, HR data is not readily available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' One workaround is to leverage a self-supervised learning framework to synthesise low-resolution images from high- resolution data, thereby mitigating the requirement of image registration88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Image denoising models improve SNR post-acquisition72,74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Two common approaches to achieve image denoising are to either directly synthesise the denoised image, or to synthesise the residual from which the final denoised image can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' In the first approach, the models are trained on pairs of noisy/clean images to optimize for image quality whilst avoiding blurring artifacts and retaining the anatomical structures present in the original image89–92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' An alternative method is to obtain the final denoised image from the difference of the original input image and the predicted residual93,94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Next, artifact reduction or removal models improve image quality by partially or completely correcting MR image artifacts that might otherwise interfere with diagnosis or reduce image quality52,95–97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Finally, contrast-synthesis models enable performing a limited MR exam whilst still obtaining the same diagnostic information as from a comprehensive MR exam, by generating the missing contrasts98,99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' They can also enable performing contrast-enhanced MR examinations with reduced dosages of the exogenous contrast agents100,101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Model-based image reconstruction involves transforming undersampled data into fully-sampled reconstructed images102–105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' One primary challenge associated with image synthesis and reconstruction is hallucination40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' This relates to the addition of features that are not present in the input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Since the model’s representations are learned implicitly, hallucinations typically tend to reflect the characteristics of the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' The challenge of distinguishing true image signals from hallucinated signals is exacerbated in the task of contrast-synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Existing explainable AI approaches applicable to other tasks are not amenable to image synthesis tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Consequently, mitigation strategies to avoid hallucinations are an active area of research in the broader DL community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' For image reconstruction, embedding data-consistency steps into the reconstruction process is a viable strategy to mitigate hallucinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' 19 Radiological reporting: The typical workflow of a radiologist involves identifying, localizing, and characterizing the pathology of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' This is labour-intensive, and recent DL implementations have attempted to alleviate this burden on the radiologist106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Examples range from predicting diagnosis from input images, to generating a text-based radiological report from input images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' The superior performance of DL methods on identification and classification tasks lends itself to the automated detection of findings from acquired images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Furthermore, several works have also demonstrated a potential for automated interpretation of findings107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Finally, assisting clinical decision support systems could improve quality of care108,109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' However, the attribution of clinical decisions that were assisted by DL systems is an unresolved problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Along with other ethical and legal challenges such as those related to data sharing and bias (refer to sections on prerequisites and GMLP), these bottlenecks need to be addressed prior to a potential deployment in a real-world scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' discuss the entire workflow of medical imaging: from tomographic raw data/features to reconstructed images and then extracted diagnostic features/readings7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Figure 4 briefly captures the broader challenges and opportunities associated with employing DL in medical imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' In general, DL methods present other challenges and opportunities apart from the application-specific ones discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' First, the current state-of-the-art DL qualifies as narrow intelligence since it lacks global context108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' This results in severe performance degradation when tackling out of distribution data (OOD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Furthermore, it is not trivial to identify whether unseen data is OOD110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' This problem is exacerbated in diagnostic healthcare imaging because the generated data is heterogeneous, noisy, and incomplete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' This can be attributed to the differences in vendor hardware and software, and the plethora of component configurations111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Second, the lack of interpretability of DL models does not allow clinical users to develop trust in the models’ predictions, resulting in stymied adoption and deployment in healthcare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Third, training DL models to achieve the level of robustness necessary to handle this variety requires an ImageNet-like breakthrough in the medical imaging community at large112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Recent works such as 20 RadImageNet112 are encouraging, and can potentially facilitate such advancements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' However, with ever-growing scales of data collection, the closely coupled and critical task of data curation grows in complexity, at least for supervised learning frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' This relates to the fifth challenge, which involves ensuring bias-free data curation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' The performance of any DL model is directly dependent on the quality of the data it was trained on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' To avoid any biases in the output which could potentially compound in downstream analyses, the training dataset has to be free of all confounding factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Training DL models on large-scale datasets requires prohibitively expensive hardware setups to provide the required compute, coupled with extremely long training durations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Consequently, this time-, cost-, and resource-intensive workflow raises the development barrier thereby mostly limiting research efforts to well-funded organizations and institutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' However, the recently increasing availability of commercial cloud solutions by Amazon, Microsoft, Google, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=', unlocks cost-effective compute that is globally accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' In addition to the pre-existing heterogeneity of the data, acquisition methods are constantly evolving, introducing another dimension of variability to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' This will require deployed DL models to be capable of online training to avoid incorrect or irrelevant predictions, or potential misdiagnoses in downstream analyses when encountering OOD data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Any development workflow lag, regardless of the duration, will result in incorrect treatment planning until updated models are deployed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' On the other hand, disengaging the models until newer versions are available will result in workflow interruptions and throughput degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Lastly, the ethical and legal uncertainties involved critically need to be resolved prior to any potential deployments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Different countries enforce different medical data custody laws, necessitating region-specific modifications to the DL tool and the data pipeline to ensure compliance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Most importantly, the ownership of a DL-assisted clinical decision is an open question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Despite these challenges, the ability to automate tasks such as image interpretation and diagnosis will alleviate the immense burden on healthcare providers, allowing them to focus on other important tasks whilst improving the quality of their work lives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Providing more accurate and timely diagnosis, reduced costs, increased efficiency, and tailored 21 treatments to individual patients based on their specific characteristics and needs all result in improved patient outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' These are strong motivators to strategize immediate or near-future adoption of existing DL methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Directing research efforts to explore opportunities and simultaneously addressing existing issues will aid in the wider adoption and improved realization of DL’s potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Along with addressing weaknesses and leveraging strengths, incorporating the GMLP principles (Section 4) across the development lifecycle of DL-assisted medical applications will aid in maximizing safety, efficiency, and quality during clinical deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Conclusion Our literature review indicates an increase in DL models for brain MRI tasks related to the acquisition, reconstruction, image analysis, and reporting in the last five years across neuropathologies such as tumors, stroke, and Alzheimer’s disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' These studies were summarized as a suggestive DL pipeline for brain MRI studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Importantly, the proportion of studies that adhere to GMLP principles and contain XAI components are significantly low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' This DL neuro-MRI GMLP checklist in this review is motivated by this gap and emanates from the ten- point guidelines espoused by the accreditation agencies for these principles tailored to brain MRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Finally, our assessment of the opportunities and challenges in DL studies on brain MRI indicates that the inclusion of the GMLPs significantly reduces the challenges associated with cost, and lack of interpretability, bias in the training data among others (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Overcoming these challenges will unlock the potential to improve multiple aspects of neuroimaging using MRI through the successful deployment of accreditation agency-approved DL models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' References: [1] World Health Organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Global atlas of medical devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Published online 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content='who.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' RadImageNet: An Open Radiologic Deep Learning Research Dataset for Effective Transfer Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Radiol Artif Intell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content='4(5):e210315.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' 31 Figures and Tables: Figure 1: Publication trend in the past five years for deep learning in MRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' We used the PubMed database with the following keywords - regression MRI deep learning, classification MRI deep learning, and MRI deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Figure 2: Steps for creating a dataset for the development of a deep learning model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Studydesign DataQA Data Database andIRB and privacyand creation curation security and sharing ApprovalData Data Data acquisition annotation augmentation orretrievallearningMRl publications 2018 2019 2020 2021 2022 Year of Publication Publications Abstracts containing 1000- \'classification" Deep # 500-Deep learning MRl publications 2000 Abstracts containing32 Figure 3: Mountain represents the number of publications getting reduced as we focus our scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' At the base of the mountain, we have a number of publications on AI in MRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' As we move up the mountain, we specialize more into accreditation agency approved XAI models in MRI following GMLPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' We can see that a small fraction of all publications actually make it to the top of the mountain, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=', follow all the requirements of the accreditation agency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Many of the publications follow black box approach which doesn’t explain the decision making methodology of their models and thus end up nowhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=" CAM Artificial Intelligence in MRI (6486publications)imaging-392 Guided Backpropagation LRP Black BoxApproach-Doesn'tanswer Occlusion Sensitivity why, what and how GradCAMAccreditation agency approved XAl models in MRI following Good MachineLearningPractices(GMLPs)-35 Accreditation agency approved XAl models in MRl-166 Accreditation agency approved XAl models in medical33 Figure 4: Challenges and opportunities of employing deep learning (DL) in neuroimaging." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Developing DL methods poses several challenges, such as (clockwise from top) ensuring bias- free data curation and annotation, difficulty in assembling datasets reflective of real-world heterogeneity, black-box models stymieing development of trust in predictions, region-specific ethical and legal requirements, and managing the massive compute requirements to train and deploy models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' However, unrealized opportunities such as (clockwise from top) more timely and accurate clinical decisions, augmenting available human expertise to alleviate the burden on skilled personnel, increased throughput and the associated reduction in operating costs are strong motivators to work toward a successful deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=" andradiologists Improved Lack oftrust inmodels'predictions throughput dueto lack of interpretabilitydecisions $$$ $$$ Ethical& legal Growing compute Lackof Reduced costs Augment expertise requirements requirementsto robustness due from increased to alleviate burden train and deploy tovariegate data efficiency on MRtechniciansCHALLENGES OPPORTUNITIES Bias-free data curation and Improved annotation clinical34 Checklist of GMLPs for brain MRI 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Are neuroradiologists, neuroimaging scientists, MR technician and data scientist working together throughout the whole life cycle of the product?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=" Is the patient's personal information anonymous in the brain MR images?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Is the metadata being filled for each patient scan with proper details of all parameters?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Does training and testing MR datasets contain different scans?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' There shouldn’t be any common scan in both datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Does reference MR dataset for validation of model have completely unique scans with same parameters as training and testing dataset?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Are you using the model for segmenting brain structures from the specific contrast for which it has been trained for?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' If so, don’t use it for other contrasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Is the output of the model accepted and readable by the neuroradiologist?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Has the model been tested in the neuroradiology department under the supervision of an expert neuroradiologist before deployment?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Are the precautions and potential dangers of using the model explicitly mentioned?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Is the model being updated frequently for incorporating the changes in the new scans that may occur naturally?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' Table 1: This checklist represents the 10 guiding principles framed by the FDA for medical device development known as Good Machine Learning Practices (GMLPs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' This is a simpler version of those principles rephrased specifically for experts working in brain MRI, such as neuroradiologists, neuroimaging scientists and associated data scientists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} +page_content=' In order to get approved by the FDA, the AI model developed for MRI must fulfill all these questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf'} diff --git a/_dAzT4oBgHgl3EQfhPzZ/content/tmp_files/2301.01483v1.pdf.txt b/_dAzT4oBgHgl3EQfhPzZ/content/tmp_files/2301.01483v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d64830d3f72ed81b6c551977a9d3905b37934c98 --- /dev/null +++ b/_dAzT4oBgHgl3EQfhPzZ/content/tmp_files/2301.01483v1.pdf.txt @@ -0,0 +1,640 @@ +Proceedings of the 9th International and 49th National Conference on Fluid Mechanics and Fluid Power (FMFP) +December 14-16, 2022, IIT Roorkee, Roorkee-247667, Uttarakhand, India +FMFP2022-594 +Performance enhancement of a 100 watts class Tesla turbine +Arindam Mandal1, Rajosik Adak1, and Sandeep Saha1 +1Department of Aerospace Engineering, IIT Kharagpur, Kharagpur 721302, India +ABSTRACT Tesla turbines are an attractive but less +explored +area +in +low-power +applications. +This +article +presents an experimental investigation of a centimeter +scale +Tesla +turbine +in +bi +and +uni-directional +outlet +configuration with compressed air at 6 bar. The turbine’s +performance is enhanced by ≈ 38% for a uni-directional +outlet configuration. Furthermore, we also investigate the +electrical power of the turbine in a bi-directional outlet +configuration by coupling the turbine with a generator. +Despite achieving higher performance in uni-directional +outlet configuration, we observe substantial losses at the +inlet we use for the experiment. To illustrate and improve +the losses, we numerically investigate the turbine inlet +at a total pressure and temperature difference of 2 bar +and +50◦C, +respectively. +Subsequently, +we +design +two +more nozzles and compare their performance with the +nozzle we used in our experiment. Our findings suggest +that nozzle 3 performs the best in delivering the highest +Mach no and uniformity across the slits. This observation +would help optimize the nozzle suitable for the Tesla turbine. +Keywords: Tesla turbine, Friction turbine, Low power +application, Slit nozzle, +I. +INTRODUCTION +Small scale low microturbines are crucial since there is a +growing need for them in numerous applications. Examples +of notable applications include waste heat recovery, pico- +hydro power, organic Rankine cycle technologies, biomass, +waste heat recovery, micro GT, and many more. The increas- +ing need for energy harvesting at these scales poses a variety +of challenges to the performance and manufacturability of +the turbine due to their compact sizes, high rotational speeds, +and increased viscous losses. An alternative expansion de- +vice addressing these challenges can be highly beneficial +regarding its techno-economic feasibility. The tesla turbine +is one of its kind, which has a uniquely simple design and +a unique mechanism of momentum transfer. The turbine +rotor consists of several co-axial discs closely packed with +each other. Each of the rotor discs has outlet ports near the +center. A casing covering the rotors helps guide the flow +from the inlet. The turbine shaft is attached to the rotor- +casing configuration with the help of two bearings. Figure 1 +shows the different components of the turbine. After being +injected through the inlet system of the turbine, the fluid +flows spirally inward and exits through the ports located +near the shaft in the axial direction. The momentum transfer +from the fluid to the rotor takes place using the adhesion +and viscosity of the fluid. This mechanism makes the turbine +attractive at small scales where the dominance of the viscous +force becomes significant. +This turbine was conceptualized by Nicola Tesla [39] +in the year 1907. Initially, the device was unable to attract +market attention because of the no potential requirement of +low-power harvesting technologies. After that, till the twen- +tieth century, a considerable amount of thrust was given to +the possible design modifications [27], [28], [12], improved +seal designs [2] and loss analysis [6], [30] of the turbine +and its ancillary components for enhancing performance. In +addition, a few simplified theoretical models were developed +[1], [21] to get an insight into the influence of the parameters +associated to the turbine. +Figure 1: Schematic of the exploded view of the turbine. +a: Casing, b: Rotors, c: Inlet configurations (i,ii and iii), +d: Shaft, e. Bearing. (f) shows the fluid flow between two +consecutive corotating discs +More recently, there has been a surge in interest in +exploring this technology due to its several advantages +over conventional energy harvesting technologies. Due +to its simple design and flow mechanics, the turbine can +handle particle-laden fluids and two-phase expansion with +minimal damage. Numerical simulations of the gas flow +and mass transfer between two coaxially co-rotating discs +were performed by Sandilya et al. [32] where they found a +satisfactory match between the numerical and experimental +value +of +mass +transfer +coefficient. +Using +numerical +simulations, Ladino [18] presented the load coefficient +curve, efficiency, and degree of reaction variation after +maintaining a constant rotational speed. Upon developing +an analytical model, Deam et al. [7] scaled down the turbine +in millimeter size to achieve better efficiency. Lemma et al. +[22] investigated the Parasitic, viscous, and other dissipative +losses in the bearings and end walls to mitigate their effect +on decaying the performance. +An explicit description of a flexible test rig was de- +1 +arXiv:2301.01483v1 [physics.flu-dyn] 4 Jan 2023 + +(i) +(ii) +(ii) +(c) +(p) +(b) +(a) +f +(e)veloped by Hoya et al. [14] where they calculated the +low torque at high RPM using the angular acceleration +method. A comparative study in the efficiency offered by +Tesla and small bladed microturbine in a micro power plant +was addressed by Lampart et al. [20], [19] to establish +the competitiveness of a Tesla turbine. To understand the +transport phenomena inside the turbine rotor, a number +of numerical and analytical models solving the Navier- +Stokes equations using different approaches are present in +the literature repository [15], [9], [36], [10], [31], [35], [34], +[5]. Aside from that, flow diagnostics using particle tracking +velocimetry by Schosser et al. [33] provides an insight into +the component-wise velocity profiles inside the rotor gaps at +different radial locations. In recent years, A wide range of +application based studies of Tesla turbine related to Micro- +air vehicles [23], Organic Rankine cycle [37], [38], [8], +[25], [24], Combined heat plant [3], Pico-hydro applications +[13], [4], [16], [17], Ammonia synthesys [11] have been +investigated or under investigation [26], [29]. +The recent resurgence in interest indicates the impor- +tance of investigating the turbine further to mitigate the +losses due to the nonuniformity and disturbances associated +with the inflow to rotor and rotor to outflow interaction. +This article presents the experimental investigation of a +centimeter-scaled Tesla turbine in uni and bi-directional +outlet configuration with compressed air. We compare the +performance in terms of Mechanical power output for the +two configurations. Furthermore, we investigate the losses in +the inlet due to the sharp divergent and the slit configuration +at the inlet rotor junction. Finally, we compare the nozzles’ +performance by looking at the peak discharge Mach number +and channel-wise disparity in flow injection to the rotor. Our +numerical results can be beneficial in coming up with better +inlet configurations for extracting maximum energy from the +fluid, which leaves a further scope for further investigation. +II. +METHODOLOGY AND EXPERIMENT +A lab scale turbine prototype (in fig. 2a) is fabricated +for experimentation in the Aerospace Engineering depart- +ment, IIT Kharagpur. The turbine consists of 10 consecutive +corotating discs having an outer diameter of 10 cm and a +thickness of 2 mm. The gap between the consecutive rotating +discs is 2 mm, and the distance between two extreme discs to +the adjacent casing wall is 1 mm. The turbine has four outlet +holes near the shaft, having a center distance of 2 cm from +the shaft center. The shaft and the exhaust ports’ diameters +are 1.5 cm and 1 cm, respectively. The distance between the +shroud and the disc’s edge is 1 mm. The inlet system of the +turbine is designed to connect the compressor outlet port of +6mm dia to the turbine having an inlet of 4 cm thickness. +The area of the inlet is 0.4 cm2, which is segregated using +slit configurations to guide air to each gap with minimum +interaction with the peripheral walls of the rotor. +The turbine inlet is connected to the compressor by +Polyeurathane pneumatic pipes with an installed pres- +sure gauge. The RPM of the turbine is measured using +an Arduino-based tachometer. The angular acceleration- +deceleration approach computes the net accelerating and de- +celerating torque.The experiment also uses a digital tachome- +(a) +(b) +Figure 2: (a) Fabricated turbine and (b) the inlet system +ter to validate the turbine’s RPM. We conduct our experiment +at 6 bar of inlet pressure for uni-and bi-directional outlet +configuration. The detail of the experimental setup can be +seen in the figure 3. +(a) +(b) +Figure 3: Details of the experimental set up. 1- Compres- +sor discharge, 2- Tachometer, 3- Turbine, 4- Arduino +based Tachometer, 5- Pressure gauge, 6- PC for data +acquisition, 7- Camera, 8- Generator, 9- Multimeters, +10- Rheostat, 11- Electrical circuit +III. +RESULTS AND DISCUSSION +A. Performance characteristics +We tabulate the RPM with time with the help of a serial +monitor and calculate the angular acceleration as a function +of RPM. Once the turbine reaches its stable RPM, we shut +off the compressed air sypply and continue to perform data +acquisition until the turbine becomes stationary. We continue +the similar process for a uni-directional outlet case. We +observe from figure 4 that the turbine with a uni-directional +outlet accelerates faster than the turbine with a bi-directional +outlet configuration. In addition, for a supply pressure of +6 bar, we achieve a maximum RPM of 13019 and 11124 +for uni and bi-directional outlet configurations, respectively. +Figure 4 shows the power variation due to accelerating +and braking torque. There is an increment of ≈ 38% in +power due to accelerating torque for a uni-directional outlet +configuration. +We integrate the turbine with a 100 W class Fedus +RS-775 DC electric motor to measure the electrical power +output. However, the motor is used as a generator to measure +the output voltage and current. The electrical circuit is +attached to a rheostat, and the experiment is performed with +2 + +t (sec) +0 +15 +30 +45 +RPM +0 +5000 +10000 +15000 +RPM +0 +4500 +9000 +13500 +Power (Watts) +0 +40 +80 +120 +(a) +(b) +Figure 4: Distribution of (a) RPM with t; (b) Power +due to accelerating torque with RPM for bi-directional +(dashed line) and uni-directional (solid line) outlet. Dot- +ted line represents the power due to braking torque. +the 5-ohm load resistance. The figure 5 illustrates the motor’s +voltage and power variation at various RPM. The turbine- +generator can produce a maximum of 78 watts of electrical +power at 7800 RPM for bi-directional outlet configuration. +RPM +0 +4000 +8000 +PowerE (Watts) +0 +40 +80 +80 +0 +40 +Voltage (Volts) +Figure 5: Experimental results of voltage (dashed line) +and electrical power (solid line) at different RPM +B. Inlet design +Designing the inlet is the most crucial part of the turbine +where the maximum loss occurs. Notably, the compressor +and back pressure at the inlet rotor connection point regu- +lates the flow across the nozzle. In the present article, we +only consider the inlet section for analysis. The details of +the nozzle dimensions are in figure 6. +1) Numerical methodology and grid Grid sensitivity: +The Inlet section of the nozzle is a pressure inlet boundary +where the total pressure is at 6 bar. We consider the outlet +of the nozzle is at 4 bar. The temperature at the compressor +discharge and the inlet-rotor junction are 450K and 400K, +respectively. The surface of the nozzle is a no-slip type +boundary. Considering these boundary conditions, we solve +governing compressible Reynolds-averaged Navier-Stokes +equations using the K − ω SST turbulence model in the +commercial CFD package Fluent 2021 R2. The numerical +domain is discretized using the body-fitted tetrahedral grids, +maintaining a wall y+ ≈ O(1). The table 1 below presents +the grid sensitivity study. As the desired output shows a +Figure 6: Schematic diagram of the nozzles. The dotted, +dashed and solid lines represent nozzle 1, nozzle 2 and +nozzle 3, respectively. +variation ≤ 2%, We conduct the subsequent numerical +simulations using grids with ≈ 1M elements. The inlet +Table 1: Grid sensitivity of the three inlet configurations +Grid type +Elements +Outlet area averaged Mach no +Nozzle 1 +Nozzle 2 +Nozzle 3 +Coarse +≈ 0.5 M +.5093 +.4993 +.5296 +Medium +≈ 1 M +.5089 +.4979 +.5289 +Fine +≈ 2 M +.5080 +.4971 +.5284 +design presented in this article is based on a two-pronged +objective. (a) To Maximize the Mach number peak (b) To +minimize the disparity in the Mach number peaks through +every slit. To understand the loss mechanism and the flow +behavior, we conduct a series of numerical simulations +Where Nozzle 1 is the replica of the inlet nozzle considered +for the experiment. Due to the abrupt divergent section and +the presence of a large recirculation zone enclosed by a +strong shear layer, we detect significant losses in Mach no. +The dividing streamlines seen in figure 7 (a) are considered +when designing the following two inlet systems. The second +nozzle we design is to eliminate the effect of recirculation. +We gradually increase the inlet nozzle area until we reach +the section where the flow bends in the previous observation. +Despite the improvement in the average peak Mach number +observed from figure 7 (b), the disparity in the discharge +Mach number through the slits increased. We consider +designing the third nozzle as a converging-diverging type +nozzle where we place the throat section at x/L ≈ 0.6 to +provide sufficient scope for flow to bend. Figure 8 represents +the peak discharge Mach number through the nozzles and +the associated % disparity from the mean peak Mach no +through each slit. The comparison shows that the third nozzle +offers maximum peak Mach number along with minimum +% deviation from the mean peak Mach no. It is necessary +to note that the differential pressure between the upstream +and downstream sections, the fluid, and the fluid’s thermo- +physical characteristics significantly influence the nozzle +design. The nozzle’s optimal size and shape might differ +depending on these conditions. +3 + +10mm +40 mm +4mm +8 +40 +ww +20mm +mm +mm +15 +mm +2 +60mm(a) +(b) +(c) +Figure 7: Distribution of the Mach number for (a) nozzle +1 (b) nozzle 2 (c) nozzle 3. +IV. +CONCLUSIONS +The article presents a preliminary experimental investiga- +tion of a centimeter scaled Tesla turbine using compressed +air as a working medium and suggests an improved inlet +design that could deliver higher achievable power compared +to the inlet nozzle considered for this experiment. The key +findings of the article are as follows; +1) +Experimental investigation of the turbine conducted +for uni and bi-directional outlet configuration at 6 +bar inlet pressure. +2) +uni-directional outlet configuration offered a peak +power output of ≈ 110 watts at ≈ 7000 RPM. +Whereas, bi-directional outlet configuration a peak +output of 80 watts at 6500 RPM. +3) +The turbine-generator configuration produced a +peak electrical power output of 78 watts at 7800 +-0.02 +-0.01 +0 +0.01 +0.02 +M +0 +0.4 +0.8 +(a) +%M/ +-20 +-10 +0 +10 +20 +%M' +-30 +-20 +-10 +0 +10 +20 +%M' +-15 +-10 +-5 +0 +5 +(b) +(c) +(d) +Figure 8: (a) Comparison of peak Mach number across +the slits. The dotted, dashed and solid lines represent +First, second and third nozzle, respectively. (b, c, d) The +% disparity in peak Mach no at discharge for three +nozzles. +RPM. +4) +While assessing the inlet, we observe that nozzle 2 +offers better peak Mach number than nozzle 1, but +due to the losses accounted by the sharp bend, the +disparity in % deviation of the peak mach numbers +from mean peak Mach no is nearly 30%. +5) +Nozzle 3 performs the best among all three nozzles. +It offers a maximum peak Mach no with minimum +% deviation from mean peak Mach no is reduced +to 12%. +The turbine’s efficiency depends on several factors, e.g., +RPM, disc gap, rotor radius, rotor to casing clearance, +outlet configurations, fluid properties, and many more. +Designing an efficient Tesla turbine could bring down the +cost of a turbine substantially as compared to the other +existing technologies because of its simplistic design. The +above findings presented in this article could be helpful in +the design improvement, thus leaving a plausible scope for +further investigation. +NOMENCLATURE +t +time +[sec] +x/L +Non-dimensional-axial location +[–] +k +Turbulent kinetic energy +[J/kg] +ω +Specific dissipation rate +[1/sec] +M +Mach no +– +M ′ +% Deviation from peak mean Mach no +– +4 + +0.050.1 +0.150.2 +0.25 +0.30.350.40.450.50.55 +0.60.650.7 +0.75 +0.8 +0.85REFERENCES +[1] +James Hal Armstrong, An investigation of the performance of a +modified tesla turbine, Ph.D. thesis, Georgia Institute of Technology, +1952. +[2] +John Spencer Caldwell, The efficiency of a viscous flow compressor, +Ph.D. thesis, Georgia Institute of Technology, 1973. +[3] +Van P Carey, Assessment of tesla turbine performance for small scale +rankine combined heat and power systems, Journal of Engineering +for Gas Turbines and Power 132 (2010), no. 12, 122301. +[4] +Tan Wee Choon, AA Rahman, Foo Shy Jer, and Lim Eng Aik, +Optimization of tesla turbine using computational fluid dynamics +approach, Industrial Electronics and Applications (ISIEA), 2011 +IEEE Symposium on, IEEE, 2011, pp. 477–480. +[5] +L Ciappi, D Fiaschi, PH Niknam, and L Talluri, Computational +investigation of the flow inside a tesla turbine rotor, Energy 173 +(2019), 207–217. +[6] +ME Crawford and W Rice, Calculated design data for the multiple- +disk pump using incompressible fluid, ASME J. 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+no. 3, 285–305. +[33] +Constantin Schosser, Thomas Fuchs, Rainer Hain, Stefan Lecheler, +and C Kahler, Three–dimensional particle tracking velocimetry in a +tesla turbine rotor using a non–intrusive calibration method, 18th +International Symposium on the Application of Laser and Imaging +Techniques to Fluid Mechanics, 2016. +[34] +Constantin Schosser, Stefan Lecheler, and Michael Pfitzner, Ana- +lytical and numerical solutions of the rotor flow in tesla turbines, +Periodica Polytechnica Mechanical Engineering 61 (2017), no. 1, +12–22. +[35] +Constantin Schosser and Michael Pfitzner, A numerical study of the +three-dimensional incompressible rotor airflow within a tesla turbine, +Conference of Modelling Fluid Flow CMFF, 2015, pp. 1–4. +[36] +Sayantan Sengupta and Abhijit Guha, A theory of tesla disc turbines, +Proceedings of the Institution of Mechanical Engineers, Part A: +Journal of Power and Energy 226 (2012), no. 5, 650–663. +[37] +L Talluri, D Fiaschi, G Neri, and L Ciappi, Design and optimization +of a tesla turbine for orc applications, Applied energy 226 (2018), +300–319. +[38] +Lorenzo Talluri, Olivier Dumont, Giampaolo Manfrida, Vincent +Lemort, and Daniele Fiaschi, Experimental investigation of an or- +ganic rankine cycle tesla turbine working with r1233zd (e), Applied +Thermal Engineering 174 (2020), 115293. +[39] +Nikola Tesla, Turbine., May 6 1913, US Patent 1,061,206. +5 + diff --git a/_dAzT4oBgHgl3EQfhPzZ/content/tmp_files/load_file.txt b/_dAzT4oBgHgl3EQfhPzZ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2af158e0e733503257a36fdad7c1817bdc041e44 --- /dev/null +++ b/_dAzT4oBgHgl3EQfhPzZ/content/tmp_files/load_file.txt @@ -0,0 +1,240 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf,len=239 +page_content='Proceedings of the 9th International and 49th National Conference on Fluid Mechanics and Fluid Power (FMFP) December 14-16, 2022, IIT Roorkee, Roorkee-247667, Uttarakhand, India FMFP2022-594 Performance enhancement of a 100 watts class Tesla turbine Arindam Mandal1, Rajosik Adak1, and Sandeep Saha1 1Department of Aerospace Engineering, IIT Kharagpur, Kharagpur 721302, India ABSTRACT Tesla turbines are an attractive but less explored area in low-power applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' This article presents an experimental investigation of a centimeter scale Tesla turbine in bi and uni-directional outlet configuration with compressed air at 6 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' The turbine’s performance is enhanced by ≈ 38% for a uni-directional outlet configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' Furthermore, we also investigate the electrical power of the turbine in a bi-directional outlet configuration by coupling the turbine with a generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' Despite achieving higher performance in uni-directional outlet configuration, we observe substantial losses at the inlet we use for the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' To illustrate and improve the losses, we numerically investigate the turbine inlet at a total pressure and temperature difference of 2 bar and 50◦C, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' Subsequently, we design two more nozzles and compare their performance with the nozzle we used in our experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' Our findings suggest that nozzle 3 performs the best in delivering the highest Mach no and uniformity across the slits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' This observation would help optimize the nozzle suitable for the Tesla turbine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' Keywords: Tesla turbine, Friction turbine, Low power application, Slit nozzle, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' INTRODUCTION Small scale low microturbines are crucial since there is a growing need for them in numerous applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' Examples of notable applications include waste heat recovery, pico- hydro power, organic Rankine cycle technologies, biomass, waste heat recovery, micro GT, and many more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' The increas- ing need for energy harvesting at these scales poses a variety of challenges to the performance and manufacturability of the turbine due to their compact sizes, high rotational speeds, and increased viscous losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' An alternative expansion de- vice addressing these challenges can be highly beneficial regarding its techno-economic feasibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' The tesla turbine is one of its kind, which has a uniquely simple design and a unique mechanism of momentum transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' The turbine rotor consists of several co-axial discs closely packed with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' Each of the rotor discs has outlet ports near the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' A casing covering the rotors helps guide the flow from the inlet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' The turbine shaft is attached to the rotor- casing configuration with the help of two bearings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' Figure 1 shows the different components of the turbine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' After being injected through the inlet system of the turbine, the fluid flows spirally inward and exits through the ports located near the shaft in the axial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' The momentum transfer from the fluid to the rotor takes place using the adhesion and viscosity of the fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' This mechanism makes the turbine attractive at small scales where the dominance of the viscous force becomes significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' This turbine was conceptualized by Nicola Tesla [39] in the year 1907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' Initially, the device was unable to attract market attention because of the no potential requirement of low-power harvesting technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' After that, till the twen- tieth century, a considerable amount of thrust was given to the possible design modifications [27], [28], [12], improved seal designs [2] and loss analysis [6], [30] of the turbine and its ancillary components for enhancing performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' In addition, a few simplified theoretical models were developed [1], [21] to get an insight into the influence of the parameters associated to the turbine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' Figure 1: Schematic of the exploded view of the turbine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' a: Casing, b: Rotors, c: Inlet configurations (i,ii and iii), d: Shaft, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' Bearing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' (f) shows the fluid flow between two consecutive corotating discs More recently, there has been a surge in interest in exploring this technology due to its several advantages over conventional energy harvesting technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' Due to its simple design and flow mechanics, the turbine can handle particle-laden fluids and two-phase expansion with minimal damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' Numerical simulations of the gas flow and mass transfer between two coaxially co-rotating discs were performed by Sandilya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' [32] where they found a satisfactory match between the numerical and experimental value of mass transfer coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' Using numerical simulations, Ladino [18] presented the load coefficient curve, efficiency, and degree of reaction variation after maintaining a constant rotational speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' Upon developing an analytical model, Deam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' [7] scaled down the turbine in millimeter size to achieve better efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' Lemma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' [22] investigated the Parasitic, viscous, and other dissipative losses in the bearings and end walls to mitigate their effect on decaying the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' An explicit description of a flexible test rig was de- 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content='01483v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content='flu-dyn] 4 Jan 2023 (i) (ii) (ii) (c) (p) (b) (a) f (e)veloped by Hoya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' [14] where they calculated the low torque at high RPM using the angular acceleration method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' A comparative study in the efficiency offered by Tesla and small bladed microturbine in a micro power plant was addressed by Lampart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' [20], [19] to establish the competitiveness of a Tesla turbine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' To understand the transport phenomena inside the turbine rotor, a number of numerical and analytical models solving the Navier- Stokes equations using different approaches are present in the literature repository [15], [9], [36], [10], [31], [35], [34], [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' Aside from that, flow diagnostics using particle tracking velocimetry by Schosser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' [33] provides an insight into the component-wise velocity profiles inside the rotor gaps at different radial locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' In recent years, A wide range of application based studies of Tesla turbine related to Micro- air vehicles [23], Organic Rankine cycle [37], [38], [8], [25], [24], Combined heat plant [3], Pico-hydro applications [13], [4], [16], [17], Ammonia synthesys [11] have been investigated or under investigation [26], [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' The recent resurgence in interest indicates the impor- tance of investigating the turbine further to mitigate the losses due to the nonuniformity and disturbances associated with the inflow to rotor and rotor to outflow interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' This article presents the experimental investigation of a centimeter-scaled Tesla turbine in uni and bi-directional outlet configuration with compressed air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' We compare the performance in terms of Mechanical power output for the two configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' Furthermore, we investigate the losses in the inlet due to the sharp divergent and the slit configuration at the inlet rotor junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' Finally, we compare the nozzles’ performance by looking at the peak discharge Mach number and channel-wise disparity in flow injection to the rotor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' Our numerical results can be beneficial in coming up with better inlet configurations for extracting maximum energy from the fluid, which leaves a further scope for further investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' METHODOLOGY AND EXPERIMENT A lab scale turbine prototype (in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' 2a) is fabricated for experimentation in the Aerospace Engineering depart- ment, IIT Kharagpur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' The turbine consists of 10 consecutive corotating discs having an outer diameter of 10 cm and a thickness of 2 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' The gap between the consecutive rotating discs is 2 mm, and the distance between two extreme discs to the adjacent casing wall is 1 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' The turbine has four outlet holes near the shaft, having a center distance of 2 cm from the shaft center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' The shaft and the exhaust ports’ diameters are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content='5 cm and 1 cm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' The distance between the shroud and the disc’s edge is 1 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' The inlet system of the turbine is designed to connect the compressor outlet port of 6mm dia to the turbine having an inlet of 4 cm thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' The area of the inlet is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content='4 cm2, which is segregated using slit configurations to guide air to each gap with minimum interaction with the peripheral walls of the rotor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' The turbine inlet is connected to the compressor by Polyeurathane pneumatic pipes with an installed pres- sure gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' The RPM of the turbine is measured using an Arduino-based tachometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' The angular acceleration- deceleration approach computes the net accelerating and de- celerating torque.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content='The experiment also uses a digital tachome- (a) (b) Figure 2: (a) Fabricated turbine and (b) the inlet system ter to validate the turbine’s RPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' We conduct our experiment at 6 bar of inlet pressure for uni-and bi-directional outlet configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' The detail of the experimental setup can be seen in the figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' (a) (b) Figure 3: Details of the experimental set up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' 1- Compres- sor discharge, 2- Tachometer, 3- Turbine, 4- Arduino based Tachometer, 5- Pressure gauge, 6- PC for data acquisition, 7- Camera, 8- Generator, 9- Multimeters, 10- Rheostat, 11- Electrical circuit III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' RESULTS AND DISCUSSION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' Performance characteristics We tabulate the RPM with time with the help of a serial monitor and calculate the angular acceleration as a function of RPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' Once the turbine reaches its stable RPM, we shut off the compressed air sypply and continue to perform data acquisition until the turbine becomes stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' We continue the similar process for a uni-directional outlet case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' We observe from figure 4 that the turbine with a uni-directional outlet accelerates faster than the turbine with a bi-directional outlet configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' In addition, for a supply pressure of 6 bar, we achieve a maximum RPM of 13019 and 11124 for uni and bi-directional outlet configurations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' Figure 4 shows the power variation due to accelerating and braking torque.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' There is an increment of ≈ 38% in power due to accelerating torque for a uni-directional outlet configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' We integrate the turbine with a 100 W class Fedus RS-775 DC electric motor to measure the electrical power output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' However, the motor is used as a generator to measure the output voltage and current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' The electrical circuit is attached to a rheostat, and the experiment is performed with 2 t (sec) 0 15 30 45 RPM 0 5000 10000 15000 RPM 0 4500 9000 13500 Power (Watts) 0 40 80 120 (a) (b) Figure 4: Distribution of (a) RPM with t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' (b) Power due to accelerating torque with RPM for bi-directional (dashed line) and uni-directional (solid line) outlet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' Dot- ted line represents the power due to braking torque.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' the 5-ohm load resistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' The figure 5 illustrates the motor’s voltage and power variation at various RPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' The turbine- generator can produce a maximum of 78 watts of electrical power at 7800 RPM for bi-directional outlet configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' RPM 0 4000 8000 PowerE (Watts) 0 40 80 80 0 40 Voltage (Volts) Figure 5: Experimental results of voltage (dashed line) and electrical power (solid line) at different RPM B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' Inlet design Designing the inlet is the most crucial part of the turbine where the maximum loss occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' Notably, the compressor and back pressure at the inlet rotor connection point regu- lates the flow across the nozzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' In the present article, we only consider the inlet section for analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' The details of the nozzle dimensions are in figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' 1) Numerical methodology and grid Grid sensitivity: The Inlet section of the nozzle is a pressure inlet boundary where the total pressure is at 6 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' We consider the outlet of the nozzle is at 4 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' The temperature at the compressor discharge and the inlet-rotor junction are 450K and 400K, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' The surface of the nozzle is a no-slip type boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' Considering these boundary conditions, we solve governing compressible Reynolds-averaged Navier-Stokes equations using the K − ω SST turbulence model in the commercial CFD package Fluent 2021 R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' The numerical domain is discretized using the body-fitted tetrahedral grids, maintaining a wall y+ ≈ O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' The table 1 below presents the grid sensitivity study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' As the desired output shows a Figure 6: Schematic diagram of the nozzles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' The dotted, dashed and solid lines represent nozzle 1, nozzle 2 and nozzle 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' variation ≤ 2%, We conduct the subsequent numerical simulations using grids with ≈ 1M elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' The inlet Table 1: Grid sensitivity of the three inlet configurations Grid type Elements Outlet area averaged Mach no Nozzle 1 Nozzle 2 Nozzle 3 Coarse ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content='5 M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content='5093 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content='4993 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content='5296 Medium ≈ 1 M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content='5089 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content='4979 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content='5289 Fine ≈ 2 M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content='5080 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content='4971 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content='5284 design presented in this article is based on a two-pronged objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' (a) To Maximize the Mach number peak (b) To minimize the disparity in the Mach number peaks through every slit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' To understand the loss mechanism and the flow behavior, we conduct a series of numerical simulations Where Nozzle 1 is the replica of the inlet nozzle considered for the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' Due to the abrupt divergent section and the presence of a large recirculation zone enclosed by a strong shear layer, we detect significant losses in Mach no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' The dividing streamlines seen in figure 7 (a) are considered when designing the following two inlet systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' The second nozzle we design is to eliminate the effect of recirculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' We gradually increase the inlet nozzle area until we reach the section where the flow bends in the previous observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' Despite the improvement in the average peak Mach number observed from figure 7 (b), the disparity in the discharge Mach number through the slits increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' We consider designing the third nozzle as a converging-diverging type nozzle where we place the throat section at x/L ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content='6 to provide sufficient scope for flow to bend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' Figure 8 represents the peak discharge Mach number through the nozzles and the associated % disparity from the mean peak Mach no through each slit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' The comparison shows that the third nozzle offers maximum peak Mach number along with minimum % deviation from the mean peak Mach no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' It is necessary to note that the differential pressure between the upstream and downstream sections, the fluid, and the fluid’s thermo- physical characteristics significantly influence the nozzle design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' The nozzle’s optimal size and shape might differ depending on these conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' 3 10mm 40 mm 4mm 8 40 ww 20mm mm mm 15 mm 2 60mm(a) (b) (c) Figure 7: Distribution of the Mach number for (a) nozzle 1 (b) nozzle 2 (c) nozzle 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' CONCLUSIONS The article presents a preliminary experimental investiga- tion of a centimeter scaled Tesla turbine using compressed air as a working medium and suggests an improved inlet design that could deliver higher achievable power compared to the inlet nozzle considered for this experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' The key findings of the article are as follows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' 1) Experimental investigation of the turbine conducted for uni and bi-directional outlet configuration at 6 bar inlet pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' 2) uni-directional outlet configuration offered a peak power output of ≈ 110 watts at ≈ 7000 RPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' Whereas, bi-directional outlet configuration a peak output of 80 watts at 6500 RPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' 3) The turbine-generator configuration produced a peak electrical power output of 78 watts at 7800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content='01 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content='02 M 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content="8 (a) %M/ 20 10 0 10 20 %M' 30 20 10 0 10 20 %M' 15 10 5 0 5 (b) (c) (d) Figure 8: (a) Comparison of peak Mach number across the slits." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' The dotted, dashed and solid lines represent First, second and third nozzle, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' (b, c, d) The % disparity in peak Mach no at discharge for three nozzles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' RPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' 4) While assessing the inlet, we observe that nozzle 2 offers better peak Mach number than nozzle 1, but due to the losses accounted by the sharp bend, the disparity in % deviation of the peak mach numbers from mean peak Mach no is nearly 30%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' 5) Nozzle 3 performs the best among all three nozzles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' It offers a maximum peak Mach no with minimum % deviation from mean peak Mach no is reduced to 12%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' The turbine’s efficiency depends on several factors, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=', RPM, disc gap, rotor radius, rotor to casing clearance, outlet configurations, fluid properties, and many more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' Designing an efficient Tesla turbine could bring down the cost of a turbine substantially as compared to the other existing technologies because of its simplistic design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' The above findings presented in this article could be helpful in the design improvement, thus leaving a plausible scope for further investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' NOMENCLATURE t time [sec] x/L Non-dimensional-axial location [–] k Turbulent kinetic energy [J/kg] ω Specific dissipation rate [1/sec] M Mach no – M ′ % Deviation from peak mean Mach no – 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content='050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content='150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content='350.' metadata={'source': 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rankine cycle tesla turbine working with r1233zd (e), Applied Thermal Engineering 174 (2020), 115293.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' [39] Nikola Tesla, Turbine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=', May 6 1913, US Patent 1,061,206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} +page_content=' 5' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfhPzZ/content/2301.01483v1.pdf'} diff --git a/atAyT4oBgHgl3EQfv_nZ/content/tmp_files/2301.00642v1.pdf.txt b/atAyT4oBgHgl3EQfv_nZ/content/tmp_files/2301.00642v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0961538f87eee801732333cb7de1705581c455fe --- /dev/null +++ b/atAyT4oBgHgl3EQfv_nZ/content/tmp_files/2301.00642v1.pdf.txt @@ -0,0 +1,1820 @@ +arXiv:2301.00642v1 [math.CA] 30 Dec 2022 +SPECIAL POLYNOMIALS AND NEW REAL-ROOTEDNESS RESULTS +AURELIEN GRIBINSKI +EPFL +Abstract. In this paper, we show that for some orthogonal polynomials P (z) +n (x) showing up +in physics, namely Laguerre and Gegenbauer, P (z) +n (x) are realrooted in z for x in the support of +orthogonality. As an application we show realrootedness in x and interlacing properties of ∂k +z P (z) +n (x) +for k ≤ n for z > 0. +1. +Introduction +1 +2. +General strategy +1 +3. +Laguerre polynomials +2 +4. +Gegenbauer polynomials +7 +5. +Applications to realrootedness in x +13 +References +16 +1. Introduction +Orthogonal polynomials like generalized Laguerre and Gegenbauer polynomials have long been +studied, and show up in all fields of maths and physics. However little has been said about the +properties of such polynomials when we vary the underlying parameter (see [1]). We study families +of generalized orthogonal polynomials P (z) +n (x) depending on a parameter z from a new angle, that +is, as bivariate polynomials Pn(x, z). We fix the usual variable x and consider them instead as +polynomials in z. We show that they are real-rooted in z for x in the support of the underlying +measure of orthogonality, and monotonous. Furthermore we show that when we differentiate these +orthogonal polynomials with respect to z > 0 and consider them as polynomials in x, then they are +realrooted in x. Such polynomials (derivatives with respect to the parameter) seem to have many +nice properties similar to their corresponding orthogonal polynomials from which they are derived +and yet have never been studied. +2. General strategy +Consider P (z) +n (x), a family of polynomials depending on a parameter z. We want to show that they +are real-rooted in z for a fixed x in a given interval. +The strategy is as follows +• We check that for x at one of the extreme point of the interval it is real-rooted. +• We show that locally around this extreme point the roots in z are all monotonous. +• We show that there can’t be a shared zero in z for ∂xPn(x, z) and Pn(x, z) or equivalently +for Pn−1(x, z) and Pn(x, z). +• The roots in z are therefore monotonous when they are well-defined. +Date: January 3, 2023. +1 + +2 +AURELIEN GRIBINSKI EPFL +• We show that the roots in z of Pn(x, z) and simultaneously Pn−1(x, z) and ∂xPn(x, z) inter- +lace as long as they are well-defined. +• We extend the local properties by exhibiting an ODE to which all roots in z are solutions +and show that there has to be explosion at the other extreme point of the interval, all roots +evolving monotonously along the way. +First we derive a way to locally prove the existence of rel-rootedness if it is known at some extreme +point of the interval. +Lemma 2.1 (Local existence of the roots and smoothness). Take a ∈ R. Assume P(x, z) is a +bivariate polynomial such that P(a, z) is realrooted in z of degree j and has only simple roots in +z (let’s call them zi(a) for i = 1....j). Assume that P(x, z) has degree less than j for all x.Then +for x in a neighborhood of a, P(x, z) is also realrooted in z of degree j with simple roots zi(x). +Furthermore, the roots zi(x) are C∞ functions of x. +Proof. Consider the equation P(x, z) = 0, around the points +� +a, zi(a) +� +. We have ∂zP(x, z)|x=a,z=zi(a) ̸= +0 as the roots are simple (can’t be a root of the derivative in z). Then using the implicit function +theorem, we can find in the neighborhood of each point a C∞ function zi(x) which will be the only +solution to the equation P(x, z) = 0 on this neighborhood. Therefore we have found j roots, and +it is the maximal number of roots for a fixed x, as the polynomial is of degree less than j. +□ +3. Laguerre polynomials +Let L(z) +n (x) be the Laguerre polynomials with complex parameter z. It is a polynomial in R[x, z]. +Theorem 3.1. For fixed x0 ∈ [0, +∞[, L(z) +n (x0) is a real-rooted polynomial in z of degree n. Fur- +thermore, its roots in z are strictly increasing to +∞ when x0 moves along [0, +∞[. +Proof. Let’s recall the hypergeometric confluent definition +L(z) +n (x) = +�n + z +n +� +M(−n, z + 1, x) += +n +� +l=1 +z + n + 1 − l +l +n +� +k=0 +(−1)kn! +(n − k)! �k−1 +l=0 (z + 1 + k) +xk += +n +� +k=0 +(−1)k �n +l=1(z + l) +(n − k)! �k +l=1(z + l) +xk += +n +� +k=0 +(−1)k �n +j=k+1(z + j) +(n − k)! +xk += 1 +n!zn + +��n +j=1 j +n! +− +x +(n − 1)! +� +zn−1 + Rn−2(z, x) +where Rn−2(z, x) is of degree lower than n−2 in z. We will henceforth write Ln(x, z) as it is clearly +a bivariate polynomial of degree n in z with front coefficient +1 +n!. +Let’s furthermore decompose +Ln(x, z) using a priori complex roots λi(x): +Ln(x, z) = 1 +n! +n +� +i=1 +� +z − λi(x) +� +where we order the roots by decreasing module: |λ1(x)| ≥ |λ2(x)| ≥ ... ≥ |λn(x)|. +Lemma 3.2 (Local realrootedness). Ln(x, z) is real rooted of degree n in z with simple roots in a +neighborhood of x = 0. + +SPECIAL POLYNOMIALS AND NEW REAL-ROOTEDNESS RESULTS +3 +Proof. We have +Ln(0, z) = +�n +l=1(z + l) +n! +so that we can apply Lemma 2.1 with a = 0. +□ +Lemma 3.3 (Local increasing property). The roots of Ln(x, z) in z are all strictly increasing when +x is, in a neighborhood of x = 0. +Proof. To prove this, we need some information on the derivatives with respect to x of the roots in +the neighborhood of 0, which we know are going to be real by the previous lemma. +Lemma 3.4. dlλi(x) +dxl +|x=0 = 0 for 1 ≤ l < i, and diλi(x) +dxi +|x=0 > 0. +Proof. We get, for all 1 ≤ l ≤ n, +∂l +xLn(x, z)|x=0 = l!(−1)l +�n +j=l+1(z + j) +(n − l)! +Notice that λi(0) = −i for i = 1...n, so we see that for l ≤ i − 1, +∂l +xLn +� +0, λi(0) +� += l!(−1)l +�n +j=l+1(λi(0) + j) +(n − l)! += 0 +And +∂i +xLn +� +0, λi(0) +� += i!(−1)i +�n +j=i+1(λi(0) + j) +(n − i)! += i!(−1)i +On the other hand, +∂zLn +� +0, λi(0) +� += 1 +n! +n +� +l=1,l̸=i +(−i + l) = i!(−1)i−1(n − i)! +n! +Now we have Ln +� +x, λi(x) +� += 0 for all i = 1...n, by definition, so differentiating with respect to x, +we get +dλi(x) +dx += −∂xLn +∂zLn +� +x, λi(x) +� +Note that the denominator is nonzero as the roots in z are simple at 0 ( so they won’t be roots of +the derivative in z). Using Leibniz’s formula and induction on l, we get for i > l ≥ 1, +dlλi(x) +dxl +|x=0 += 0 +And +diλi(x) +dxi +|x=0 += −∂i +xLn +∂zLn +� +− 1, λi(0) +� += +n! +(n − i)! > 0 +□ +We conclude by a Taylor expansion around 0 as +λi(x) = −i + xi +i! +n! +(n − i)! + o(xi) +□ +Lemma 3.5 (Distinct roots, degree and derivative wise). Assume λi(x) is real for x ∈]0, bi[, bi > 0. +Then ∂xLn(x, λi(x)) can’t be zero for x ∈]0, bi[. Therefore it has a constant sign on this interval. +Equivalently, Ln−1(x, λi(x)) can’t be zero either: that is we can’t have a nontrivial shared real root +for Ln−1(x, z) and Ln(x, z). + +4 +AURELIEN GRIBINSKI EPFL +Proof. Traditional results on simplicity of the roots can’t be used because they are true only for +z ≥ 0. By definition, Ln +� +x, λi(x) +� += 0. Then the usual differential euqation still holds +(1) +x∂xLn(x, z) = nLn(x, z) − (n + z)Ln−1(x, z) +Let’s assume by contradiction that ∂xLn +� +x0, λi(x0) +� += 0 for some i and x0 ∈]0, bi[. As ∂xLn(x, λi(x)) +is nonzero in a neighborhood of x = 0, x > 0(local monotonicity above), then we can assume x0 is +the smallest x > 0 such that ∂xLn +� +x, λi(x) +� += 0. Therefore as +dλi(x) +dx += −∂xLn +∂zLn +� +x, λi(x) +� +we have that on ]0, x0], λi(x) is strictly increasing in x. As λi(0) ≥ −n = λn(0) for all i, then +� +n − λi(x0) +� +> 0, and we get that the statement is equivalent to Ln−1 +� +x0, λi(x0) +� += 0 using +Equation 1. But then, as the following recurrence relations are still valid +(n + 1)Ln+1(x, z) = (−x + 2(n + 1) + z)Ln(x, z) − (n + z)Ln−1(x, z) +we also get by induction Ln+k(x0, λi(x0) + 1) = 0 for all k ∈ N. Using then the equality +∂xLn+k(x, z) = −Ln+k−1(x, z + 1) +we get that Ln+k−1(x0, λi(x0) + 1) = 0 for all k ∈ N. +Using induction, applying successively +the previous recurrence equations, we then get that Ln+k−1(x0, λi(x0) + j) = 0 for all j ∈ N. +For j ≤ n, the polynomials Ln+k−1(x, λi(x) + j) are standard Laguerre polynomials (positive +parameter). It would mean that successive Laguerre polynomials of parameter λi(x0) + j have +the root x0 in common, so that their derivatives share this root too, which is absurd as the roots +of Laguerre polynomials are simple by classical orthogonality. +We conclude that Ln−1(x, λi(x)) +as well as ∂xLn +� +x0, λi(x0) +� +can’t be zero, and therefore ∂xLn +� +x0, λi(x0) +� +has a constant sign for +x ∈]0, bi[. +□ +Corollary 3.6 (Extended monotonicity). Assume λi(x) is real for x ∈]0, bi[, bi > 0. Then it +follows from the previous proof that for x ∈]0, bi[ +dλi(x) +dx +> 0 +Theorem 3.7 (Interlacing roots, degreewise, simple roots). Consider an interval I = [0, b[ such +that Ln(x, z) has real roots in z on I, then the same will be true of Ln−1(x, z) and their roots +interlace. Furthermore, the interlacing is strict for x > 0 and both polynomials have simple roots +on I. +Proof. Let’s write +Ln(x, z) = 1 +n! +n +� +i=1 +� +z − λn +i (x) +� +Ln−1(x, z) = +1 +(n − 1)! +n−1 +� +i=1 +� +z − λn−1 +i +(x) +� +and show that all x ∈ I, all i ≤ n − 1: +λn +i (x) ≥ λn−1 +i +(x) ≥ λn +i+1(x) +with strict inequalities for x > 0. We first check the property locally, that is a neighborhood of 0. +λn +i (0) = −i +λn−1 +i +(0) = −i +λn +i+1(0) = −(i + 1) + +SPECIAL POLYNOMIALS AND NEW REAL-ROOTEDNESS RESULTS +5 +We have +diλn +i (x) +dxi +|x=0 += +n! +(n − i)! += +n +n − i +(n − 1)! +(n − 1 − i)! += +n +n − i +diλn−1 +i +(x) +dxi +|x=0 +As +n +n−i > 1, we conclude that for all i ≤ n − 1, diλn +i (x) +dxi +|x=0 > diλn−1 +i +(x) +dxi +|x=0. +We can then do a Taylor expansion around x = 0: +λn +i (x) = −i + (x + 1)i +i! +diλn +i (x) +dxi +|x=−1 ++ o((x + 1)i) +λn−1 +i +(x) = −i + (x + 1)i +i! +diλn−1 +i +(x) +dxi +|x=−1 ++ o((x + 1)i) +It is then clear that in a neighborhood of 0, λn +i (x) > λn−1 +i +(x). +As λn−1 +i +(−1) − λn +i+1(−1) = 1, we also get λn−1 +i +(x) > λn +i+1(x) in a neighborhood of 0. Now as +for all i λn +i (x), λn−1 +i +(x), λn +i+1(x) are continuous functions of x, if by contradiction such inequalities +where to fail for some x ∈ I, then there would exist x0 such that λn +i (x0) = λn−1 +i +(x0) or λn−1 +i +(x0) = +λn +i+1(x0).But then this would mean that λn−1 +i +(x0) is a root of Gn(x0, z) and Gn−1(x0, z), which is +impossible by Lemma 4.5. Therefore we conclude that the inequality +λn +i (x) > λn−1 +i +(x) > λn +i+1(x) +holds for allx ∈ I, x > 0 and i ≤ n − 1. +□ +Theorem 3.8 (Interlacing roots, derivative). Consider an interval I = [0, b[ such that Ln(x, z) has +real roots in z on I, then the same will be true of ∂xLn(x, z) and the roots of the two polynomials +interlace and are simple. +Proof. We bring ourselves back to a variant of the previous theorem by using the equality +∂xLn(x, z) = −Ln−1(x, z + 1) +The roots being simple results from Lemma 3.7. So it amounts to proving that Ln−1(x, z + 1) and +Ln(x, z) interlace. We want to show more precisely that for all x ∈ I, with x > 0, all i ≤ n − 1 +λn +i (x) ≥ λn−1 +i +(x) − 1 ≥ λn +i+1(x) +With strict inequalities for x > 0. First we check such inequalities in a neighborhood of 0. We +can check that the inequality λn +i (x) > λn−1 +i +(x) − 1 is going to be true in a neighborhood of 0 as +λn +i (0) = λn−1 +i +(0). So the nontrivial one is the other one, λn−1 +i +(x) − 1 > λn +i+1(x) for x > 0. We have +equality at the origin as λi(0) − 1 = λi+1(0). Then we look at the Taylor expansions around x = 0: +λn−1 +i +(x) − 1 = λi+1(0) + (x + 1)i +i! +diλn−1 +i +(x) +dxi +|x=0 ++ o((x + 1)i) +λn +i+1(x) = λi+1(0) + (x + 1)i+1 +(i + 1)! +di+1λn +i+1(x) +dxi+1 +|x=0 ++ o((x + 1)i+1) +It is clear then that locally λn−1 +i +(x) − 1 > λn +i+1(x) as (x + 1)i+1 << (x + 1)i. We extend the +inequality to the whole interval I by noticing again that if the inequalities where not valid anymore, +then there would have to be some equality λn +i (x) = λn−1 +i +(x) − 1 or λn−1 +i +(x) − 1 = λn +i+1(x), which +would mean ∂xLn(x, λn−1 +i +(x)) = 0 and as Ln(x, λn−1 +i +(x)) = 0, we would again get a contradiction +by Lemma 4.5. +□ + +6 +AURELIEN GRIBINSKI EPFL +Lemma 3.9 (Global extension through ODE). The local property is in fact true over the whole +interval: Ln(x, z) is real rooted in z with simple (distinct) roots for x ∈ [0, +∞[ , and the roots are +all increasing to +∞ when x goes to +∞. +Proof. Denote by Fn(x, z) := − ∂xLn +∂zLn +� +x, z +� +. Consider a rectangular domain D such that ∂zLn(x, z) +is nonzero on the domain. Fn is continuous in x and z in the the domain D. Indeed, it is a rational +fraction whose denominator is nonzero and it is therefore C∞ in both variables by theorem of +composition. As Ln(0, z) is realrooted in z with simple roots, ∂zLn +� +0, λi(0) +� +̸= 0 and by continuity +we can find small rectangles Di := [0, ǫ] × [λi(0) − δ, λi(0) + δ] such that Ln(x, z) is nonzero on +Di. A strong version of Picard’s theorem tells us that there is a maximal interval Imax +i += [0, ηi +max[ +(where ηi +max ∈ +¯ +R+) for which the roots λi(x) (i = 1, 2...n) are the unique solutions of the initial +value ODE +dz +dx = Fn(x, z), +z(−1) = −i +Note that on Imax +i +, ∂zLn +� +x, λi(x) +� +̸= 0 (the denominator is nonzero, so that the differential equa- +tion is well defined). +Let’s prove that Imax +i += [0, +∞[ (for all i) and that there is explosion at +∞ (roots going to +infinity), the roots increasing constantly to +∞. +By Corollary 3.6, Fn(x, λi(x)) > 0 on Imax +i +. +According to Picard’s theorem, we either have +λi(x) →x→ηimax +∞ (explosion), or ηi +max is such that limx→ηimax Fn(x, λi(x)) is not well defined +(we leave the domain of definition). +Now, explosion can’t happen if ηi +max < +∞. Indeed, we have using the hypergeometric expansion +above +n +� +i=1 +λi(x) = −n! +�n(n + 1) +2 +1 +n! − +x +(n − 1)! +� += n +� +− n + 1 +2 ++ x +� +so the sum of roots is bounded above for x < ηi +max and there can be no explosion (necessarily to+∞ +by monotonicity). +We can leave the domain of definition only if limx→ηimax ∂zLn +� +x, λi(x) +� += 0. If this is the case +and if by contradiction ηi +max < +∞, ∂zLn(ηi +max, z) would be of degree n in z. +Therefore it +means that limx→ηimax λi(x) = µ where µ is a root of ∂zLn(ηi +max, z). +But then it means that +we can extend by continuity λi(x) at x = ηi +max with λi(ηi +max) = µ. We check by continuity that +Ln +� +ηi +max, λi(ηi +max) +� += ∂zLn +� +ηi +max, λi(ηi +max) +� += 0 so that in fact λi(ηi +max) is a real double root in z +of Ln +� +ηi +max, z +� +. Using Lemma 3.8, as there is a root of ∂xLn(x, z) between any two roots of Ln(x, z) +in z by interlacing, it follows that necessarily ∂xLn +� +ηi +max, λi(ηmax) +� += 0. But this is impossible +according to Lemma 4.5. Therefore, we have necessarily ηi +max = +∞ for all i = 1...n. +Furthermore, assume by contradiction that there is no explosion for some index i at +∞. As λi(x) +is monotonous for x ∈ [0, +∞[, then we have necessarily that limx→+∞ λi(x) = µ exists and is +finite. By continuity we have Ln(x, µ) ∼x→∞ (−1)nxn, and ∂xLn(x, µ) ∼x→∞ n(−1)nxn−1, as well +as ∂zLn(x, µ) ∼x→∞ (−1)n−1xn−1 using +L(z) +n (x) = +n +� +k=0 +(−1)k �n +j=k+1(z + j) +(n − k)! +xk +so that +dλi(x) +dx +→x→+∞ n +and clearly we would have λi(x) → +∞, which is a contradiction. + +SPECIAL POLYNOMIALS AND NEW REAL-ROOTEDNESS RESULTS +7 +□ +□ +4. Gegenbauer polynomials +Let G(z) +n (x) be the Gegenbauer polynomial with complex parameter z. It is a polynomial in R[x, z]. +Theorem 4.1. For fixed x0 ∈ [−1, 1], G(z) +n (x0) is a real-rooted polynomial in z of degree at most n +(exactly n except for x0 = 0). Furthermore, its roots in z then they are increasing for x ∈ [−1, 0[, +and decreasing for x ∈]0, 1], with an explosion to infinity at 0. +Proof. As the Gegenbauer polynomials are even or odd in x, that is G(z) +n (−x) = (−1)nG(z) +n (x), it is +enough to prove our statement for x ∈ [−1, 0[. We prove it in an incremental way moving x from +−1 to 0. Let’s recall the hypergeometric expression: +G(z) +n (x) = +n−1 +� +l=0 +(2z + l) +n +� +k=0 +(−1)k +1 +k!(n − k)! +�k−1 +i=0 (2z + n + i) +�k−1 +i=0 (z + 1/2 + i)2k (1 − x)k += +n +� +k=0 +(−1)k 2n�n +k +� +n! +�n+k−1 +i=0 +(z + i/2) +�k−1 +i=0 (z + (2i + 1)/2) +(1 − x)k += +n +� +k=0 +2n�n +k +� +n! +n+k−1 +� +i=2k +(z + i/2) +k−1 +� +i=0 +(z + i)(x − 1)k +We will henceforth write Gn(x, z) as it is clear from the previous expression that it is indeed a +bivariate polynomial and not a rational fraction in z. +We start dealing with the extreme boundary. We have +(2) +Gn(x, z) = (−1)nGn(−x, z) = +n +� +k=0 +2n�n +k +� +n! +(−1)n+k +n+k−1 +� +i=2k +(z + i/2) +k−1 +� +i=0 +(z + i)(x + 1)k +So that Gn(−1, z) = (−1)n 2n +n! +�n−1 +i=0 (z +i/2), which is clearly realrooted in z with simple roots. We +can also check this property for x = 0: +Gn(0, z) = +Γ(n/2 + z) +Γ(z)Γ(n/2 + 1) = +1 +(n/2)! +j=n/2−1 +� +j=0 +(z + j) +when n is even +Gn(0, z) = 0 +when n is odd +Also, each bivariate polynomial in the sum is of degree n in z so the sum is of degree at most n in +z. It is in fact of degree exactly n for x ̸= 0 by inspection of the coefficient of zn in the sum, which +is equal to +(−1)n 2n +n! +n +� +k=0 +�n +k +� +(−1)k(1 + x)k = (−1)n 2n +n! +� +1 − (1 + x) +�n = (−1)n 2n +n! (−x)n = (2x)n +n! +Notice that we can write, if n is even, +Gn(x, z) = +� n/2−1 +� +j=0 +(z + j) +� +˜ +Gn(x, z) +and if n is odd, +Gn(x, z) = +� (n−1)/2 +� +j=0 +(z + j) +� +˜ +Gn(x, z) + +8 +AURELIEN GRIBINSKI EPFL +So that in all cases for x ̸= 0, we can write +Gn(x, z) = +� ⌈n/2⌉−1 +� +j=0 +(z + j) +� +˜ +Gn(x, z) += +� ⌈n/2⌉−1 +� +j=0 +(z − µj) +�(2x)n +n! +⌊n/2⌋ +� +i=1 +� +z − λi(x) +� +where µj = −j and the λi(x) are a priori complex roots defined only forx ̸= 0. But it simplifies +greatly for x = −1: +˜ +Gn(x, z)|x=−1 = 2n +n! (−1)n +⌊n/2⌋ +� +i=1 +(z + 1/2 + i − 1) +so that λi(−1) = −1/2−(i−1) for i = 1, 2...⌊n/2⌋ are all real and distinct. That is, approximately +half of the roots in z, depending on the oddness, are constant when x is moving. We can therefore +investigate instead the evolution of the roots of Ln(x, z) to avoid considering roots that remain +constant (and real). First, let’s explain why roots will be indeed smooth and real for x close to −1. +Corollary 4.2 (Local realrootedness). +˜ +Gn(x, z) is real rooted of degree ⌊n/2⌋ in z with simple roots +in a neighborhood of x = −1. +Proof. We have +˜ +Gn(x, z)|x=−1 = 2n +n! (−1)n +⌊n/2⌋ +� +i=1 +(z + 1/2 + i − 1) +which has ⌈n/2⌉ simple roots in z, so we just have to apply Lemma 2.1 with a = −1. +□ +Lemma 4.3 (Local increasing property). The roots of ˜ +Gn(x, z) in z are all strictly increasing when +x is, in a neighborhood of x = −1. +To prove this, we need some information on the derivatives with respect to x of the roots in the +neighborhood of −1. +Lemma 4.4. If we denote by λi(x) the roots of ˜ +Gn(x, z) in decreasing order, then dlλi(x) +dxl +|x=−1 = 0 +for 1 ≤ l < i, and diλi(x) +dxi +|x=−1 > 0. +Proof. Using Equation 2 we get that +∂l +xGn(x, z)|x=−1 = l!2n�n +l +� +n! +(−1)n+l +n+l−1 +� +j=2l +(z + j/2) +l−1 +� +j=0 +(z + j) +So that +∂l +x ˜ +Gn(x, z)|x=−1 = 2n�n +l +� +l! +n! +(−1)n+l +⌈n/2⌉−1 +� +j=l +(z + 1/2 + j) +n+l−1 +� +j=n+1 +(z + j/2) +So we see that +∂l +x ˜ +Gn(x, z) +� +− 1, λi(−1) +� += 0 for i ≥ l + 1 +and +(−1)n+i∂i +x ˜ +Gn(x, z) +� +− 1, λi(−1) +� += 2n�n +i +� +i! +n! +⌈n/2⌉−1 +� +j=i +� +j − (i − 1) +� n+i−1 +� +j=n+1 +� +(j − 1)/2 − (i − 1) +� +> 0 +as i ≤ ⌊n/2⌋. + +SPECIAL POLYNOMIALS AND NEW REAL-ROOTEDNESS RESULTS +9 +Now we have ˜ +Gn +� +x, λi(x) +� += 0 for all i, by definition, so differentiating with respect to x, we get: +dλi(x) +dx += −∂x ˜ +Gn +∂z ˜ +Gn +� +x, λi(x) +� +Note that the denominator is nonzero as the roots in z are simple at −1 ( so they won’t be roots +of the derivative in z). Using Leibniz’s formula and induction on l, we get for i > l ≥ 1, +dlλi(x) +dxl +|x=−1 += 0 +And +diλi(x) +dxi +|x=−1 += −∂i +x ˜ +Gn +∂z ˜ +Gn +� +− 1, λi(−1) +� +Now, we have ˜ +Gn(x, z)|x=−1 = (−1)n 2n +n! +�⌈n/2⌉−1 +j=0 +(z + 1/2 + j) so that +∂z ˜ +Gn(x, z) +� +−1, λi(−1) +� += (−1)n 2n +n! +⌈n/2⌉−1 +� +j=0,j̸=i−1 +� +j−(i−1) +� += (−1)n 2n +n! (−1)i−1 +i−2 +� +j=0 +� +(i−1)−j +� ⌈n/2⌉−1 +� +j=i +� +j−(i−1) +� +and it follows that (−1)n+i−1∂z ˜ +Gn(x, z) +� +− 1, λi(−1) +� +> 0. Therefore +diλi(x) +dxi +|x=−1 +> 0 +as claimed. +□ +Then Lemma 4.3 follows easily by Taylor expansion of the roots in x around 0 , as by some +asymptotic expansion at x = −1, +λi(x) = λi(−1) + (x + 1)i +i! +diλi(x) +dxi +|x=−1 ++ o((x + 1)i) +We have proved the ”initial condition” : now we need to look at the evolution of roots from a +differential equation point of view. First, let’s prove some intermediate results. +Lemma 4.5 (Simple roots). Assume λi(x) is real for x ∈] − 1, bi[, bi ≤ 0 ( such a bi > −1 exists +according to the previous local existence). Then ∂x ˜ +Gn(x, λi(x)) and a fortiori ∂xGn(x, λi(x)) ( for +i = 1, 2...⌊n/2⌋) can’t be zero for x ∈] − 1, bi[. Therefore it has a constant sign on this interval. +Equivalently, Gn−1(x, λi(x)) can’t be zero either: that is we can’t have a nontrivial shared root for +Gn−1(x, z) and Gn(x, z). +Proof. We have ∂xGn(x, λi(x)) = �⌈n/2⌉−1 +j=0 +� +λi(x) + j +� +∂x ˜ +Gn +� +x, λi(x) +� +. We can’t use directly the +results on the monotonicity of the roots of Gegenbauer polynomials when the paremeter is mov- +ing, or the simplicity of the roots in x, because the parameter here is negative and orthogonal- +ity results don’t apply. Let’s assume by contradiction that ∂x ˜ +Gn(x0, λi(x0)) = 0, and therefore +∂xGn(x0, λi(x0)) = 0 for some i and x0 ∈] − 1, bi[. As ∂xGn(x, λi(x)) is nonzero in a neighborhood +of x = −1, x > −1(local monotonicity), then we can assume x0 is the smallest x > −1 such that +∂xGn(x, λi(x)) = 0. Therefore on ] − 1, x0], λi(x) is strictly increasing in x because +dλi(x) +dx += −∂x ˜ +Gn +∂z ˜ +Gn +� +x, λi(x) +� +As λi(−1) ≥ −(n − 1)/2 for all i, then (n + 2λi(x0) − 1) > 0, and using the differential equation +(1 − x2 +0)∂xGn(x0, λi(x0)) = −nxGn(x0, λi(x0)) + (n + 2λi(x0) − 1)Gn−1(x, λi(x0)) + +10 +AURELIEN GRIBINSKI EPFL +and the fact that Gn(x0, λi(x0)) = 0 (by definition), it would lead us to Gn−1(x0, λi(x0)) = 0. Then, +using the recurrence relation (we have a fortiori 2n + 2λi(x0) > 0) +n + 1 +2n + 2λi(x0)Gn+1(x0, λi(x0)) = xGn(x, λi(x)) − n + 2λi(x0) − 1 +2n + 2λi(x0) Gn−1(x0, λi(x0)) +we get successively by induction that Gn+k(x0, λi(x0)) = 0 for all k ∈ N, and using again the +differential equation we get that ∂xGn+k(x0, λi(x0)) = 0 for all k ∈ N. But we have +∂xGn+k(x0, λi(x0)) = 2λi(x0)Gn+k−1(x0, λi(x0) + 1) +so that Gn+k−1(x0), λi(x0) + 1) = 0 for all k ∈ N. Then it is easy to show by induction that +Gn+k−1(x0, λi(x0) + j) = 0 for all j ∈ N, and for j larger than (n − 1)/2, the parameter is +positive, and we are brought back to classical Gegenbauer polynomials. This means that successive +Gegenbauer polynomials with parameter λi(x0)+j have a root in common, so that their derivatives +share these roots too, which is absurd as their roots are simple by orthogonality. We conclude that +∂xGn(x, λi(x)) has a constant sign for all x ∈] − 1, bi[. +□ +Theorem 4.6 (Interlacing roots, degree). Consider an interval I = [−1, b[ such that ˜ +Gn(x, z) has +simple real roots in z on I, then the same will be true of ˜Gn−1(x, z) and their roots interlace. +Proof. Let’s write +˜ +Gn(x, z) = (2x)n +n! +⌊n/2⌋ +� +i=1 +� +z − λn +i (x) +� +˜Gn−1(x, z) = (2x)n−1 +(n − 1)! +⌊(n−1)/2⌋ +� +i=1 +� +z − λn−1 +i +(x) +� +and show that for all x ∈ I, all i ≤ ⌊(n − 1)/2⌋, λn +i (x) > λn−1 +i +(x) > λn +i+1(x). We first check the +property locally, that is a neighborhood of −1, above −1. We have +diλn +i (x) +dxi +|x=−1 += −∂i +x ˜ +Gn +∂z ˜ +Gn +� +− 1, λi(−1) +� += − +(−1)n+i 2n(n +i)i! +n! +�⌈n/2⌉−1 +j=i +� +j − (i − 1) +� �n+i−1 +j=n+1 +� +(j − 1)/2 − (i − 1) +� +2n +n! (−1)n+i−1 �i−2 +j=0 +� +(i − 1) − j +� �⌈n/2⌉−1 +j=i +� +j − (i − 1) +� += +�n +i +� +i! �n+i−1 +j=n+1 +� +(j − 1)/2 − (i − 1) +� +�i−2 +j=0 +� +(i − 1) − j +� += +n +n − i +� +(n + i − 1)/2 − (i − 1) +� +� +(n − 1)/2 − (i − 1) +� +�n − 1 +i +� +i! +�n+i−2 +j=n +� +(j − 1)/2 − (i − 1) +� +�i−2 +j=0 +� +(i − 1) − j +� += +n +n − i +� +(n + i − 1)/2 − (i − 1) +� +� +(n − 1)/2 − (i − 1) +� +diλn−1 +i +(x) +dxi +|x=−1 +As +n +n−i +� +(n+i−1)/2−(i−1) +� +� +(n−1)/2−(i−1) +� +> 1, we conclude that for all i, diλn +i (x) +dxi +|x=−1 > diλn−1 +i +(x) +dxi +|x=−1. +As λn +i (−1) = λn−1 +i +(−1) = λi(−1) = −1/2 − (i − 1), we can do a Taylor expansion around x = −1: +λn +i (x) = λi(−1) + (x + 1)i +i! +diλn +i (x) +dxi +|x=−1 ++ o((x + 1)i) +λn−1 +i +(x) = λi(−1) + (x + 1)i +i! +diλn−1 +i +(x) +dxi +|x=−1 ++ o((x + 1)i) +It is then clear that in a neighborhood of −1 and above −1, λn +i (x) > λn−1 +i +(x). As λn−1 +i +(−1) − +λn +i+1(−1) = 1, we also get λn−1 +i +(x) > λn +i+1(x) in a neighborhood of −1. Now as for all i λn +i (x), λn−1 +i +(x), λn +i+1(x) +are continuous functions of x, if by contradiction such inequalities where to fail for some x ∈ I, +then there would exist x0 such that λn +i (x0) = λn−1 +i +(x0) or λn−1 +i +(x0) = λn +i+1(x0).But then this would + +SPECIAL POLYNOMIALS AND NEW REAL-ROOTEDNESS RESULTS +11 +mean that λn−1 +i +(x0) is a root of Gn(x0, z) and Gn−1(x0, z), which is impossible by Lemma 4.5. +Therefore we conclude that the inequality +λn +i (x) > λn−1 +i +(x) > λn +i+1(x) +holds for allx ∈ I and i ≤ ⌊(n − 1)/2⌋. Notice that according to n, the polynomial ˜Gn(x, z) can be +of the same degree than ˜Gn−1(x, z), or of degree one more. +□ +Theorem 4.7 (Interlacing roots, derivative). Consider an interval I = [−1, b[ such that ˜Gn(x, z) +has simple real roots in z on I, then the same will be true of ∂x ˜Gn(x, z) and the roots of the two +polynomials interlace. +Proof. We bring ourselves back to a variant of the previous theorem by using the equality +∂xGn(x, z) = 2zGn−1(x, z + 1) +As +∂xGn(x, z) = +� ⌈n/2⌉−1 +� +j=0 +(z + j) +� +∂x ˜Gn(x, z) +Gn−1(x, z + 1) = +� ⌈(n−1)/2⌉−1 +� +j=0 +(z + j + 1) +� +˜Gn−1(x, z + 1) +We get +∂x ˜Gn(x, z) = +�⌈(n−1)/2⌉−1 +j=0 +(z + j + 1) +�⌈n/2⌉−1 +j=0 +(z + j) +2z ˜Gn−1(x, z + 1) +And +∂x ˜Gn(x, z) = 2(z + n/2) ˜Gn−1(x, z + 1) +if n is even +∂x ˜ +Gn(x, z) = 2 ˜Gn−1(x, z + 1) +if n is odd +So as −n/2 < mini,x λi(x) for all i and x, it amounts to proving that +˜ +Gn−1(x, z + 1) and ˜ +Gn(x, z) +interlace. +We want to show that for all x ∈ I, with x > −1, all i ≤ ⌊(n − 1)/2⌋, λn +i (x) > +λn−1 +i +(x) − 1 > λn +i+1(x). +First we check this in a neighborhood of −1. +We can check that the +inequality λn +i (x) > λn−1 +i +(x) − 1 is going to be true in a neighborhood of −1 as λn +i (−1) = λn−1 +i +(−1). +So the nontrivial one is the other one, λn−1 +i +(x) − 1 > λn +i+1(x) for x > −1. We have equality at the +origin as λn +i (−1) = λn−1 +i +(−1) := λi(−1) and λi(−1) − 1 = λi+1(−1). Then we look at the Taylor +expansions around x = −1: +λn−1 +i +(x) − 1 = λi+1(−1) + (x + 1)i +i! +diλn−1 +i +(x) +dxi +|x=−1 ++ o((x + 1)i) +λn +i+1(x) = λi+1(−1) + (x + 1)i+1 +(i + 1)! +di+1λn +i+1(x) +dxi+1 +|x=−1 ++ o((x + 1)i+1) +It is clear then that locally λn−1 +i +(x) − 1 > λn +i+1(x) as (x + 1)i+1 << (x + 1)i. We extend the +inequality to the whole interval I by noticing again that if the inequalities where not valid anymore, +then there would have to be some equality λn +i (x) = λn−1 +i +(x) − 1 or λn−1 +i +(x) − 1 = λn +i+1(x), which +would mean ∂x ˜ +Gn(x, λn−1 +i +(x)) = 0 and as ˜ +Gn(x, λn−1 +i +(x)) = 0, we would again get a contradiction +by Lemma 4.5. +□ +Lemma 4.8 (Global extension through ODE). The local property is in fact true over the whole +interval: +˜ +Gn(x, z) is real rooted in z with simple (distinct) roots for for x ∈ [−1, 0[ , and they are +all increasing to +∞ when x goes to zero. + +12 +AURELIEN GRIBINSKI EPFL +Proof. Denote by Fn(x, z) := − ∂x ˜ +Gn +∂z ˜ +Gn +� +x, z +� +. Consider a rectangular domain D such that ∂z ˜ +Gn(x, z) +is nonzero on the domain. +Fn is continuous in x and z in the the domain D. +Indeed, it is a +rational fraction whose denominator is nonzero and it is therefore C∞ in both variables by theorem +of composition. As ˜ +Gn(−1, z) is realrooted in z with simple roots, ∂z ˜ +Gn(−1, λi(−1)) ̸= 0 and by +continuity we can find small rectangles Di := [−1, −1 + ǫ] × [λi(−1) − δ, λi(−1) + δ] such that +∂z ˜ +Gn(x, z) is nonzero on Di. A strong version of Picard’s theorem tells us that there is a maximal +interval Imax +i += [−1, ηi +max[ for which the roots λi(x) (i = 1, 2...⌊n/2⌋) are the unique solutions of +the initial value ODE +dz +dx = Fn(x, z), +z(−1) = −1/2 − (i − 1) +Note that on Imax +i +, ∂z ˜ +Gn(x, λi(x)) ̸= 0 (the denominator is nonzero, so that the differential equa- +tion is well defined). Let’s prove that Imax +i += [−1, 0[ (for all i) and that there is explosion at 0 +(roots going to infinity), the roots increasing constantly to +∞. The local Lemma 4.3 tell us that +on a neighborhood of −1, Fn(x, λi(x)) > 0, and as by Lemma 4.5, the numerator is of constant +sign and the denominator doesn’t vanish, then Fn(x, λi(x)) > 0 on Imax +i +. +According to Picard’s theorem, we either have λi(x) →x→ηimax +∞ (explosion), or ηi +max is such +that lim Fn(x, λi(x)) is not well defined (we leave the domain of definition). +Now, explosion can’t happen if ηi +max < 0. Indeed, we have that +� +i +λi(x) + +� +j +µj = −Pn−1(x) +(2x)n +n! +where Pn−1(x) is a polynomial of degree n − 1 as well as the coefficient of zn−1 in the expansion +of Gn(x, z). So the sum of roots is bounded above by a constant, so there can be no explosion +(necessarily to+∞ by monotonicity). +We can leave the domain of definition only if limx→ηimax ∂z ˜ +Gn +� +x, λi(x) +� += 0. If this is the case and +if by contradiction ηi +max < 0, we have seen that ∂z ˜ +Gn(ηi +max, z) would be of degree exactly ⌊n/2⌋ − 1 +in z. Therefore it means that limx→ηimax λi(x) = µ where µ is a root of ∂z ˜ +Gn(ηi +max, z). But then +it means that we can extend by continuity λi(x) at x = ηi +max with λi(ηmax) = µ. We check by +continuity that ˜ +Gn +� +ηi +max, λi(ηmax) +� += ∂z ˜ +Gn +� +ηi +max, λi(ηmax) +� += 0 so that in fact λi(ηmax) is a real +double root in z of ˜ +Gn +� +ηi +max, z +� +. Using Lemma 4.7, as there is a root of ∂x ˜ +Gn(x, z) between any two +roots of ˜ +Gn(x, z) in z by interlacing, it follows that necessarily ∂x ˜ +Gn +� +ηi +max, λi(ηmax) +� += 0. But this +is impossible according to Lemma 4.5. Therefore, we have necessarily ηi +max = 0 for all i = 1...⌊n/2⌋. +Furthermore, assume by contradiction that there is no explosion for some index i at 0. As λi(x) is +monotonous for x ∈] − 1, 0[, then we have necessarily that limx→0 λi(x) = µ exists and is finite. By +continuity we have ˜ +Gn(0, µ) = 0. Let’s distinguish according to the parity of n. If n is even, we +have ˜ +Gn(0, z) = +Gn(0,z) +�n/2−1 +j=0 +(z+j) = +1 +(n/2)! for all z, which shows the contradiction right away. If n is odd, +then ∂z ˜ +Gn +� +x, λi(x) +� += xQn +� +x, λi(x) +� +where Qn(0, z) = 2(−1)⌊n/2⌋ +(n−1)/2! . For x ∈ [−1, 0], xFn(x, λi(x)) is +therefore bounded above as a continuous function on a compact. It is always nonpostive, and the +maximum can’t be zero because it would mean that for an x ∈ [−1, 0], ∂xLn +� +x, λi(x) +� += 0, which +is impossible according to Lemma 4.5. Therefore it is always smaller than −K < 0. +dλi(x) +dx += 1 +xxFn(x, λi(x)) > −K +x +λi(x) − λi(−1) > −K log(|x|) + +SPECIAL POLYNOMIALS AND NEW REAL-ROOTEDNESS RESULTS +13 +It would follow that λi(x) →x→0 +∞ which is contrary to the assumptions. +We conclude that there is explosion for all i = 1...⌊n/2⌋. +□ +□ +5. Applications to realrootedness in x +We start by recalling a well-known monotonicity result. In all the following we will consider z > 0. +Lemma 5.1 (Monotonicity of the roots with respect to the parameter, from [1]). If xi(z) are +the roots of L(z) +n (x) (Laguerre), then +d +dzxi(z) ≥ 0, i ≤ n. Also, the positive roots yi of G(z) +n (x) +(Gegenbauer) are such that +d +dzyi(z) ≥ 0 and the negative symmetric such that +d +dzyi(z) ≤ 0. +Lemma 5.2. For a fixed z, we have that the roots of ∂zLn(x, z) in x (of degree n − 1) are real and +interlace those of Ln(x, z). +Proof. +Ln(x, z) = +n +� +k=0 +(−1)k �n +j=k+1(z + j) +(n − k)! +xk +(3) +From this expression it follows that ∂zLn(x, z) is of degree n − 1 in x. +d +dz xi = −∂zLn +∂xLn +(xi(z), z) +∂zLn(xi(z), z) = −∂xLn(xi(z), z) d +dz xi +(4) +∂xLn(xi(z), z) changes sign when we increment i because Ln(xi(z), z) = 0 so the derivative changes +sign when we go from one root to the next. As +d +dzxi ≥ 0, We get that ∂zLn(x, z) changes sign +n − 1 times and therefore we have n − 1 real zeros between zeros of Ln. Therefore all the zeros of +∂zLn(x, z) have been found and are interlacing with the zeros of Ln(x, z). +□ +Theorem 5.3. ∂zLn(x, z) and more generally ∂k +z Ln(x, z) for all k ≤ n are real-rooted in x, and they +form an interlacing family of decreasing degree, in the sense that ∂k+1 +z +Ln(x, z) interlaces ∂k +z Ln(x, z) +and the roots are monotonously increasing in z. +Proof. We show inductively the following property: ∂k+1 +z +Ln(x, z) is realrooted, the roots of ∂k+1 +z +Ln(x, z) +interlace the roots of ∂k +z Ln(x, z) and are increasing in z. We start with the initial condition. Using +5.2, we get the real-rootedness and interlacing property for ∂zLn(x, z). Now we need to prove that +the roots ˜xi(z) (i = 1...n − 1) of ∂zLn(x, z) also share the monotonicity property. +∂zLn( ˜xi(z), z) = 0 +Which lead to +∂zLn +Ln +( ˜xi(z), z) = 0 +d(∂zLn +Ln ( ˜xi(z), z) +� +dz += ∂x +�∂zLn +Ln +( ˜xi(z), z) +�d ˜xi +dz + ∂z +�∂zLn +Ln +( ˜xi(z), z) +� += 0 +(5) +We want to show that +d ˜xi +dz ≥ 0 +On the one hand, +∂zLn +Ln +( ˜xi(z), z) = +n +� +j=1 +∂zLn +∂xLn +(xj, z) +1 +˜xi − xj +So that +∂x +�∂zLn +Ln +( ˜xi(z), z) +� += − +n +� +j=1 +∂zLn +∂xLn +(xj, z) +1 +( ˜xi − xj)2 = +n +� +j=1 +dxj +dz +1 +( ˜xi − xj)2 ≥ 0 +(6) + +14 +AURELIEN GRIBINSKI EPFL +On the other hand, using the real-rootedness in z, as ˜xi ∈ [0, +∞[ by the interlacing property, then +∂z +�∂zLn +Ln +( ˜xi(z), z) +� +≤ 0 +using Laguerre inequality for realrooted polynomials, stating that ∂zzLnLn − (∂zLn)2 ≤ 0. We +conclude by gathering the two inequalities. +The induction is proven using exactly the same method, given that +∂z +�∂k+1 +z +Ln +∂kz Ln +� +≤ 0 +Because ∂k +z Ln(x, z) is realrooted in z as the derivative of a realrooted polynomial, and x in the +appropriate interval. +□ +Lemma 5.4. 0 is a root of ∂k +z Gn(x, z) for all k ≤ n, when n is odd. +Proof. It comes directly from the formula +Gn(x, z) = +⌊n/2⌋ +� +k=0 +(−1)k Γ(n − k + z) +Γ(z)k!(n − 2k)!(2x)n−2k +□ +Lemma 5.5. For a fixed z > 0, we have that the roots of ∂zGn(x, z) in x (of degree n) are real +and the positive ones interlace those of Gn(x, z) by below (that is the largest root in module belongs +to Gn(x, z)). +Proof. +Gn(x, z) = +⌊n/2⌋ +� +k=0 +(−1)k Γ(n − k + z) +Γ(z)k!(n − 2k)!(2x)n−2k +(7) +From this expression it follows that ∂zGn(x, z) is of degree n in x, as the coefficient of xn is 2n Γ(n+z) +Γ(z)n! . +Let’s denote the roots of Gn(x, z) by yi, then by differentiating the equality Gn(x, z) = 0 with +respect to z we get +d +dz yi = −∂zGn +∂xGn +(yi(z), z) +∂zGn(yi(z), z) = −∂xGn(yi(z), z) d +dz yi +(8) +∂xGn(yi(z), z) changes sign when we increment i because Gn(yi(z), z) = 0 so the derivative −∂xGn(yi(z), z) +changes sign when we go from one root to the next (the roots are simple). Let’s distinguish between +the even and odd cases. In the even case, +d +dzyi ≥ 0 when i = 1..n/2, and it gives us by the sign rule +i = 1..n/2 − 1 positive roots. Now there is still one root missing, and we can check that the sign +of ∂zGn(yn/2(z), z) is opposite to the sign of ∂zGn(0, z), and as ∂zGn(−yn/2(z), z) has the same +sign by symmetry there has to be a root between both, and actually two, one positive and one +negative by symmetry. We get n roots overall. In the odd case, 0 is a root, and we have again two +roots missing. But if yl denotes the positive smallest root, then ∂zGn(yl(z), z) has the same sign as +∂zGn(−yl(z), z) and so as 0 is a root we need to have at least two other roots by some easy change +of sign argument. Therefore all the zeros of ∂zGn(x, z) have been found and the positive ones are +interlacing with the zeros of Gn(x, z). +□ +Theorem 5.6. ∂zGn(x, z) and more generally ∂k +z Gn(x, z) for all k ≤ n are real-rooted in x for +z > 0, and the positive roots are monotonously increasing in z, and their symmetric negative +counterpart monotonously decreasing in z. + +SPECIAL POLYNOMIALS AND NEW REAL-ROOTEDNESS RESULTS +15 +Proof. We have +Gn(−x, z) = (−1)nGn(x, z) +∂zGn(−x, z) = (−1)n∂zGn(x, z) +(9) +So that if we have a root ˜y of ∂zGn(x, z), then its symmetric −˜y will also be a root of ∂zGn(x, z). In +other terms, ∂zGn(x, z) and more generally , ∂k +z Gn(x, z) have symmetric roots. We show inductively +the following property: ∂k+1 +z +Gn(x, z) is realrooted, the positive roots of ∂k+1 +z +Gn(x, z) interlace the +positive roots of ∂k +z Gn(x, z) and the positive roots are increasing with z. We start with the initial +condition. Using 5.2, we get the real-rootedness and interlacing property for ∂zGn(x, z). Now we +need to prove that the positive roots ˜yi(z) of ∂zGn(x, z) also share the monotonicity property. +∂zGn( ˜yi(z), z) = 0 +which leads to +∂zGn +Ln +( ˜yi(z), z) = 0 +d(∂zGn +Gn ( ˜yi(z), z) +� +dz += ∂x +�∂zGn +Gn +( ˜yi(z), z) +�d ˜yi +dz + ∂z +�∂zGn +Gn +( ˜yi(z), z) +� += 0 +(10) +We want to show that for the positive roots, +d ˜yi +dz ≥ 0 +On the one hand, +∂zGn +Gn +( ˜yi(z), z) = +n +� +j=1 +∂zGn +∂xGn +(yj, z) +1 +˜yi − yj +so that +∂x +�∂zGn +Ln +( ˜yi(z), z) +� += − +n +� +j=1 +∂zGn +∂xGn +(xj, z) +1 +( ˜yi − yj)2 = +n +� +j=1 +dyj +dz +1 +( ˜yi − yj)2 +(11) += +⌊n/2⌋ +� +j=1 +dyj +dz +� +1 +( ˜yi − yj)2 − +1 +( ˜yi + yj)2 +� += 4˜yi +⌊n/2⌋ +� +j=1 +dyj +dz yj +( ˜yi − yj)2( ˜yi + yj)2 +(12) +As we have ˜yi > 0 and dyi +dz ≥ 0 as well as yj > 0 we get +∂x +�∂zGn +Ln +( ˜yi(z), z) +� +≥ 0 +On the other hand, using the real-rootedness in z, as ˜yi ∈ [−1, 1] by the interlacing property, then +∂z +�∂zGn +Gn +( ˜yi(z), z) +� +≤ 0 +using Laguerre inequality for realrooted polynomials, stating that ∂zzGnGn − (∂zGn)2 ≤ 0. We +conclude by gathering the two inequalities. +The induction is proven using exactly the same method, given that +∂z +�∂k+1 +z +Gn +∂kz Gn +� +≤ 0 +Because ∂k +z Gn(x, z) is realrooted in z as the derivative of a realrooted polynomial, and x in the +appropriate interval ([−1, 1]). +□ + +16 +AURELIEN GRIBINSKI EPFL +References +[1] G. Szego. Orthogonal polynomials. AMS, Providence, RI MR 51 (1975): 8724. + diff --git a/atAyT4oBgHgl3EQfv_nZ/content/tmp_files/load_file.txt b/atAyT4oBgHgl3EQfv_nZ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..94a5fc08712f251680c73326415772202c2229c7 --- /dev/null +++ b/atAyT4oBgHgl3EQfv_nZ/content/tmp_files/load_file.txt @@ -0,0 +1,494 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf,len=493 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='00642v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='CA] 30 Dec 2022 SPECIAL POLYNOMIALS AND NEW REAL-ROOTEDNESS RESULTS AURELIEN GRIBINSKI EPFL Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' In this paper, we show that for some orthogonal polynomials P (z) n (x) showing up in physics, namely Laguerre and Gegenbauer, P (z) n (x) are realrooted in z for x in the support of orthogonality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' As an application we show realrootedness in x and interlacing properties of ∂k z P (z) n (x) for k ≤ n for z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' General strategy 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Laguerre polynomials 2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Gegenbauer polynomials 7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Applications to realrootedness in x 13 References 16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Introduction Orthogonal polynomials like generalized Laguerre and Gegenbauer polynomials have long been studied, and show up in all fields of maths and physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' However little has been said about the properties of such polynomials when we vary the underlying parameter (see [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' We study families of generalized orthogonal polynomials P (z) n (x) depending on a parameter z from a new angle, that is, as bivariate polynomials Pn(x, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' We fix the usual variable x and consider them instead as polynomials in z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' We show that they are real-rooted in z for x in the support of the underlying measure of orthogonality, and monotonous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Furthermore we show that when we differentiate these orthogonal polynomials with respect to z > 0 and consider them as polynomials in x, then they are realrooted in x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Such polynomials (derivatives with respect to the parameter) seem to have many nice properties similar to their corresponding orthogonal polynomials from which they are derived and yet have never been studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' General strategy Consider P (z) n (x), a family of polynomials depending on a parameter z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' We want to show that they are real-rooted in z for a fixed x in a given interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' The strategy is as follows We check that for x at one of the extreme point of the interval it is real-rooted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' We show that locally around this extreme point the roots in z are all monotonous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' We show that there can’t be a shared zero in z for ∂xPn(x, z) and Pn(x, z) or equivalently for Pn−1(x, z) and Pn(x, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' The roots in z are therefore monotonous when they are well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Date: January 3, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' 1 2 AURELIEN GRIBINSKI EPFL We show that the roots in z of Pn(x, z) and simultaneously Pn−1(x, z) and ∂xPn(x, z) inter- lace as long as they are well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' We extend the local properties by exhibiting an ODE to which all roots in z are solutions and show that there has to be explosion at the other extreme point of the interval, all roots evolving monotonously along the way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' First we derive a way to locally prove the existence of rel-rootedness if it is known at some extreme point of the interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='1 (Local existence of the roots and smoothness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Take a ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Assume P(x, z) is a bivariate polynomial such that P(a, z) is realrooted in z of degree j and has only simple roots in z (let’s call them zi(a) for i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='.j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Assume that P(x, z) has degree less than j for all x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='Then for x in a neighborhood of a, P(x, z) is also realrooted in z of degree j with simple roots zi(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Furthermore, the roots zi(x) are C∞ functions of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Consider the equation P(x, z) = 0, around the points � a, zi(a) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' We have ∂zP(x, z)|x=a,z=zi(a) ̸= 0 as the roots are simple (can’t be a root of the derivative in z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Then using the implicit function theorem, we can find in the neighborhood of each point a C∞ function zi(x) which will be the only solution to the equation P(x, z) = 0 on this neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Therefore we have found j roots, and it is the maximal number of roots for a fixed x, as the polynomial is of degree less than j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Laguerre polynomials Let L(z) n (x) be the Laguerre polynomials with complex parameter z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' It is a polynomial in R[x, z].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' For fixed x0 ∈ [0, +∞[, L(z) n (x0) is a real-rooted polynomial in z of degree n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Fur- thermore, its roots in z are strictly increasing to +∞ when x0 moves along [0, +∞[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Let’s recall the hypergeometric confluent definition L(z) n (x) = �n + z n � M(−n, z + 1, x) = n � l=1 z + n + 1 − l l n � k=0 (−1)kn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' (n − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' �k−1 l=0 (z + 1 + k) xk = n � k=0 (−1)k �n l=1(z + l) (n − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' �k l=1(z + l) xk = n � k=0 (−1)k �n j=k+1(z + j) (n − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' xk = 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='zn + ��n j=1 j n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' − x (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' � zn−1 + Rn−2(z, x) where Rn−2(z, x) is of degree lower than n−2 in z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' We will henceforth write Ln(x, z) as it is clearly a bivariate polynomial of degree n in z with front coefficient 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='. Let’s furthermore decompose Ln(x, z) using a priori complex roots λi(x): Ln(x, z) = 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' n � i=1 � z − λi(x) � where we order the roots by decreasing module: |λ1(x)| ≥ |λ2(x)| ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' ≥ |λn(x)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='2 (Local realrootedness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Ln(x, z) is real rooted of degree n in z with simple roots in a neighborhood of x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' SPECIAL POLYNOMIALS AND NEW REAL-ROOTEDNESS RESULTS 3 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' We have Ln(0, z) = �n l=1(z + l) n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' so that we can apply Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='1 with a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='3 (Local increasing property).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' The roots of Ln(x, z) in z are all strictly increasing when x is, in a neighborhood of x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' To prove this, we need some information on the derivatives with respect to x of the roots in the neighborhood of 0, which we know are going to be real by the previous lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' dlλi(x) dxl |x=0 = 0 for 1 ≤ l < i, and diλi(x) dxi |x=0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' We get, for all 1 ≤ l ≤ n, ∂l xLn(x, z)|x=0 = l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' (−1)l �n j=l+1(z + j) (n − l)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Notice that λi(0) = −i for i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='n, so we see that for l ≤ i − 1, ∂l xLn � 0, λi(0) � = l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' (−1)l �n j=l+1(λi(0) + j) (n − l)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' = 0 And ∂i xLn � 0, λi(0) � = i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' (−1)i �n j=i+1(λi(0) + j) (n − i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' = i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' (−1)i On the other hand, ∂zLn � 0, λi(0) � = 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' n � l=1,l̸=i (−i + l) = i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' (−1)i−1(n − i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Now we have Ln � x, λi(x) � = 0 for all i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='n, by definition, so differentiating with respect to x, we get dλi(x) dx = −∂xLn ∂zLn � x, λi(x) � Note that the denominator is nonzero as the roots in z are simple at 0 ( so they won’t be roots of the derivative in z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Using Leibniz’s formula and induction on l, we get for i > l ≥ 1, dlλi(x) dxl |x=0 = 0 And diλi(x) dxi |x=0 = −∂i xLn ∂zLn � − 1, λi(0) � = n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' (n − i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' > 0 □ We conclude by a Taylor expansion around 0 as λi(x) = −i + xi i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' (n − i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' + o(xi) □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='5 (Distinct roots, degree and derivative wise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Assume λi(x) is real for x ∈]0, bi[, bi > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Then ∂xLn(x, λi(x)) can’t be zero for x ∈]0, bi[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Therefore it has a constant sign on this interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Equivalently, Ln−1(x, λi(x)) can’t be zero either: that is we can’t have a nontrivial shared real root for Ln−1(x, z) and Ln(x, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' 4 AURELIEN GRIBINSKI EPFL Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Traditional results on simplicity of the roots can’t be used because they are true only for z ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' By definition, Ln � x, λi(x) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Then the usual differential euqation still holds (1) x∂xLn(x, z) = nLn(x, z) − (n + z)Ln−1(x, z) Let’s assume by contradiction that ∂xLn � x0, λi(x0) � = 0 for some i and x0 ∈]0, bi[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' As ∂xLn(x, λi(x)) is nonzero in a neighborhood of x = 0, x > 0(local monotonicity above), then we can assume x0 is the smallest x > 0 such that ∂xLn � x, λi(x) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Therefore as dλi(x) dx = −∂xLn ∂zLn � x, λi(x) � we have that on ]0, x0], λi(x) is strictly increasing in x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' As λi(0) ≥ −n = λn(0) for all i, then � n − λi(x0) � > 0, and we get that the statement is equivalent to Ln−1 � x0, λi(x0) � = 0 using Equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' But then, as the following recurrence relations are still valid (n + 1)Ln+1(x, z) = (−x + 2(n + 1) + z)Ln(x, z) − (n + z)Ln−1(x, z) we also get by induction Ln+k(x0, λi(x0) + 1) = 0 for all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Using then the equality ∂xLn+k(x, z) = −Ln+k−1(x, z + 1) we get that Ln+k−1(x0, λi(x0) + 1) = 0 for all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Using induction, applying successively the previous recurrence equations, we then get that Ln+k−1(x0, λi(x0) + j) = 0 for all j ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' For j ≤ n, the polynomials Ln+k−1(x, λi(x) + j) are standard Laguerre polynomials (positive parameter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' It would mean that successive Laguerre polynomials of parameter λi(x0) + j have the root x0 in common, so that their derivatives share this root too, which is absurd as the roots of Laguerre polynomials are simple by classical orthogonality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' We conclude that Ln−1(x, λi(x)) as well as ∂xLn � x0, λi(x0) � can’t be zero, and therefore ∂xLn � x0, λi(x0) � has a constant sign for x ∈]0, bi[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='6 (Extended monotonicity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Assume λi(x) is real for x ∈]0, bi[, bi > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Then it follows from the previous proof that for x ∈]0, bi[ dλi(x) dx > 0 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='7 (Interlacing roots, degreewise, simple roots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Consider an interval I = [0, b[ such that Ln(x, z) has real roots in z on I, then the same will be true of Ln−1(x, z) and their roots interlace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Furthermore, the interlacing is strict for x > 0 and both polynomials have simple roots on I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Let’s write Ln(x, z) = 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' n � i=1 � z − λn i (x) � Ln−1(x, z) = 1 (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' n−1 � i=1 � z − λn−1 i (x) � and show that all x ∈ I, all i ≤ n − 1: λn i (x) ≥ λn−1 i (x) ≥ λn i+1(x) with strict inequalities for x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' We first check the property locally, that is a neighborhood of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' λn i (0) = −i λn−1 i (0) = −i λn i+1(0) = −(i + 1) SPECIAL POLYNOMIALS AND NEW REAL-ROOTEDNESS RESULTS 5 We have diλn i (x) dxi |x=0 = n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' (n − i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' = n n − i (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' (n − 1 − i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' = n n − i diλn−1 i (x) dxi |x=0 As n n−i > 1, we conclude that for all i ≤ n − 1, diλn i (x) dxi |x=0 > diλn−1 i (x) dxi |x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' We can then do a Taylor expansion around x = 0: λn i (x) = −i + (x + 1)i i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' diλn i (x) dxi |x=−1 + o((x + 1)i) λn−1 i (x) = −i + (x + 1)i i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' diλn−1 i (x) dxi |x=−1 + o((x + 1)i) It is then clear that in a neighborhood of 0, λn i (x) > λn−1 i (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' As λn−1 i (−1) − λn i+1(−1) = 1, we also get λn−1 i (x) > λn i+1(x) in a neighborhood of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Now as for all i λn i (x), λn−1 i (x), λn i+1(x) are continuous functions of x, if by contradiction such inequalities where to fail for some x ∈ I, then there would exist x0 such that λn i (x0) = λn−1 i (x0) or λn−1 i (x0) = λn i+1(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='But then this would mean that λn−1 i (x0) is a root of Gn(x0, z) and Gn−1(x0, z), which is impossible by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Therefore we conclude that the inequality λn i (x) > λn−1 i (x) > λn i+1(x) holds for allx ∈ I, x > 0 and i ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' □ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='8 (Interlacing roots, derivative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Consider an interval I = [0, b[ such that Ln(x, z) has real roots in z on I, then the same will be true of ∂xLn(x, z) and the roots of the two polynomials interlace and are simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' We bring ourselves back to a variant of the previous theorem by using the equality ∂xLn(x, z) = −Ln−1(x, z + 1) The roots being simple results from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' So it amounts to proving that Ln−1(x, z + 1) and Ln(x, z) interlace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' We want to show more precisely that for all x ∈ I, with x > 0, all i ≤ n − 1 λn i (x) ≥ λn−1 i (x) − 1 ≥ λn i+1(x) With strict inequalities for x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' First we check such inequalities in a neighborhood of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' We can check that the inequality λn i (x) > λn−1 i (x) − 1 is going to be true in a neighborhood of 0 as λn i (0) = λn−1 i (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' So the nontrivial one is the other one, λn−1 i (x) − 1 > λn i+1(x) for x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' We have equality at the origin as λi(0) − 1 = λi+1(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Then we look at the Taylor expansions around x = 0: λn−1 i (x) − 1 = λi+1(0) + (x + 1)i i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' diλn−1 i (x) dxi |x=0 + o((x + 1)i) λn i+1(x) = λi+1(0) + (x + 1)i+1 (i + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' di+1λn i+1(x) dxi+1 |x=0 + o((x + 1)i+1) It is clear then that locally λn−1 i (x) − 1 > λn i+1(x) as (x + 1)i+1 << (x + 1)i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' We extend the inequality to the whole interval I by noticing again that if the inequalities where not valid anymore, then there would have to be some equality λn i (x) = λn−1 i (x) − 1 or λn−1 i (x) − 1 = λn i+1(x), which would mean ∂xLn(x, λn−1 i (x)) = 0 and as Ln(x, λn−1 i (x)) = 0, we would again get a contradiction by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' □ 6 AURELIEN GRIBINSKI EPFL Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='9 (Global extension through ODE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' The local property is in fact true over the whole interval: Ln(x, z) is real rooted in z with simple (distinct) roots for x ∈ [0, +∞[ , and the roots are all increasing to +∞ when x goes to +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Denote by Fn(x, z) := − ∂xLn ∂zLn � x, z � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Consider a rectangular domain D such that ∂zLn(x, z) is nonzero on the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Fn is continuous in x and z in the the domain D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Indeed, it is a rational fraction whose denominator is nonzero and it is therefore C∞ in both variables by theorem of composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' As Ln(0, z) is realrooted in z with simple roots, ∂zLn � 0, λi(0) � ̸= 0 and by continuity we can find small rectangles Di := [0, ǫ] × [λi(0) − δ, λi(0) + δ] such that Ln(x, z) is nonzero on Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' A strong version of Picard’s theorem tells us that there is a maximal interval Imax i = [0, ηi max[ (where ηi max ∈ ¯ R+) for which the roots λi(x) (i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='n) are the unique solutions of the initial value ODE dz dx = Fn(x, z), z(−1) = −i Note that on Imax i , ∂zLn � x, λi(x) � ̸= 0 (the denominator is nonzero, so that the differential equa- tion is well defined).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Let’s prove that Imax i = [0, +∞[ (for all i) and that there is explosion at +∞ (roots going to infinity), the roots increasing constantly to +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' By Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='6, Fn(x, λi(x)) > 0 on Imax i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' According to Picard’s theorem, we either have λi(x) →x→ηimax +∞ (explosion), or ηi max is such that limx→ηimax Fn(x, λi(x)) is not well defined (we leave the domain of definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Now, explosion can’t happen if ηi max < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Indeed, we have using the hypergeometric expansion above n � i=1 λi(x) = −n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' �n(n + 1) 2 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' − x (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' � = n � − n + 1 2 + x � so the sum of roots is bounded above for x < ηi max and there can be no explosion (necessarily to+∞ by monotonicity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' We can leave the domain of definition only if limx→ηimax ∂zLn � x, λi(x) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' If this is the case and if by contradiction ηi max < +∞, ∂zLn(ηi max, z) would be of degree n in z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Therefore it means that limx→ηimax λi(x) = µ where µ is a root of ∂zLn(ηi max, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' But then it means that we can extend by continuity λi(x) at x = ηi max with λi(ηi max) = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' We check by continuity that Ln � ηi max, λi(ηi max) � = ∂zLn � ηi max, λi(ηi max) � = 0 so that in fact λi(ηi max) is a real double root in z of Ln � ηi max, z � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='8, as there is a root of ∂xLn(x, z) between any two roots of Ln(x, z) in z by interlacing, it follows that necessarily ∂xLn � ηi max, λi(ηmax) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' But this is impossible according to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Therefore, we have necessarily ηi max = +∞ for all i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Furthermore, assume by contradiction that there is no explosion for some index i at +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' As λi(x) is monotonous for x ∈ [0, +∞[, then we have necessarily that limx→+∞ λi(x) = µ exists and is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' By continuity we have Ln(x, µ) ∼x→∞ (−1)nxn, and ∂xLn(x, µ) ∼x→∞ n(−1)nxn−1, as well as ∂zLn(x, µ) ∼x→∞ (−1)n−1xn−1 using L(z) n (x) = n � k=0 (−1)k �n j=k+1(z + j) (n − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' xk so that dλi(x) dx →x→+∞ n and clearly we would have λi(x) → +∞, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' SPECIAL POLYNOMIALS AND NEW REAL-ROOTEDNESS RESULTS 7 □ □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Gegenbauer polynomials Let G(z) n (x) be the Gegenbauer polynomial with complex parameter z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' It is a polynomial in R[x, z].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' For fixed x0 ∈ [−1, 1], G(z) n (x0) is a real-rooted polynomial in z of degree at most n (exactly n except for x0 = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Furthermore, its roots in z then they are increasing for x ∈ [−1, 0[, and decreasing for x ∈]0, 1], with an explosion to infinity at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' As the Gegenbauer polynomials are even or odd in x, that is G(z) n (−x) = (−1)nG(z) n (x), it is enough to prove our statement for x ∈ [−1, 0[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' We prove it in an incremental way moving x from −1 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Let’s recall the hypergeometric expression: G(z) n (x) = n−1 � l=0 (2z + l) n � k=0 (−1)k 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' (n − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' �k−1 i=0 (2z + n + i) �k−1 i=0 (z + 1/2 + i)2k (1 − x)k = n � k=0 (−1)k 2n�n k � n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' �n+k−1 i=0 (z + i/2) �k−1 i=0 (z + (2i + 1)/2) (1 − x)k = n � k=0 2n�n k � n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' n+k−1 � i=2k (z + i/2) k−1 � i=0 (z + i)(x − 1)k We will henceforth write Gn(x, z) as it is clear from the previous expression that it is indeed a bivariate polynomial and not a rational fraction in z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' We start dealing with the extreme boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' We have (2) Gn(x, z) = (−1)nGn(−x, z) = n � k=0 2n�n k � n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' (−1)n+k n+k−1 � i=2k (z + i/2) k−1 � i=0 (z + i)(x + 1)k So that Gn(−1, z) = (−1)n 2n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' �n−1 i=0 (z +i/2), which is clearly realrooted in z with simple roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' We can also check this property for x = 0: Gn(0, z) = Γ(n/2 + z) Γ(z)Γ(n/2 + 1) = 1 (n/2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' j=n/2−1 � j=0 (z + j) when n is even Gn(0, z) = 0 when n is odd Also, each bivariate polynomial in the sum is of degree n in z so the sum is of degree at most n in z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' It is in fact of degree exactly n for x ̸= 0 by inspection of the coefficient of zn in the sum, which is equal to (−1)n 2n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' n � k=0 �n k � (−1)k(1 + x)k = (−1)n 2n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' � 1 − (1 + x) �n = (−1)n 2n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' (−x)n = (2x)n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Notice that we can write, if n is even, Gn(x, z) = � n/2−1 � j=0 (z + j) � ˜ Gn(x, z) and if n is odd, Gn(x, z) = � (n−1)/2 � j=0 (z + j) � ˜ Gn(x, z) 8 AURELIEN GRIBINSKI EPFL So that in all cases for x ̸= 0, we can write Gn(x, z) = � ⌈n/2⌉−1 � j=0 (z + j) � ˜ Gn(x, z) = � ⌈n/2⌉−1 � j=0 (z − µj) �(2x)n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' ⌊n/2⌋ � i=1 � z − λi(x) � where µj = −j and the λi(x) are a priori complex roots defined only forx ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' But it simplifies greatly for x = −1: ˜ Gn(x, z)|x=−1 = 2n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' (−1)n ⌊n/2⌋ � i=1 (z + 1/2 + i − 1) so that λi(−1) = −1/2−(i−1) for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='⌊n/2⌋ are all real and distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' That is, approximately half of the roots in z, depending on the oddness, are constant when x is moving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' We can therefore investigate instead the evolution of the roots of Ln(x, z) to avoid considering roots that remain constant (and real).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' First, let’s explain why roots will be indeed smooth and real for x close to −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='2 (Local realrootedness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' ˜ Gn(x, z) is real rooted of degree ⌊n/2⌋ in z with simple roots in a neighborhood of x = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' We have ˜ Gn(x, z)|x=−1 = 2n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' (−1)n ⌊n/2⌋ � i=1 (z + 1/2 + i − 1) which has ⌈n/2⌉ simple roots in z, so we just have to apply Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='1 with a = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='3 (Local increasing property).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' The roots of ˜ Gn(x, z) in z are all strictly increasing when x is, in a neighborhood of x = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' To prove this, we need some information on the derivatives with respect to x of the roots in the neighborhood of −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' If we denote by λi(x) the roots of ˜ Gn(x, z) in decreasing order, then dlλi(x) dxl |x=−1 = 0 for 1 ≤ l < i, and diλi(x) dxi |x=−1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Using Equation 2 we get that ∂l xGn(x, z)|x=−1 = l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='2n�n l � n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' (−1)n+l n+l−1 � j=2l (z + j/2) l−1 � j=0 (z + j) So that ∂l x ˜ Gn(x, z)|x=−1 = 2n�n l � l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' (−1)n+l ⌈n/2⌉−1 � j=l (z + 1/2 + j) n+l−1 � j=n+1 (z + j/2) So we see that ∂l x ˜ Gn(x, z) � − 1, λi(−1) � = 0 for i ≥ l + 1 and (−1)n+i∂i x ˜ Gn(x, z) � − 1, λi(−1) � = 2n�n i � i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' ⌈n/2⌉−1 � j=i � j − (i − 1) � n+i−1 � j=n+1 � (j − 1)/2 − (i − 1) � > 0 as i ≤ ⌊n/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' SPECIAL POLYNOMIALS AND NEW REAL-ROOTEDNESS RESULTS 9 Now we have ˜ Gn � x, λi(x) � = 0 for all i, by definition, so differentiating with respect to x, we get: dλi(x) dx = −∂x ˜ Gn ∂z ˜ Gn � x, λi(x) � Note that the denominator is nonzero as the roots in z are simple at −1 ( so they won’t be roots of the derivative in z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Using Leibniz’s formula and induction on l, we get for i > l ≥ 1, dlλi(x) dxl |x=−1 = 0 And diλi(x) dxi |x=−1 = −∂i x ˜ Gn ∂z ˜ Gn � − 1, λi(−1) � Now, we have ˜ Gn(x, z)|x=−1 = (−1)n 2n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' �⌈n/2⌉−1 j=0 (z + 1/2 + j) so that ∂z ˜ Gn(x, z) � −1, λi(−1) � = (−1)n 2n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' ⌈n/2⌉−1 � j=0,j̸=i−1 � j−(i−1) � = (−1)n 2n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' (−1)i−1 i−2 � j=0 � (i−1)−j � ⌈n/2⌉−1 � j=i � j−(i−1) � and it follows that (−1)n+i−1∂z ˜ Gn(x, z) � − 1, λi(−1) � > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Therefore diλi(x) dxi |x=−1 > 0 as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' □ Then Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='3 follows easily by Taylor expansion of the roots in x around 0 , as by some asymptotic expansion at x = −1, λi(x) = λi(−1) + (x + 1)i i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' diλi(x) dxi |x=−1 + o((x + 1)i) We have proved the ”initial condition” : now we need to look at the evolution of roots from a differential equation point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' First, let’s prove some intermediate results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='5 (Simple roots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Assume λi(x) is real for x ∈] − 1, bi[, bi ≤ 0 ( such a bi > −1 exists according to the previous local existence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Then ∂x ˜ Gn(x, λi(x)) and a fortiori ∂xGn(x, λi(x)) ( for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='⌊n/2⌋) can’t be zero for x ∈] − 1, bi[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Therefore it has a constant sign on this interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Equivalently, Gn−1(x, λi(x)) can’t be zero either: that is we can’t have a nontrivial shared root for Gn−1(x, z) and Gn(x, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' We have ∂xGn(x, λi(x)) = �⌈n/2⌉−1 j=0 � λi(x) + j � ∂x ˜ Gn � x, λi(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' We can’t use directly the results on the monotonicity of the roots of Gegenbauer polynomials when the paremeter is mov- ing, or the simplicity of the roots in x, because the parameter here is negative and orthogonal- ity results don’t apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Let’s assume by contradiction that ∂x ˜ Gn(x0, λi(x0)) = 0, and therefore ∂xGn(x0, λi(x0)) = 0 for some i and x0 ∈] − 1, bi[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' As ∂xGn(x, λi(x)) is nonzero in a neighborhood of x = −1, x > −1(local monotonicity), then we can assume x0 is the smallest x > −1 such that ∂xGn(x, λi(x)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Therefore on ] − 1, x0], λi(x) is strictly increasing in x because dλi(x) dx = −∂x ˜ Gn ∂z ˜ Gn � x, λi(x) � As λi(−1) ≥ −(n − 1)/2 for all i, then (n + 2λi(x0) − 1) > 0, and using the differential equation (1 − x2 0)∂xGn(x0, λi(x0)) = −nxGn(x0, λi(x0)) + (n + 2λi(x0) − 1)Gn−1(x, λi(x0)) 10 AURELIEN GRIBINSKI EPFL and the fact that Gn(x0, λi(x0)) = 0 (by definition), it would lead us to Gn−1(x0, λi(x0)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Then, using the recurrence relation (we have a fortiori 2n + 2λi(x0) > 0) n + 1 2n + 2λi(x0)Gn+1(x0, λi(x0)) = xGn(x, λi(x)) − n + 2λi(x0) − 1 2n + 2λi(x0) Gn−1(x0, λi(x0)) we get successively by induction that Gn+k(x0, λi(x0)) = 0 for all k ∈ N, and using again the differential equation we get that ∂xGn+k(x0, λi(x0)) = 0 for all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' But we have ∂xGn+k(x0, λi(x0)) = 2λi(x0)Gn+k−1(x0, λi(x0) + 1) so that Gn+k−1(x0), λi(x0) + 1) = 0 for all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Then it is easy to show by induction that Gn+k−1(x0, λi(x0) + j) = 0 for all j ∈ N, and for j larger than (n − 1)/2, the parameter is positive, and we are brought back to classical Gegenbauer polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' This means that successive Gegenbauer polynomials with parameter λi(x0)+j have a root in common, so that their derivatives share these roots too, which is absurd as their roots are simple by orthogonality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' We conclude that ∂xGn(x, λi(x)) has a constant sign for all x ∈] − 1, bi[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' □ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='6 (Interlacing roots, degree).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Consider an interval I = [−1, b[ such that ˜ Gn(x, z) has simple real roots in z on I, then the same will be true of ˜Gn−1(x, z) and their roots interlace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Let’s write ˜ Gn(x, z) = (2x)n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' ⌊n/2⌋ � i=1 � z − λn i (x) � ˜Gn−1(x, z) = (2x)n−1 (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' ⌊(n−1)/2⌋ � i=1 � z − λn−1 i (x) � and show that for all x ∈ I, all i ≤ ⌊(n − 1)/2⌋, λn i (x) > λn−1 i (x) > λn i+1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' We first check the property locally, that is a neighborhood of −1, above −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' We have diλn i (x) dxi |x=−1 = −∂i x ˜ Gn ∂z ˜ Gn � − 1, λi(−1) � = − (−1)n+i 2n(n i)i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' �⌈n/2⌉−1 j=i � j − (i − 1) � �n+i−1 j=n+1 � (j − 1)/2 − (i − 1) � 2n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' (−1)n+i−1 �i−2 j=0 � (i − 1) − j � �⌈n/2⌉−1 j=i � j − (i − 1) � = �n i � i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' �n+i−1 j=n+1 � (j − 1)/2 − (i − 1) � �i−2 j=0 � (i − 1) − j � = n n − i � (n + i − 1)/2 − (i − 1) � � (n − 1)/2 − (i − 1) � �n − 1 i � i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' �n+i−2 j=n � (j − 1)/2 − (i − 1) � �i−2 j=0 � (i − 1) − j � = n n − i � (n + i − 1)/2 − (i − 1) � � (n − 1)/2 − (i − 1) � diλn−1 i (x) dxi |x=−1 As n n−i � (n+i−1)/2−(i−1) � � (n−1)/2−(i−1) � > 1, we conclude that for all i, diλn i (x) dxi |x=−1 > diλn−1 i (x) dxi |x=−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' As λn i (−1) = λn−1 i (−1) = λi(−1) = −1/2 − (i − 1), we can do a Taylor expansion around x = −1: λn i (x) = λi(−1) + (x + 1)i i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' diλn i (x) dxi |x=−1 + o((x + 1)i) λn−1 i (x) = λi(−1) + (x + 1)i i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' diλn−1 i (x) dxi |x=−1 + o((x + 1)i) It is then clear that in a neighborhood of −1 and above −1, λn i (x) > λn−1 i (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' As λn−1 i (−1) − λn i+1(−1) = 1, we also get λn−1 i (x) > λn i+1(x) in a neighborhood of −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Now as for all i λn i (x), λn−1 i (x), λn i+1(x) are continuous functions of x, if by contradiction such inequalities where to fail for some x ∈ I, then there would exist x0 such that λn i (x0) = λn−1 i (x0) or λn−1 i (x0) = λn i+1(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='But then this would SPECIAL POLYNOMIALS AND NEW REAL-ROOTEDNESS RESULTS 11 mean that λn−1 i (x0) is a root of Gn(x0, z) and Gn−1(x0, z), which is impossible by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Therefore we conclude that the inequality λn i (x) > λn−1 i (x) > λn i+1(x) holds for allx ∈ I and i ≤ ⌊(n − 1)/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Notice that according to n, the polynomial ˜Gn(x, z) can be of the same degree than ˜Gn−1(x, z), or of degree one more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' □ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='7 (Interlacing roots, derivative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Consider an interval I = [−1, b[ such that ˜Gn(x, z) has simple real roots in z on I, then the same will be true of ∂x ˜Gn(x, z) and the roots of the two polynomials interlace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' We bring ourselves back to a variant of the previous theorem by using the equality ∂xGn(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' z) = 2zGn−1(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' z + 1) As ∂xGn(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' z) = � ⌈n/2⌉−1 � j=0 (z + j) � ∂x ˜Gn(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' z) Gn−1(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' z + 1) = � ⌈(n−1)/2⌉−1 � j=0 (z + j + 1) � ˜Gn−1(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' z + 1) We get ∂x ˜Gn(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' z) = �⌈(n−1)/2⌉−1 j=0 (z + j + 1) �⌈n/2⌉−1 j=0 (z + j) 2z ˜Gn−1(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' z + 1) And ∂x ˜Gn(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' z) = 2(z + n/2) ˜Gn−1(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' z + 1) if n is even ∂x ˜ Gn(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' z) = 2 ˜Gn−1(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' z + 1) if n is odd So as −n/2 < mini,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='x λi(x) for all i and x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' it amounts to proving that ˜ Gn−1(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' z + 1) and ˜ Gn(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' z) interlace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' We want to show that for all x ∈ I, with x > −1, all i ≤ ⌊(n − 1)/2⌋, λn i (x) > λn−1 i (x) − 1 > λn i+1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' First we check this in a neighborhood of −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' We can check that the inequality λn i (x) > λn−1 i (x) − 1 is going to be true in a neighborhood of −1 as λn i (−1) = λn−1 i (−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' So the nontrivial one is the other one, λn−1 i (x) − 1 > λn i+1(x) for x > −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' We have equality at the origin as λn i (−1) = λn−1 i (−1) := λi(−1) and λi(−1) − 1 = λi+1(−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Then we look at the Taylor expansions around x = −1: λn−1 i (x) − 1 = λi+1(−1) + (x + 1)i i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' diλn−1 i (x) dxi |x=−1 + o((x + 1)i) λn i+1(x) = λi+1(−1) + (x + 1)i+1 (i + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' di+1λn i+1(x) dxi+1 |x=−1 + o((x + 1)i+1) It is clear then that locally λn−1 i (x) − 1 > λn i+1(x) as (x + 1)i+1 << (x + 1)i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' We extend the inequality to the whole interval I by noticing again that if the inequalities where not valid anymore, then there would have to be some equality λn i (x) = λn−1 i (x) − 1 or λn−1 i (x) − 1 = λn i+1(x), which would mean ∂x ˜ Gn(x, λn−1 i (x)) = 0 and as ˜ Gn(x, λn−1 i (x)) = 0, we would again get a contradiction by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='8 (Global extension through ODE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' The local property is in fact true over the whole interval: ˜ Gn(x, z) is real rooted in z with simple (distinct) roots for for x ∈ [−1, 0[ , and they are all increasing to +∞ when x goes to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' 12 AURELIEN GRIBINSKI EPFL Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Denote by Fn(x, z) := − ∂x ˜ Gn ∂z ˜ Gn � x, z � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Consider a rectangular domain D such that ∂z ˜ Gn(x, z) is nonzero on the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Fn is continuous in x and z in the the domain D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Indeed, it is a rational fraction whose denominator is nonzero and it is therefore C∞ in both variables by theorem of composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' As ˜ Gn(−1, z) is realrooted in z with simple roots, ∂z ˜ Gn(−1, λi(−1)) ̸= 0 and by continuity we can find small rectangles Di := [−1, −1 + ǫ] × [λi(−1) − δ, λi(−1) + δ] such that ∂z ˜ Gn(x, z) is nonzero on Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' A strong version of Picard’s theorem tells us that there is a maximal interval Imax i = [−1, ηi max[ for which the roots λi(x) (i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='⌊n/2⌋) are the unique solutions of the initial value ODE dz dx = Fn(x, z), z(−1) = −1/2 − (i − 1) Note that on Imax i , ∂z ˜ Gn(x, λi(x)) ̸= 0 (the denominator is nonzero, so that the differential equa- tion is well defined).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Let’s prove that Imax i = [−1, 0[ (for all i) and that there is explosion at 0 (roots going to infinity), the roots increasing constantly to +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' The local Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='3 tell us that on a neighborhood of −1, Fn(x, λi(x)) > 0, and as by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='5, the numerator is of constant sign and the denominator doesn’t vanish, then Fn(x, λi(x)) > 0 on Imax i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' According to Picard’s theorem, we either have λi(x) →x→ηimax +∞ (explosion), or ηi max is such that lim Fn(x, λi(x)) is not well defined (we leave the domain of definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Now, explosion can’t happen if ηi max < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Indeed, we have that � i λi(x) + � j µj = −Pn−1(x) (2x)n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' where Pn−1(x) is a polynomial of degree n − 1 as well as the coefficient of zn−1 in the expansion of Gn(x, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' So the sum of roots is bounded above by a constant, so there can be no explosion (necessarily to+∞ by monotonicity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' We can leave the domain of definition only if limx→ηimax ∂z ˜ Gn � x, λi(x) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' If this is the case and if by contradiction ηi max < 0, we have seen that ∂z ˜ Gn(ηi max, z) would be of degree exactly ⌊n/2⌋ − 1 in z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Therefore it means that limx→ηimax λi(x) = µ where µ is a root of ∂z ˜ Gn(ηi max, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' But then it means that we can extend by continuity λi(x) at x = ηi max with λi(ηmax) = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' We check by continuity that ˜ Gn � ηi max, λi(ηmax) � = ∂z ˜ Gn � ηi max, λi(ηmax) � = 0 so that in fact λi(ηmax) is a real double root in z of ˜ Gn � ηi max, z � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='7, as there is a root of ∂x ˜ Gn(x, z) between any two roots of ˜ Gn(x, z) in z by interlacing, it follows that necessarily ∂x ˜ Gn � ηi max, λi(ηmax) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' But this is impossible according to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Therefore, we have necessarily ηi max = 0 for all i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='⌊n/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Furthermore, assume by contradiction that there is no explosion for some index i at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' As λi(x) is monotonous for x ∈] − 1, 0[, then we have necessarily that limx→0 λi(x) = µ exists and is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' By continuity we have ˜ Gn(0, µ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Let’s distinguish according to the parity of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' If n is even, we have ˜ Gn(0, z) = Gn(0,z) �n/2−1 j=0 (z+j) = 1 (n/2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' for all z, which shows the contradiction right away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' If n is odd, then ∂z ˜ Gn � x, λi(x) � = xQn � x, λi(x) � where Qn(0, z) = 2(−1)⌊n/2⌋ (n−1)/2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' For x ∈ [−1, 0], xFn(x, λi(x)) is therefore bounded above as a continuous function on a compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' It is always nonpostive, and the maximum can’t be zero because it would mean that for an x ∈ [−1, 0], ∂xLn � x, λi(x) � = 0, which is impossible according to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Therefore it is always smaller than −K < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' dλi(x) dx = 1 xxFn(x, λi(x)) > −K x λi(x) − λi(−1) > −K log(|x|) SPECIAL POLYNOMIALS AND NEW REAL-ROOTEDNESS RESULTS 13 It would follow that λi(x) →x→0 +∞ which is contrary to the assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' We conclude that there is explosion for all i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='⌊n/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' □ □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Applications to realrootedness in x We start by recalling a well-known monotonicity result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' In all the following we will consider z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='1 (Monotonicity of the roots with respect to the parameter, from [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' If xi(z) are the roots of L(z) n (x) (Laguerre), then d dzxi(z) ≥ 0, i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Also, the positive roots yi of G(z) n (x) (Gegenbauer) are such that d dzyi(z) ≥ 0 and the negative symmetric such that d dzyi(z) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' For a fixed z, we have that the roots of ∂zLn(x, z) in x (of degree n − 1) are real and interlace those of Ln(x, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Ln(x, z) = n � k=0 (−1)k �n j=k+1(z + j) (n − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' xk (3) From this expression it follows that ∂zLn(x, z) is of degree n − 1 in x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' d dz xi = −∂zLn ∂xLn (xi(z), z) ∂zLn(xi(z), z) = −∂xLn(xi(z), z) d dz xi (4) ∂xLn(xi(z), z) changes sign when we increment i because Ln(xi(z), z) = 0 so the derivative changes sign when we go from one root to the next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' As d dzxi ≥ 0, We get that ∂zLn(x, z) changes sign n − 1 times and therefore we have n − 1 real zeros between zeros of Ln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Therefore all the zeros of ∂zLn(x, z) have been found and are interlacing with the zeros of Ln(x, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' □ Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' ∂zLn(x, z) and more generally ∂k z Ln(x, z) for all k ≤ n are real-rooted in x, and they form an interlacing family of decreasing degree, in the sense that ∂k+1 z Ln(x, z) interlaces ∂k z Ln(x, z) and the roots are monotonously increasing in z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' We show inductively the following property: ∂k+1 z Ln(x, z) is realrooted, the roots of ∂k+1 z Ln(x, z) interlace the roots of ∂k z Ln(x, z) and are increasing in z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' We start with the initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Using 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='2, we get the real-rootedness and interlacing property for ∂zLn(x, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Now we need to prove that the roots ˜xi(z) (i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='n − 1) of ∂zLn(x, z) also share the monotonicity property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' ∂zLn( ˜xi(z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' z) = 0 Which lead to ∂zLn Ln ( ˜xi(z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' z) = 0 d(∂zLn Ln ( ˜xi(z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' z) � dz = ∂x �∂zLn Ln ( ˜xi(z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' z) �d ˜xi dz + ∂z �∂zLn Ln ( ˜xi(z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' z) � = 0 (5) We want to show that d ˜xi dz ≥ 0 On the one hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' ∂zLn Ln ( ˜xi(z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' z) = n � j=1 ∂zLn ∂xLn (xj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' z) 1 ˜xi − xj So that ∂x �∂zLn Ln ( ˜xi(z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' z) � = − n � j=1 ∂zLn ∂xLn (xj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' z) 1 ( ˜xi − xj)2 = n � j=1 dxj dz 1 ( ˜xi − xj)2 ≥ 0 (6) 14 AURELIEN GRIBINSKI EPFL On the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' using the real-rootedness in z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' as ˜xi ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' +∞[ by the interlacing property,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' then ∂z �∂zLn Ln ( ˜xi(z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' z) � ≤ 0 using Laguerre inequality for realrooted polynomials,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' stating that ∂zzLnLn − (∂zLn)2 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' We conclude by gathering the two inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' The induction is proven using exactly the same method, given that ∂z �∂k+1 z Ln ∂kz Ln � ≤ 0 Because ∂k z Ln(x, z) is realrooted in z as the derivative of a realrooted polynomial, and x in the appropriate interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' □ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' 0 is a root of ∂k z Gn(x, z) for all k ≤ n, when n is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' It comes directly from the formula Gn(x, z) = ⌊n/2⌋ � k=0 (−1)k Γ(n − k + z) Γ(z)k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' (n − 2k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' (2x)n−2k □ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' For a fixed z > 0, we have that the roots of ∂zGn(x, z) in x (of degree n) are real and the positive ones interlace those of Gn(x, z) by below (that is the largest root in module belongs to Gn(x, z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Gn(x, z) = ⌊n/2⌋ � k=0 (−1)k Γ(n − k + z) Γ(z)k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' (n − 2k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' (2x)n−2k (7) From this expression it follows that ∂zGn(x, z) is of degree n in x, as the coefficient of xn is 2n Γ(n+z) Γ(z)n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Let’s denote the roots of Gn(x, z) by yi, then by differentiating the equality Gn(x, z) = 0 with respect to z we get d dz yi = −∂zGn ∂xGn (yi(z), z) ∂zGn(yi(z), z) = −∂xGn(yi(z), z) d dz yi (8) ∂xGn(yi(z), z) changes sign when we increment i because Gn(yi(z), z) = 0 so the derivative −∂xGn(yi(z), z) changes sign when we go from one root to the next (the roots are simple).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Let’s distinguish between the even and odd cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' In the even case, d dzyi ≥ 0 when i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='.n/2, and it gives us by the sign rule i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='.n/2 − 1 positive roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Now there is still one root missing, and we can check that the sign of ∂zGn(yn/2(z), z) is opposite to the sign of ∂zGn(0, z), and as ∂zGn(−yn/2(z), z) has the same sign by symmetry there has to be a root between both, and actually two, one positive and one negative by symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' We get n roots overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' In the odd case, 0 is a root, and we have again two roots missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' But if yl denotes the positive smallest root, then ∂zGn(yl(z), z) has the same sign as ∂zGn(−yl(z), z) and so as 0 is a root we need to have at least two other roots by some easy change of sign argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Therefore all the zeros of ∂zGn(x, z) have been found and the positive ones are interlacing with the zeros of Gn(x, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' □ Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' ∂zGn(x, z) and more generally ∂k z Gn(x, z) for all k ≤ n are real-rooted in x for z > 0, and the positive roots are monotonously increasing in z, and their symmetric negative counterpart monotonously decreasing in z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' SPECIAL POLYNOMIALS AND NEW REAL-ROOTEDNESS RESULTS 15 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' We have Gn(−x, z) = (−1)nGn(x, z) ∂zGn(−x, z) = (−1)n∂zGn(x, z) (9) So that if we have a root ˜y of ∂zGn(x, z), then its symmetric −˜y will also be a root of ∂zGn(x, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' In other terms, ∂zGn(x, z) and more generally , ∂k z Gn(x, z) have symmetric roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' We show inductively the following property: ∂k+1 z Gn(x, z) is realrooted, the positive roots of ∂k+1 z Gn(x, z) interlace the positive roots of ∂k z Gn(x, z) and the positive roots are increasing with z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' We start with the initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Using 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content='2, we get the real-rootedness and interlacing property for ∂zGn(x, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Now we need to prove that the positive roots ˜yi(z) of ∂zGn(x, z) also share the monotonicity property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' ∂zGn( ˜yi(z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' z) = 0 which leads to ∂zGn Ln ( ˜yi(z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' z) = 0 d(∂zGn Gn ( ˜yi(z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' z) � dz = ∂x �∂zGn Gn ( ˜yi(z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' z) �d ˜yi dz + ∂z �∂zGn Gn ( ˜yi(z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' z) � = 0 (10) We want to show that for the positive roots,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' d ˜yi dz ≥ 0 On the one hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' ∂zGn Gn ( ˜yi(z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' z) = n � j=1 ∂zGn ∂xGn (yj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' z) 1 ˜yi − yj so that ∂x �∂zGn Ln ( ˜yi(z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' z) � = − n � j=1 ∂zGn ∂xGn (xj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' z) 1 ( ˜yi − yj)2 = n � j=1 dyj dz 1 ( ˜yi − yj)2 (11) = ⌊n/2⌋ � j=1 dyj dz � 1 ( ˜yi − yj)2 − 1 ( ˜yi + yj)2 � = 4˜yi ⌊n/2⌋ � j=1 dyj dz yj ( ˜yi − yj)2( ˜yi + yj)2 (12) As we have ˜yi > 0 and dyi dz ≥ 0 as well as yj > 0 we get ∂x �∂zGn Ln ( ˜yi(z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' z) � ≥ 0 On the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' using the real-rootedness in z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' as ˜yi ∈ [−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' 1] by the interlacing property,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' then ∂z �∂zGn Gn ( ˜yi(z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' z) � ≤ 0 using Laguerre inequality for realrooted polynomials,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' stating that ∂zzGnGn − (∂zGn)2 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' We conclude by gathering the two inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' The induction is proven using exactly the same method, given that ∂z �∂k+1 z Gn ∂kz Gn � ≤ 0 Because ∂k z Gn(x, z) is realrooted in z as the derivative of a realrooted polynomial, and x in the appropriate interval ([−1, 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' □ 16 AURELIEN GRIBINSKI EPFL References [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Szego.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' Orthogonal polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} +page_content=' AMS, Providence, RI MR 51 (1975): 8724.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfv_nZ/content/2301.00642v1.pdf'} diff --git a/b9E1T4oBgHgl3EQfdgQp/content/tmp_files/2301.03195v1.pdf.txt b/b9E1T4oBgHgl3EQfdgQp/content/tmp_files/2301.03195v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..63ae5f8cb907bd698d1faf39f672147987dbe38c --- /dev/null +++ b/b9E1T4oBgHgl3EQfdgQp/content/tmp_files/2301.03195v1.pdf.txt @@ -0,0 +1,1409 @@ +Thermodynamics +in Stochastic Conway’s Game of Life +Krzysztof Pomorski1A,2 and Dariusz Kotula1B +1 Cracow University of Technology +A: Faculty of Electrical and Computer Engineering +B: Faculty of Computer Science and Telecommunications +2 Quantum Hardware Systems (www.quantumhardwaresystems.com) +Abstract. Cellular automata can simulate many complex physical phenomena using the +power of simple rules. The presented methodological platform expresses the concept of pro- +grammable matter in which Newton’s laws of motion are one of examples. Energy has been +introduced as the equivalent of the ”Game of Life” mass, which can be treated as first level of +approximation. The temperature presence and propagation was calculated for various lattice +topology and boundary conditions by using the Shannon entropy measure. The conducted +study provides strong evidence that despite not fulfillment the principle of mass and energy +conservation, the entropy, mass distribution and temperatures approaches thermodynamic +equilibrium. In addition, the described cellular automata system transits from positive to a +negative temperatures that stabilizes and can be treated as a signature of system dynami- +cal equilibrium. Furthermore, the system dynamics was presented in case of few species of +cellular automata competing for maximum presence on given lattice with different boundary +conditions. +1 +Introduction to Classical Conway’s Game of Life (CCGoL) +A cellular automaton is a system consisting of cells arranged most often on a one, two or three +dimensional regular lattice, which at a given moment are in one of M-th possible states expressing +M-valued logic. The dynamics of the model depends on the definition of individual cell states and +the rules of transitions between them [13]. One of the simplest examples is a one-dimensional cellular +automaton. Suppose that the cells placed on the lattice can be in one of two states, which are marked +with white (default assigned to dead state or logical zero) or black color (default assigned to alive +state or logical one). We define a rule that if a given cell is black, then the cell to the right of it will +change its state. This situation is depicted in a Figure 1. We can observe how the system changes +in the subsequent steps of the simulation. The parameter determining the change of the cell state +is the state of the left neighbour of a given cell. There are many other possibilities to choose such a +parameter, e.g. the condition of the state of both neighboring cells or having the nearest neighbors +with opposite states. In order to determine system dynamics, we must have a defined initial cellula +automata states (information about initial system dynamical state) and a specific set of deterministic +or probabilistic rules. +arXiv:2301.03195v1 [nlin.CG] 9 Jan 2023 + +Fig. 1: Evolution of a one-dimensional cellular automaton in successive cycles with left side partial +logical negation rule (if the state of nearest left cell is alive then given automata cell change its state +to opposite). +The Conway’s Game of Life [3] is an example of a cellular automaton with a deterministic rules. It +was proposed in 1970 by John Conway and cellular automaton system consists of cells located on +a two-dimensional lattice, which can be in one of two states: alive or dead. The rules specify the +required number of neighbors and cell states that are taken into account to determine their state in +the next cycle. Given the neighborhood through the 8 closest cells, we can write the 3 main rules of +the Classical Conway’s Game of Life (CCGoL): +1. If a dead cell has exactly 3 neighbors, it comes alive in the next cycle. +2. If a living cell has 2 or 3 neighbors, it survives in the next cycle. +3. If a cell has a different number of neighbors than stated above, it will be dead in the next cycle. +The rules defined in this way allow for the generation of various types of structure topologies with +automaton state set to 1, as shown in Figure 2. The most common are ”unstable structures”, which +change in successive cycles, but do not return to their initial state. A single cell cannot survive on the +lattice, because it has less than 2 neighbors. Dead cell surrounded by alife cells cannot come to life, +because it has number of neighbors is different from 3. If the simulation is continuing sufficiently +long time, on the lattice usually remain structures that are unchanging in time - ”still lifes” (an +example is the ”block” shown in the Figure 2) or changing in time in periodic way, so they return to +their original shape after k cycles - ”oscillators” (an example is the ”blinker” shown in the Figure 2). +There are also structures that move in a certain direction - ”gliders” (Figure 3) that are analogical +to 1st Newton dynamics preserving momentum (speed and direction of propagation in this case), +leave a trace of ”blinkers” - ”star ships” and objects, which periodically eject ”gliders” - ”guns” [8]. +Figure 4 shows one among many existing in the Conway’s Game of Life oscillators during successive +iterations of the system simulation. +The ”toad” oscillator has a period of 2, which means that it has continuous switching between 2 +different fixed configurations. Each 2-dimensional discrete lattice field has a specified number that +indicates the number of neighbors of the given cell. If the number is green, the cell will be alive in +the next cycle. If the number is blue, that cell will be dead in the next cycle. + +step 0 +step 1 +step 2 +step 3Fig. 2: Evolution of various topologies of cellular automata structures in time with deterministic +rules of CCGoL. One can identify two dynamically unstable structures and two structures that have +dynamical stability with time. +(a) +(b) +(c) +(d) +(e) +Fig. 3: Evolution of the ”glider” configuration of cellular automata having propagation property with +time in deterministic CCGoL. +Fig. 4: Evolution of the ”toad” configuration of cellular automata being an oscillator in successive +cycles in CCGoL. + +step 0 +step 1 +step 2 +Unstable structure 1 +Unstable structure 2 +Still life "block" +Oscilator "blinker"3 +1 +2 +2 +3 +2 +2 +2 +3 +2 +3 +3 +2 +3 +3 +4 +3 +3 +m +2 +2 +2 +3 +2 +2 +2 +72 +Introduction to Stochastic Classical Conway’s Game of Life (SCCGoL) +The Stochastic Classical Conway’s Game of Life (SCCGoL) was created by an addition of a cell spon- +taneous change probability to the rules that were initially deterministic, so stochastic determinism +was achieved. With a given prefixed spontaneous probability value, the state of the cell can change +regardless of the number of neighbors. Cell states have values between 0 and 1 and are called mass. +Due to the fact that SCCGoL have different rules from CCGoL, cells have almost never exactly two +or three neighbors. A condition for given cell to come to life from a dead cell state (creationism of +alife cell) is by having a number of neighbors in a certain range of values. Similar rules applies to +a living cell justifying its alife or dead state in next time iteration. By setting standard intervals of +allowed/forbidden number of neighbours values, in which cell is alife/dead and by adding the addi- +tional spontaneous rule probability for cell to change in the next iteration (probability of changing +the state of a cell regardless of the number of neighbors), we are able to create a simulator, where the +cells never die or it is very difficult from a probabilistic point of view for a given cell to stay alive. +If, maintaining standard neighbours condition for being dead/alife in previously defined intervals +and having a selected initial cell alife automata configuration, we can systematically increase the +spontaneous probability level from 0% to 100%, and we result in the graph as depicted in Figure +5. Simulations were conducted for a lattice size 10 by 10 with the initial condition of a cellular +(a) +(b) +(c) +Fig. 5: (a) Schematic view of initial conditions in SCCGoL. (b-c) Dependence of automata population +average cycle lifetime (over 1000 trials) on probability of spontaneous change of cell state from life +to dead and reversely with preservation of standard rules in Conway’s Game of Life. +automaton 2 by 2 (Figure 5a) with limited maximum number of cycles set to 1000 in conducted +simulations. For a probability of changing the state of a cell regardless of the number of neighbors +equal to zero, there is full determinism, therefore it is a situation of CCGoL with changed numbers of +neighbors. As the probability increases (in range from 0 to 2%), the life expectancy of the population +decreases due to too few neighbors. Starting simulation from a probability level close to two percent + +Life expectancy of the population +cycleofl +Aoverage denth +240 +0 +0 +24 +4 +140 +The probability of changing the stabe of a cell +regardless of the number of neighbors (%]Life expectancy of the population +Aaverage deeth cycle of the populatior +10F +102 +10~2 +10- +100 +101 +The probability pf changing the stabe of a cell +regardless of the number of neighbor's (%](probability of changing the state of a cell regardless of the number of neighbors), the average life +expectancy of the population begins to rise, which is caused by the more frequent appearances of +living cells. Conducting the simulation with a probability level above eighteen percent we observe +that the population practically never dies and keeps average cycle life time being at least 1000 or +more time iterations. +3 +Generalization of Stochastic Classical Conway’s Game of Life +(SCCGoL) to the case of N competing cellular automata species +The created simulation platform enables to create N different cellular automata species that com- +petes between themselves for the resources enabling them to replicate. The given automata species +has better replication properties within its own community and worse replication properties in case +of neighbors being of different species. Such situation is normally encountered in human population +when people of one homogenous identity prefer to collaborate more than in the case of people hav- +ing much different identity (being from different tribes). Due to that fact soft antagonistic relation +between different species is introduced indirectly by means of higher level of tolerance (or effective- +ness of replication) towards neighbors of own species than neighbors of other species (other different +species are treated in the same way). In real way this is an analogy of members collaboration of a +given nation or culture within given culture or community versus different culture or community. +Let us consider N = 4 (number of different automata species), so we have the following determined +formula for number of effective existing neighbors Neeffective,k for given k-th tribes as: +Neeffective,k = a1,kNe1 + a2,kNe2 + ... + ak,kNek + ... + ak,NNeN. +(1) +Previously defined replication rules promoting its own species can be formally expressed by following +condition (max(as,k) = ak,k and ak,k > as,k if s ̸= k). In the conducted simulations all tribes have +assigned the same value (a1,1 = a2,2 = a3,3 = a4,4 = 1 +2) and another same value for (a1,2 = a1,3 = +a1,4 = a2,1 = a2,3 = a2,4 = a3,1 = a3,2 = a3,4 = a4,1 = a4,2 = a4,3 = +1 +3) what simply means +that automata tribes promotes its own tribe and only partly promotes other tribes with no special +distinction on which different tribe it is pointing to. Cellular automata species distribution for k-th +tribe across two-dimensional lattice is generated with use of a two-dimensional Gaussian distribution +given as follows: +f(x, y)k = Ake +− +� +(x−x0,k)2 +2σ2 +x,k ++ +(y−y0,k)2 +2σ2 +y,k +� +, +(2) +where f(x, y)k is a mass function depending on the cell coordinates, (x0,k; y0,k) the center of the +given cellular automaton species. Given parameters A0,k and σx,k, σy,k can control the mass of +species and spread across the ”Globe” as depicted in Figure 6. In the case of several species on +the lattice, SCCGoL simulation results promote the formation of new cells among the species that +has the most mass in the neighborhood as in accordance in the Figure 7. During life cycle always +newly formed cell has a mass that is more dependent on cells of the same species than other species +of its neighbors. We observed that automata species fight each other in order to spread across the +lattice and exclude other species what is an expression of some form of biological Darwinism being +a consequence of formula 1. + +(a) +(b) +Fig. 6: (a) Case of mass distribution of cellular automata of one species using a two-dimensional +Gaussian function being isotropic. (b) Example of initial distribution of three cellular automata tribes +with the case of boundary existence between each separate tribe with a rule that one geometrical +place on the lattice is occupied by cellular automata of given species with dominant mass. +(a) +(b) +Fig. 7: Evolution of map distribution of 4 cellular automata populations of different species with 100 +by 100 lattice size. One can spot effective tendency of each species to occupy maximum possibly +territory at the cost of other species in 4-species SCCGoL (SCCGoL4S) what can be understand as +occurrence of weak antagonistic relation +4 +Methodology of description of SCCGoL dynamics by tools of classical +statistical physics +Adding probability to the initially deterministic Game of Life and changing the interaction rules +between neighboring automata brought the necessity for a new quantity called mass that has con- +tinuous real values (other than 0 and 1 present in deterministic Game of Life). At this stage vividness +of the whole cellular automata population can be understood and approximated by the population +total energy (where simply mass is equal to energy). The level of the automata distribution order is +expressed by the entropy. Entropy was introduced by Rudolf Clausius in 1865 as a thermodynamic +state function [11]. If the entropy at the initial state and the entropy of the final state are devoted +by Si and Sf respectively, then we have: +Sf − Si = +� f +i +dE +T , dS +dE = 1 +T +(3) + +Very last formula gives definition of a temperature under assumption that energy and entropy is +defined that is the case. We use instead of thermodynamic entropy the Shannon entropy measure +given as: +SShannon = − +� +i,j +p(xi, yj) log p(xi, yj) = E [− log p(x, y)] +(4) +In order to calculate the probability in the equation 4, the SCCGoL simulation is repeated several +hundred times, so one determines an average population cell mass in successive cycles. The calcula- +tion of the temperature, which is a measure of thermal state was achieved with an equation 3 with +a changed form: +T(x, y, t) = +dE(x,y,t) +dt +dSShannon(x,y,t) +dt += +dE(x, y, t) +dSShannon(x, y, t) +(5) +The temperature defined by last formula can be calculated by two different methods. The first +method relies on the calculation of the energy and entropy for the entire system, and then numerical +calculation of the derivative of energy in function of entropy. The second calculation approach is +to differentiate mass and entropy with respect to simulation time for each cell and as a result of +corresponding ratio we obtain a temperature map. The example of evolution of mass and entropy +for SCCGoL is depicted in Figure 8 and shows maximization and saturation of entropy, what is +confirmation of Second Thermodynamic Law that is valid in physical systems, while we are dealing +with cellular automata system. On another hand the mass of a system saturates and increases to +certain critical value as given in left part of Figure 8. +Fig. 8: Evolution of mass and entropy in successive cycles in SCCGoL manifesting maximization with +saturation and approaching stationary thermodynamical equilibrium with initial condition depicted +in Figure 9. +5 +Numerical analysis of SCCGoL dynamics with methodology of +classical statistical physics +Numerical simulations were carried out for 4 topologies cellular automata (case of Figures 9a, 11a, +13a, 15a). We preinpose such a rule that given alive cell has 20% probability of changing its state +to dead state and that initially dead cell has 20% probability of changing its state to alive with +a mass choose randomly from an interval value in (0,0.5). Still given cell state is depended on its +neighbors, since it has 80% probability of changing its state due to state of neighbors. In order +to obtain cellular automata probability map dynamics, simulations average of cellular automata +positions was conducted. Figure 10 shows results that were obtained for a lattice size 100 by 100 +with a one cell alive as initial lattice state (case of Figure 9a). With subsequent cycles, the cells + +Mass +Entropy +25000 +800 +20000 +600 +15000 +400 +10000 +200 +5000 +0 +0 +0 +25 +50 +75 +100 +125 +150 +175 +200 +0 +25 +50 +75 +100 +125 +150 +175 +200 +Cycle +Cycle(a) +(b) +Fig. 9: Evolution of diffusion process in cellular automata system for a limited lattice size 100 by 100 +in SCCGoL (averaged over 1000 trials). It shows final saturation of mass and entropy (as depicted +in Figure 8) what implies approaching thermodynamical equilibrium with characteristic fluctuations +of mass and entropy around effective stationary values. +occupy more and more space on the lattice, which can be seen as increase the mass of the entire +system (possible mass creationism is inherent feature of Conway’s Game of Life). The sum of the +masses of all cells in successive cycles is depicted in Figure 8 with a comparison of the entropy +change of the entire system. As we observe in Figure 10e, high entropy occurs at the edges of the +population, which is caused by the entropy wave that meets area with almost zero cell occurrence. +Before equilibrium is established, the entropy of the system slightly decreases due to the loss of this +extra entropy at the edges as can be seen in the right part of Figure 8. Having established those two +quantities, we conduct their differentiating with respect to time, and by formula 3 one can establish +the whole effective temperature of system by dividing change of mass at given time by change of +the entropy. Figure 10k shows the greater susceptibility of the entropy change to the constraints +associated with the limited size of the simulation lattice that is ended with impenetrable walls. The +dependence of mass derivative with respect to time simulation (case of Figure 10j) does not have +such large oscillations as in the case of entropy derivative with simulation simulation (case of Figure +10k). The temperature calculated by such procedure gives values just above zero (slightly positive) +up to the 50’th cycle. We can see a large down peak caused by a slowdown in entropy increase. +From about the 75’th cycle, the temperature of the system goes from positive to negative values. +A Figure 10l) describes anomalous thermalization process in SCCGoL with a case of approaching +temperature and entropy equilibrium. Surprisingly, the thermodynamic equilibrium is achieved for +a case for negative temperatures. A second possible approach in determination the temperature is +by use the derivatives of the mass and entropy with respect to time of the individual cells and by +obtaining a temperature map of all cells. The temperature depicted in Figure 10i is mostly negative +and steady, what corresponds to a situation where mass and entropy have come to equilibrium. We +can distinguish two regions in the simulation, the first one with no temperature gradient and zero +negative temperature and the second region with non-zero temperature gradient that also include +positive temperatures. The place, where time derivatives of mass and entropy have noticeable values +is at the edges of automata population, where we observe slightly positive temperature, which +corresponds to the situation in Figure 10l before the 75’th cycle. + +■(a) Mass at t=4 +(b) Mass at t=26 +(c) Mass at t=92 +(d) Entropy at t=4 +(e) Entropy at t=26 +(f) Entropy at t=92 +(g) T(x,y,t=4) +(h) T(x,y,t=26) +(i) T(x,y,t=92) +(j) dm +dt with time +(k) dS +dt with time +(l) Temperature with time +Fig. 10: Dynamics of thermodynamical parameters (mass, entropy and temperature) with simula- +tion time in SCCGoL (lattice size 100 by 100) also given in Figure 8. Achieved thermodynamical +equilibrium is accompanied with final distribution of negative temperature (starting from positive +temperature distribution as in accordance with formula 3) as experimentally observed necessary +criteria for final thermodynamical stability. One can spot various similarities of statistical behaviour +of SCCGoL with physical systems described by classical thermodynamics (maximization and satu- +ration of entropy, decay of temperature gradients and final thermalization, uniform distribution of +mass and energy). + +0 +0.10 +20 +0.08 +40 +0.06 +y +60 +0.04 +80 +0.02 +0.00 +0 +20 +40 +60 +08 +x0 +0.10 +20 +0.08 +40 +0.06 +60 +0.04 +80 +0.02 +0.00 +0 +20 +40 +60 +80 +X0.10 +20 +0.08 +40 +0.06 +60 +0.04 +80 +0.02 +20 +40 +60 +80 +X0 +5 +20 +4 +40 +3 +y +60 +2 +80 +1 +20 +40 +09 +0 +0 +80 +x0 +8 +7 +20 +6 +40 +5 +4 +60 +3 +2 +80 +1 +20 +40 +60 +80 +0 +X0 +7 +20 +6 +40 +5 +60 +4 +80 +3 +0 +20 +40 +60 +80 +X0 +0.00 +20 +-0.02 +40 +-0.04 +y +60 +-0.06 +80 +-0.08 +0 +20 +40 +09 +80 +x0 +0.00 +20 +-0.02 +-0.04 +40 +y +-0.06 +60 +-0.08 +80 +-0.10 +0 +20 +40 +60 +80-0.02 +20 +-0.04 +40 +-0.06 +60 +-0.08 +80 +-0.10 +0 +20 +40 +60 +80 +XMass derivative with respect to time +15.0 +12.5 +dm/dt +14.0 +7.5 +5.D +25 +MAAy +0.0 +2.5 +0 +21 +41 +140 +120 +CycleEntropy derivative with respect to time +40 +3P/SP +240 +0- +200 +21 +4 +140 +120 +CycleTemperature +0.5 +Temperature +0.D +0.5 +1.0 +1.5 +-2.0 +0 +21 +140 +120 +Cycle6 +Numerical study of one species cellular automata with various +boundary conditions +Next family of simulations were conducted with use of barriers impenetrable by cellular automata +cells via which cellular automata cannot interact with each other. We consider a single cellular +(a) +(b) +(c) +Fig. 11: Diffusion process in SCCGoLp20L100b100 (lattice size 100 by 100, 20% probability of +spontaneous change of cell state from life to dead and reversely with preservation of standard rules +in Conway’s Game of Life as also given in Figure 5) case of system of two weekly interconnected +chambers by means of two small holes in barrier. Two stages of diffusion can be spotted in mass and +entropy dynamics that has two consecutive processes: full diffusion in the left chamber is leading to +full diffusion in the right chamber. Further details of thermodynamical parameters space dependence +evolution with time are given by Figure 12. +automata (single automata seed) placed in empty chamber with impenetrable walls with two small +holes that link it to another empty chamber as depicted in the Figure 11a. We observe a diffusion of +cellular automata with simulation time that consists of two main processes: creation and diffusion +of cellular automata in the first chamber and diffusion of cellular automata from the first chamber +into second chamber accompanied with creation new automata in the second chamber. Those two +consequent processes are accompanied by effective slowdown in diffusion that is seen in left part +of Figure 11c. At the same time we observe slowdown in entropy increase as in an accordance +to the right part of Figure 11c. Once mass saturation was obtained in the simulation we observe +maximization and small drop in entropy that later stabilizes and saturates, as in the right part of +Figure 11c. Entropy unlike mass is characterized by a large variation of values in the middle of the +population and at the edges of the cellular automata population. In the situation with barrier (case of + +Mass +Entropy +25000 +1200 +1000 +20000 +800 +15000 +600 +10000 +400 +5000 +200 +0 +0 +0 +25 +50 +75 +100 +125 +150 +175 +200 +0 +25 +50 +75 +100 +125 +150 +175 +200 +Cycle +Cycle(a) Mass at t=5 +(b) Mass at t=70 +(c) Mass at t=100 +(d) Entropy at t=5 +(e) Entropy at t=70 +(f) Entropy at t=100 +(g) T(x,y,t=5) +(h) T(x,y,t=70) +(i) T(x,y,t=100) +(j) dm +dt with time +(k) dS +dt with time +(l) Temperature with time +Fig. 12: Dynamics of thermodynamical parameters with simulation time in SCCGoLp20L100b100 +with two weekly interconnected chambers by two small holes in barrier that were initially depicted +in Figure 11. + +0 +0.14 +20 +0.12 +0.10 +40 +0.08 +y +60 +0.06 +0.04 +80 +0.02 +40 +60 +0.00 +20 +80 +x0.175 +20 +0.150 +0.125 +40 +0.100 +60 +0.075 +0.050 +80 +0.025 +000'0 +20 +40 +60 +800.200 +0.175 +20 +-0.150 +40 +0.125 +0.100 +60 +0.075 +0.050 +80 +0.025 +000'0 +600 +20 +40 +60 +80 +0 +20 +40 +60 +8020 +40 +60 +80 +0 +20 +40 +8020 +40 +60 +80 +20 +40 +500 +0.025 +20 +0.000 +0.025 +40 +y +0.050 +60 +0.075 +0.100 +80 +0.125 +0 +20 +40 +60 +80 +x0.00 +0.02 +20 +0.04 +0.06 +40 +0.08 +60 +0.10 +0.12 +80 +0.14 +0.16 +90 +60 +800.000 +0.025 +20 +0.050 +40 +0.075 +0.100 +60 +0.125 +80 +0.150 +0.175 +80vass derivative with resoect to time +15 +1 +dm/dt +0 +25 +54 +S +14D +125 +150 +175 +CycleEntropy derivative with respect to time +500 +310 +p/Sp +240 +10 +0- +100 +200 +0 +25 +50 +75 +140 +125 +150 +175 +CycleTemperature +3.D +25 +2D +15 +LD +0.5 +0.0 +0.5 +1.0 +0 +25 +50 +75 +140 +125 +150 +175 +CycleFigure 12e) this is particularly evident after the cells pass through the gaps. This particular process +is due to the logarithm function dependence of Von Neumann entropy, which tends to negative +infinity for arguments going to zero from the right. Observed criteria of an equilibrium is fact that +mass and entropy have steady values and that temperature have negative value in thermodynamical +equilibrium. Almost always before the equilibrium moment is achieved, time derivatives of mass and +entropy have positive values, and still the temperature is positive. From the moment the equilibrium +is achieved, we are dealing with small fluctuations of entropy and temperature. We observe the +correlation stating that slight increase in mass results in slight decrease in entropy and vice versa. In +an analogical way to the system with one barrier, simulations were conducted on the system with two +barriers and the initial structure as depicted in Figure 13a. We consider a single cellular automata +(a) +(b) +(c) +Fig. 13: Dynamics of diffusion process for a system SCCGoLp20L100b100 with two barriers with +two small holes in each barrier (generalization of situation from Figure 12) creating three weakly +interconnected chambers perturbed by mutual interactions mediated by holes. Monotonicity in in- +crease of entropy is twice shortly interrupted by small decline, what is associated with automata cells +”colliding” with barriers and experiencing short lasting slowing down in its propagation. Details on +space depended thermodynamical parameter evolution with time are given by Figure 14. +placed in empty chamber with impenetrable walls with two small holes that link it to the second +empty chamber, which is connected to a third empty chamber by impenetrable walls with two small +holes as depicted in the Figure 13a. We observe a diffusion of cellular automata with simulation time +that consists of three main processes: creation and diffusion of cellular automata in the first chamber, +diffusion of cellular automata from the first chamber into second chamber accompanied with creation +new automata in the second chamber and diffusion of cellular automata from the second chamber +into third chamber accompanied with creation new automata in the third chamber. Twice cells try + +Mass +Entropy +1200 +25000 +1000 +20000 +800 +15000 +600 +10000 +400 +5000 +200 +0 +0 +25 +50 +75 +100 +125 +150 +175 +200 +0 +25 +50 +75 +100 +125 +150 +175 +200 +Cycle +Cycleto reach impenetrable barriers, we observe small drops in entropy. After the second time entropy +stabilizes and saturates, as in the right part of Figure 13c. In a system with two barriers, it lasts +longer for mass and entropy to reach equilibrium than in the case of a system with only one barrier. +As depicted in Figure 14k, due to the cells approaching the barriers and losing the extra entropy at +the edges of the population, we observe significant fluctuations in the time derivative of the entropy. +This results in large peaks seen in the Figure 14l. + +(a) Mass at t=30 +(b) Mass at t=70 +(c) Mass at t=110 +(d) Entropy at t=30 +(e) Entropy at t=70 +(f) Entropy at t=110 +(g) T(x,y,t=30) +(h) T(x,y,t=70) +(i) T(x,y,t=110) +(j) dm +dt with time +(k) dS +dt with time +(l) Temperature with time +Fig. 14: Space depended dynamics of thermodynamical parameters with simulation time in SCC- +GoLp20L100b100 with three weekly interconnected chambers by four small holes also depicted in +Figure 13. + +0 +0.175 +0.150 +20 +0.125 +40 +0.100 +y +60 +0.075 +0.050 +80 +0.025 +0.000 +0 +20 +40 +09 +08 +x0.175 +20 +0.150 +0.125 +40 +0.100 +60 +0.075 +0.050 +80 +0.025 +0.000 +0 +20 +40 +60 +80 +X0.175 +20 +0.150 +0.125 +40 +0.100 +60 +0.075 +0.050 +80 +0.025 +0.000 +20 +40 +60 +80 +X0 +6 +20 +5 +40 +4 +3 +60 +2 +80 +1 +20 +40 +60 +80 +0 +X0 +6 +20 +5 +40 +4 +3 +60 +2 +80 +1 +0 +20 +40 +60 +80 +Xh +20 +5 +40 +4 +60 +3 +80 +0 +20 +40 +60 +80 +X0 +0.00 +-0.02 +20 +-0.04 +-0.06 +40 +-0.08 +60 +-0.10 +-0.12 +80 +-0.14 +-0.16 +0 +20 +40 +60 +80 +x0.00 +-0.02 +20 +-0.04 +40 +-0.06 +-0.08 +60 +-0.10 +-0.12 +80 +-0.14 +0 +20 +40 +60 +80 +X0.00 +-0.02 +20 +-0.04 +0.06 +40 +-0.08 +60 +-0.10 +-0.12 +80 +-0.14 +0.16 +0 +20 +40 +60 +80vass derivative with resoect to time +15 +dm/dt +25 +54 +75 +140 +125 +150 +175 +CycleEntropy derivative with respect to time +D +P/SP +24D +200 +0 +25 +50 +14D +125 +150 +175 +CycleTemperature +t0 +Temperature +0.2 +0.0 +-0.2 +0.4 +0.6 +0.8 +1.0 +0 +25 +50 +75 +140 +125 +150 +175 +Cycle7 +Numerical study of two species cellular automata in perturbative +interaction by narrow constriction +Further simulations for the case of two species cellular automata were carried out for a system divided +into two reservoirs separated by two impenetrable barriers with one small hole (case of Figure 15a) +that implies perturbative interaction between tribes. In the left upper corner of the first part of +the system there have been located cells of the first cellular automata tribe. At a closer distance +to the hole in impenetrable wall, but in the second right reservoir there have been located cells of +the second cellular automata tribe. As depicted in Figure 15c, we observe similar final masses and +their dynamics in case of both tribes, but in accordance to Figure 15d, different entropy dynamics. +Very last is due to the distance of the cellular automata tribes from the small hole in impenetrable +wall: a tribe located further from that hole needs more time to propagate and occupy its natural +neighborhood and first left chamber of the system, and thus this tribe has a lower probability of +taking over the territory of the other tribe. However the noticeable fact is that mass and entropy +of both tribes achieves equilibrium and finally tribes end up in bit same geometrical and dynamical +situation. As depicted in Figure 15f, the right tribe closer to the small hole occupies its nearest +neighborhood territory more quickly, resulting in an attempt to occupy a rival tribe territory. As +depicted in Figure 16 we observe large oscillations in the time derivatives of mass and entropy of +both cellular automata tribes. In contrast to previously conducted simulations, the temperature of +the system after reaching equilibrium is not only characterized by negative values. Tribes existential +competition is the reason of occurrence of both positive and negative temperatures. + +(a) +(b) +(c) +(d) +(e) Mass at t=3 +(f) Mass at t=34 +(g) Mass at t=200 +Fig. 15: Diffusion process in case of system with two cellular automata tribes weekly interacting +with each other via a small hole in double barrier as depicted in (a), where initial configuration +is presented. After long thermodynamic equilibrium is achieved as given by (b) so two cellular +automata tribes coexist in two different geometrical domains effectively geographically separated. +In case of both tribes mass and entropy saturates having tendency to oscillate in thermodynamical +equilibrium. + +Mass of the first tribe +Mass of the second tribe +140 +140 +120 +120 +100 +100 +80 +80 +60 +60 +40 +40 +20 +20 +0- ++0 +25 +50 +75 +100 +125 +150 +175 +200 +0 +25 +50 +75 +100 +125 +150 +175 +200 +Cycle +CycleEntropy of the first tribe +Entropy of the second tribe +7000 +8000 +6000 +5000 +6000 +4000 +4000 +000m +2000 +2000 +1000 +0 +0 +25 +50 +75 +100 +125 +150 +175 +200 +0 +25 +50 +75 +100 +125 +150 +175 +200 +Cycle +Cycle0 +10 +20 +y +30 +40 +0 +10 +20 +30 +40 +x0 +10 +20 +y +30 +40 +0 +10 +20 +30 +40 +x0 +10 +20 +30 +40 +0 +10 +20 +30 +40(a) dm +dt with time for first and second automata tribe +(b) dS +dt with time for first and second automata tribe +(c) Temperature with time for first and second automata tribe +Fig. 16: Dynamics of thermodynamical variables for the case of two competing cellular automata +tribes (first tribe is placed on the left and second tribe is placed on the right). + +Derivative of mass with resoect to time of the first tribe +Derivative of mass with resoect to time of the second tribe +5 +3 +4 +2 +3 +2 +0 ++ +-1 +-1 +25 +50 +75 +100 +125 +150 +175 +200 +0 +25 +50 +75 +100 +125 +150 +175 +200 +Cycle +CycleDerivative of entropy with respect to time of the first tribe +Derivative ofentropy with respect to time of the second tribe +400 +300 +200 +200 +100 +0 +0 +-200 +-100 +-400 +-200 +0 +25 +50 +75 +100 +125 +150 +175 +200 +0 +25 +50 +75 +100 +125 +150 +175 +200 +Cycle +CycleTemperature of the first tribe +Temperature of the second tribe +1.4 +2 +1.2 +0 +1.0 +-2 +0.8 +-4 +0.6 +-6 +0.4 +-8 +0.2 +-10 +0.0 +-12 +0.2 +-14 +0 +25 +50 +75 +100 +125 +150 +175 +200 +0 +25 +50 +75 +100 +125 +150 +175 +200 +Cycle +Cycle8 +Conclusions and future perspectives +Cellular automata can simulate many complex physical phenomena using the power of simple rules +as it was shown in the case of cellular automata diffusion dynamics confirmed by various simulations +for one and many automata species. Certain type of automata Darwinism was spotted by studying 4 +automata species dynamics as given by Fig.7. The SCCGoL dynamics study provides strong evidence +that despite the fact that the principle of conservation of mass is not fulfilled, since we have creation- +ism and annihilation of automata, the entropy and temperature comes to equilibrium. In conducted +various simulations of Stochastic Conway’s Game of Life dynamics we report transition from positive +to negative values of temperatures and we are aware that there is maximum level of mass and energy +density allowed for cellular automata, since otherwise they would die due to overpopulation. The fact +that the temperature can be negative is known in condensed matter physics, but with assumption +that the energy is top-bounded. In most ”normal” situations this is impossible, but in a rare cases +in solid state physics approximately it can be achieved by inverting the population state. Obviously +there is a such limitation on top-bounded energy value (mass density value) in Stochastic Conway’s +Game of Life. Therefore, it is still consistent with thermodynamics methodology, as it was pointed +by professor Adam Bednorz (Faculty of Physics, University of Warsaw). +Following conclusions were derived basing on conducted simulations and described methodology: +1. Identification of thermodynamically defined temperature as proper measure of system evolution +with ’-’ sign (case of Figures 10, 12, 14). +2. Identification of mass as effective energy of system (in first approximation) (case of Figures 8, +11, 13, 15). +3. Identification of Shannon Entropy as effective system entropy (in first approximation) (case of +Figures 8, 11, 13, 15). +4. Generalization of Stochastic Conway Game of Life of N tribes (approximated analogy to N-body +Quantum Physics can be conceptionally drawn) as depicted in Figures 7, 15. +5. Confirmation validity of second law of thermodynamics in SCCGoL (entropy maximises and +saturates, case of Figures 8, 11, 13). +6. Identification of short lasting Shannon entropy peak that later minimizes and saturates in SGoL +(case of Figures 8, 11). Monotonicity in increase of entropy is twice shortly interrupted by small +decline, what is associated with automata cells ”colliding” with barriers and experiencing short +lasting slowing down in its propagation (case of Figure 13). +Conducted analysis of Stochastic Game of Life allows to treat such system as mathematical object +well described by methodology of classical statistical physics. Obtained numerical results by various +simulations suggest that we shall introduce another definition of temperature in Stochastic Conway’s +Game of Life system by adding ’minus’ sign to temperature known in statistical physics, so we obtain +the following formula: +TemperatureConway−P omorski−Kotula = −dE +dS +(6) +TemperatureStatisticalP hysics = +dE +dS +Having such a definition of Conway-Pomorski-Kotula temperature we can use tools of statistical +physics in Stochastic Game of Life preserving most classical physics thermodynamical intuition about +various situations we can come across. There are well-known inherent analogous between classical +statistical physics [12][7][2][5][4] and quantum mechanics [1]. Therefore further research perspectives +in study of Stochastic Classical Conway’s Game of Life assume the usage of quantum mechanics + +being able to simulate classical statistical physics (as expressed by epidemic model or stochastic +finite-state machine) as explicitly represented by tight-binding model [9][10] or Schroedinger model +directly proposing structures implemented in semiconductor single-electron devices. Noticeable var- +ious obtained map of probability of Stochastic Conway’s Game of Life can be parameterized by +non-linear Schrodinger equation and especially by Ginzburg-Landau model [6], what can be the +base for quantization of Stochastic Classical Conway’s Game of Life. +9 +Acknowledgment +We would like to express our acknowledgment to professor Adam Bednorz (University of Warsaw), +Adam Chochla (Cracow University of Technology) and to doctor Lukasz Stepien (The Pedagogical +University in Cracow). The consultations with them on manuscript has allowed to improve it. The +Authors have no conflict of interests and equally contributed to this work with 50 percent of contri- +bution on each side. First Author proposed methodological and conceptual framework for this work, +while Second Author conducted all numerical simulations. The interpretations of obtained results is +equally assigned to each Author. +References +1. Baez, J.C., Pollard, B.S.: Quantropy. arXiv:1311.0813 (2013) +2. Feynman, R.P.: Statistical mechanics. Westview (1972) +3. Gardner, M.: Mathematical games - the fantastic combinations of john conway’s new solitaire game +”life”. Scientific American (1970) +4. Huang, K.: Statistical mechanics. John Wiley & Sons (1963) +5. Huang, K.: Introduction to statistical physics. CRC Press (2001) +6. Kotula, D., Pomorski, K.: Thermodynamics of Stochastic Conway Game of Life. ShanghaiAI Lectures, +https://youtu.be/kLOB9VlF-R4 (2022) +7. Mishin, Y.: Thermodynamic theory of equilibrium fluctuations. Elsevier (2015) +8. Peitgen, H.O., J¨urgens, H., Saupe, D.: Chaos and Fractals. Springer (1983) +9. Pomorski, K.: Equivalence between classical epidemic model and quantum tight-binding model. Springer +(2022) +10. Pomorski, K.: Equivalence between finite state stochastic machine, non-dissipative and dissipative tight- +binding and schroedinger model. arXiv:2208.09758 (2022) +11. Shannon, C.: A mathematical theory of communication. Bell System Technical Journal (1948) +12. Velazquez Abad, L.: Principles of classical statistical mechanics: A perspective from the notion of com- +plementarity. Annals of Physics 327(6), 1682–1693 (2012) +13. Wolfram, S.: Statistical mechanics of cellular automata. Reviews of Modern Physics 55 (1983) + diff --git a/b9E1T4oBgHgl3EQfdgQp/content/tmp_files/load_file.txt b/b9E1T4oBgHgl3EQfdgQp/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9d774a6c49fb37966f2a7bdafc29083137032414 --- /dev/null +++ b/b9E1T4oBgHgl3EQfdgQp/content/tmp_files/load_file.txt @@ -0,0 +1,633 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf,len=632 +page_content='Thermodynamics in Stochastic Conway’s Game of Life Krzysztof Pomorski1A,2 and Dariusz Kotula1B 1 Cracow University of Technology A: Faculty of Electrical and Computer Engineering B: Faculty of Computer Science and Telecommunications 2 Quantum Hardware Systems (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='quantumhardwaresystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='com) Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Cellular automata can simulate many complex physical phenomena using the power of simple rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' The presented methodological platform expresses the concept of pro- grammable matter in which Newton’s laws of motion are one of examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Energy has been introduced as the equivalent of the ”Game of Life” mass, which can be treated as first level of approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' The temperature presence and propagation was calculated for various lattice topology and boundary conditions by using the Shannon entropy measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' The conducted study provides strong evidence that despite not fulfillment the principle of mass and energy conservation, the entropy, mass distribution and temperatures approaches thermodynamic equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' In addition, the described cellular automata system transits from positive to a negative temperatures that stabilizes and can be treated as a signature of system dynami- cal equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Furthermore, the system dynamics was presented in case of few species of cellular automata competing for maximum presence on given lattice with different boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' 1 Introduction to Classical Conway’s Game of Life (CCGoL) A cellular automaton is a system consisting of cells arranged most often on a one, two or three dimensional regular lattice, which at a given moment are in one of M-th possible states expressing M-valued logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' The dynamics of the model depends on the definition of individual cell states and the rules of transitions between them [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' One of the simplest examples is a one-dimensional cellular automaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Suppose that the cells placed on the lattice can be in one of two states, which are marked with white (default assigned to dead state or logical zero) or black color (default assigned to alive state or logical one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' We define a rule that if a given cell is black, then the cell to the right of it will change its state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' This situation is depicted in a Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' We can observe how the system changes in the subsequent steps of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' The parameter determining the change of the cell state is the state of the left neighbour of a given cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' There are many other possibilities to choose such a parameter, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' the condition of the state of both neighboring cells or having the nearest neighbors with opposite states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' In order to determine system dynamics, we must have a defined initial cellula automata states (information about initial system dynamical state) and a specific set of deterministic or probabilistic rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='03195v1 [nlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='CG] 9 Jan 2023 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' 1: Evolution of a one-dimensional cellular automaton in successive cycles with left side partial logical negation rule (if the state of nearest left cell is alive then given automata cell change its state to opposite).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' The Conway’s Game of Life [3] is an example of a cellular automaton with a deterministic rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' It was proposed in 1970 by John Conway and cellular automaton system consists of cells located on a two-dimensional lattice, which can be in one of two states: alive or dead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' The rules specify the required number of neighbors and cell states that are taken into account to determine their state in the next cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Given the neighborhood through the 8 closest cells, we can write the 3 main rules of the Classical Conway’s Game of Life (CCGoL): 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' If a dead cell has exactly 3 neighbors, it comes alive in the next cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' If a living cell has 2 or 3 neighbors, it survives in the next cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' If a cell has a different number of neighbors than stated above, it will be dead in the next cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' The rules defined in this way allow for the generation of various types of structure topologies with automaton state set to 1, as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' The most common are ”unstable structures”, which change in successive cycles, but do not return to their initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' A single cell cannot survive on the lattice, because it has less than 2 neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Dead cell surrounded by alife cells cannot come to life, because it has number of neighbors is different from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' If the simulation is continuing sufficiently long time, on the lattice usually remain structures that are unchanging in time - ”still lifes” (an example is the ”block” shown in the Figure 2) or changing in time in periodic way, so they return to their original shape after k cycles - ”oscillators” (an example is the ”blinker” shown in the Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' There are also structures that move in a certain direction - ”gliders” (Figure 3) that are analogical to 1st Newton dynamics preserving momentum (speed and direction of propagation in this case), leave a trace of ”blinkers” - ”star ships” and objects, which periodically eject ”gliders” - ”guns” [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Figure 4 shows one among many existing in the Conway’s Game of Life oscillators during successive iterations of the system simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' The ”toad” oscillator has a period of 2, which means that it has continuous switching between 2 different fixed configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Each 2-dimensional discrete lattice field has a specified number that indicates the number of neighbors of the given cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' If the number is green, the cell will be alive in the next cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' If the number is blue, that cell will be dead in the next cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' step 0 step 1 step 2 step 3Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' 2: Evolution of various topologies of cellular automata structures in time with deterministic rules of CCGoL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' One can identify two dynamically unstable structures and two structures that have dynamical stability with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' (a) (b) (c) (d) (e) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' 3: Evolution of the ”glider” configuration of cellular automata having propagation property with time in deterministic CCGoL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' 4: Evolution of the ”toad” configuration of cellular automata being an oscillator in successive cycles in CCGoL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' step 0 step 1 step 2 Unstable structure 1 Unstable structure 2 Still life "block" Oscilator "blinker"3 1 2 2 3 2 2 2 3 2 3 3 2 3 3 4 3 3 m 2 2 2 3 2 2 2 72 Introduction to Stochastic Classical Conway’s Game of Life (SCCGoL) The Stochastic Classical Conway’s Game of Life (SCCGoL) was created by an addition of a cell spon- taneous change probability to the rules that were initially deterministic, so stochastic determinism was achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' With a given prefixed spontaneous probability value, the state of the cell can change regardless of the number of neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Cell states have values between 0 and 1 and are called mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Due to the fact that SCCGoL have different rules from CCGoL, cells have almost never exactly two or three neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' A condition for given cell to come to life from a dead cell state (creationism of alife cell) is by having a number of neighbors in a certain range of values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Similar rules applies to a living cell justifying its alife or dead state in next time iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' By setting standard intervals of allowed/forbidden number of neighbours values, in which cell is alife/dead and by adding the addi- tional spontaneous rule probability for cell to change in the next iteration (probability of changing the state of a cell regardless of the number of neighbors), we are able to create a simulator, where the cells never die or it is very difficult from a probabilistic point of view for a given cell to stay alive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' If, maintaining standard neighbours condition for being dead/alife in previously defined intervals and having a selected initial cell alife automata configuration, we can systematically increase the spontaneous probability level from 0% to 100%, and we result in the graph as depicted in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Simulations were conducted for a lattice size 10 by 10 with the initial condition of a cellular (a) (b) (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' 5: (a) Schematic view of initial conditions in SCCGoL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' (b-c) Dependence of automata population average cycle lifetime (over 1000 trials) on probability of spontaneous change of cell state from life to dead and reversely with preservation of standard rules in Conway’s Game of Life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' automaton 2 by 2 (Figure 5a) with limited maximum number of cycles set to 1000 in conducted simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' For a probability of changing the state of a cell regardless of the number of neighbors equal to zero, there is full determinism, therefore it is a situation of CCGoL with changed numbers of neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' As the probability increases (in range from 0 to 2%), the life expectancy of the population decreases due to too few neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Starting simulation from a probability level close to two percent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='Life expectancy of the population ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='cycleofl ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='Aoverage denth ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='240 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='140 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='The probability of changing the stabe of a cell ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='regardless of the number of neighbors (%]Life expectancy of the population ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='Aaverage deeth cycle of the populatior ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='10F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='10~2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='10- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='The probability pf changing the stabe of a cell ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content="regardless of the number of neighbor's (%](probability of changing the state of a cell regardless of the number of neighbors)," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' the average life expectancy of the population begins to rise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' which is caused by the more frequent appearances of living cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Conducting the simulation with a probability level above eighteen percent we observe that the population practically never dies and keeps average cycle life time being at least 1000 or more time iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' 3 Generalization of Stochastic Classical Conway’s Game of Life (SCCGoL) to the case of N competing cellular automata species The created simulation platform enables to create N different cellular automata species that com- petes between themselves for the resources enabling them to replicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' The given automata species has better replication properties within its own community and worse replication properties in case of neighbors being of different species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Such situation is normally encountered in human population when people of one homogenous identity prefer to collaborate more than in the case of people hav- ing much different identity (being from different tribes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Due to that fact soft antagonistic relation between different species is introduced indirectly by means of higher level of tolerance (or effective- ness of replication) towards neighbors of own species than neighbors of other species (other different species are treated in the same way).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' In real way this is an analogy of members collaboration of a given nation or culture within given culture or community versus different culture or community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Let us consider N = 4 (number of different automata species), so we have the following determined formula for number of effective existing neighbors Neeffective,k for given k-th tribes as: Neeffective,k = a1,kNe1 + a2,kNe2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' + ak,kNek + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' + ak,NNeN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' (1) Previously defined replication rules promoting its own species can be formally expressed by following condition (max(as,k) = ak,k and ak,k > as,k if s ̸= k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' In the conducted simulations all tribes have assigned the same value (a1,1 = a2,2 = a3,3 = a4,4 = 1 2) and another same value for (a1,2 = a1,3 = a1,4 = a2,1 = a2,3 = a2,4 = a3,1 = a3,2 = a3,4 = a4,1 = a4,2 = a4,3 = 1 3) what simply means that automata tribes promotes its own tribe and only partly promotes other tribes with no special distinction on which different tribe it is pointing to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Cellular automata species distribution for k-th tribe across two-dimensional lattice is generated with use of a two-dimensional Gaussian distribution given as follows: f(x, y)k = Ake − � (x−x0,k)2 2σ2 x,k + (y−y0,k)2 2σ2 y,k � , (2) where f(x, y)k is a mass function depending on the cell coordinates, (x0,k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' y0,k) the center of the given cellular automaton species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Given parameters A0,k and σx,k, σy,k can control the mass of species and spread across the ”Globe” as depicted in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' In the case of several species on the lattice, SCCGoL simulation results promote the formation of new cells among the species that has the most mass in the neighborhood as in accordance in the Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' During life cycle always newly formed cell has a mass that is more dependent on cells of the same species than other species of its neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' We observed that automata species fight each other in order to spread across the lattice and exclude other species what is an expression of some form of biological Darwinism being a consequence of formula 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' 6: (a) Case of mass distribution of cellular automata of one species using a two-dimensional Gaussian function being isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' (b) Example of initial distribution of three cellular automata tribes with the case of boundary existence between each separate tribe with a rule that one geometrical place on the lattice is occupied by cellular automata of given species with dominant mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' 7: Evolution of map distribution of 4 cellular automata populations of different species with 100 by 100 lattice size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' One can spot effective tendency of each species to occupy maximum possibly ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='territory at the cost of other species in 4-species SCCGoL (SCCGoL4S) what can be understand as ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='occurrence of weak antagonistic relation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='Methodology of description of SCCGoL dynamics by tools of classical ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='statistical physics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='Adding probability to the initially deterministic Game of Life and changing the interaction rules ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='between neighboring automata brought the necessity for a new quantity called mass that has con- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='tinuous real values (other than 0 and 1 present in deterministic Game of Life).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' At this stage vividness of the whole cellular automata population can be understood and approximated by the population total energy (where simply mass is equal to energy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' The level of the automata distribution order is expressed by the entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Entropy was introduced by Rudolf Clausius in 1865 as a thermodynamic state function [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' If the entropy at the initial state and the entropy of the final state are devoted by Si and Sf respectively, then we have: Sf − Si = � f i dE T , dS dE = 1 T (3) Very last formula gives definition of a temperature under assumption that energy and entropy is defined that is the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' We use instead of thermodynamic entropy the Shannon entropy measure given as: SShannon = − � i,j p(xi, yj) log p(xi, yj) = E [− log p(x, y)] (4) In order to calculate the probability in the equation 4, the SCCGoL simulation is repeated several hundred times, so one determines an average population cell mass in successive cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' The calcula- tion of the temperature, which is a measure of thermal state was achieved with an equation 3 with a changed form: T(x, y, t) = dE(x,y,t) dt dSShannon(x,y,t) dt = dE(x, y, t) dSShannon(x, y, t) (5) The temperature defined by last formula can be calculated by two different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' The first method relies on the calculation of the energy and entropy for the entire system, and then numerical calculation of the derivative of energy in function of entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' The second calculation approach is to differentiate mass and entropy with respect to simulation time for each cell and as a result of corresponding ratio we obtain a temperature map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' The example of evolution of mass and entropy for SCCGoL is depicted in Figure 8 and shows maximization and saturation of entropy, what is confirmation of Second Thermodynamic Law that is valid in physical systems, while we are dealing with cellular automata system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' On another hand the mass of a system saturates and increases to certain critical value as given in left part of Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' 8: Evolution of mass and entropy in successive cycles in SCCGoL manifesting maximization with saturation and approaching stationary thermodynamical equilibrium with initial condition depicted in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' 5 Numerical analysis of SCCGoL dynamics with methodology of classical statistical physics Numerical simulations were carried out for 4 topologies cellular automata (case of Figures 9a, 11a, 13a, 15a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' We preinpose such a rule that given alive cell has 20% probability of changing its state to dead state and that initially dead cell has 20% probability of changing its state to alive with a mass choose randomly from an interval value in (0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Still given cell state is depended on its neighbors, since it has 80% probability of changing its state due to state of neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' In order to obtain cellular automata probability map dynamics, simulations average of cellular automata positions was conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Figure 10 shows results that were obtained for a lattice size 100 by 100 with a one cell alive as initial lattice state (case of Figure 9a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' With subsequent cycles, the cells Mass Entropy 25000 800 20000 600 15000 400 10000 200 5000 0 0 0 25 50 75 100 125 150 175 200 0 25 50 75 100 125 150 175 200 Cycle Cycle(a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' 9: Evolution of diffusion process in cellular automata system for a limited lattice size 100 by 100 in SCCGoL (averaged over 1000 trials).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' It shows final saturation of mass and entropy (as depicted in Figure 8) what implies approaching thermodynamical equilibrium with characteristic fluctuations of mass and entropy around effective stationary values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' occupy more and more space on the lattice, which can be seen as increase the mass of the entire system (possible mass creationism is inherent feature of Conway’s Game of Life).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' The sum of the masses of all cells in successive cycles is depicted in Figure 8 with a comparison of the entropy change of the entire system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' As we observe in Figure 10e, high entropy occurs at the edges of the population, which is caused by the entropy wave that meets area with almost zero cell occurrence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Before equilibrium is established, the entropy of the system slightly decreases due to the loss of this extra entropy at the edges as can be seen in the right part of Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Having established those two quantities, we conduct their differentiating with respect to time, and by formula 3 one can establish the whole effective temperature of system by dividing change of mass at given time by change of the entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Figure 10k shows the greater susceptibility of the entropy change to the constraints associated with the limited size of the simulation lattice that is ended with impenetrable walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' The dependence of mass derivative with respect to time simulation (case of Figure 10j) does not have such large oscillations as in the case of entropy derivative with simulation simulation (case of Figure 10k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' The temperature calculated by such procedure gives values just above zero (slightly positive) up to the 50’th cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' We can see a large down peak caused by a slowdown in entropy increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' From about the 75’th cycle, the temperature of the system goes from positive to negative values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' A Figure 10l) describes anomalous thermalization process in SCCGoL with a case of approaching temperature and entropy equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Surprisingly, the thermodynamic equilibrium is achieved for a case for negative temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' A second possible approach in determination the temperature is by use the derivatives of the mass and entropy with respect to time of the individual cells and by obtaining a temperature map of all cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' The temperature depicted in Figure 10i is mostly negative and steady, what corresponds to a situation where mass and entropy have come to equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' We can distinguish two regions in the simulation, the first one with no temperature gradient and zero negative temperature and the second region with non-zero temperature gradient that also include positive temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' The place, where time derivatives of mass and entropy have noticeable values is at the edges of automata population, where we observe slightly positive temperature, which corresponds to the situation in Figure 10l before the 75’th cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' ■(a) Mass at t=4 (b) Mass at t=26 (c) Mass at t=92 (d) Entropy at t=4 (e) Entropy at t=26 (f) Entropy at t=92 (g) T(x,y,t=4) (h) T(x,y,t=26) (i) T(x,y,t=92) (j) dm dt with time (k) dS dt with time (l) Temperature with time Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' 10: Dynamics of thermodynamical parameters (mass, entropy and temperature) with simula- tion time in SCCGoL (lattice size 100 by 100) also given in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Achieved thermodynamical equilibrium is accompanied with final distribution of negative temperature (starting from positive temperature distribution as in accordance with formula 3) as experimentally observed necessary criteria for final thermodynamical stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' One can spot various similarities of statistical behaviour of SCCGoL with physical systems described by classical thermodynamics (maximization and satu- ration of entropy, decay of temperature gradients and final thermalization, uniform distribution of mass and energy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' 0 0.' metadata={'source': 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with respect to time 40 3P/SP 240 0- 200 21 4 140 120 CycleTemperature 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='5 Temperature 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='0 0 21 140 120 Cycle6 Numerical study of one species cellular automata with various boundary conditions Next family of simulations were conducted with use of barriers impenetrable by cellular automata cells via which cellular automata cannot interact with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' We consider a single cellular (a) (b) (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' 11: Diffusion process in SCCGoLp20L100b100 (lattice size 100 by 100, 20% probability of spontaneous change of cell state from life to dead and reversely with preservation of standard rules in Conway’s Game of Life as also given in Figure 5) case of system of two weekly interconnected chambers by means of two small holes in barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Two stages of diffusion can be spotted in mass and entropy dynamics that has two consecutive processes: full diffusion in the left chamber is leading to full diffusion in the right chamber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Further details of thermodynamical parameters space dependence evolution with time are given by Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' automata (single automata seed) placed in empty chamber with impenetrable walls with two small holes that link it to another empty chamber as depicted in the Figure 11a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' We observe a diffusion of cellular automata with simulation time that consists of two main processes: creation and diffusion of cellular automata in the first chamber and diffusion of cellular automata from the first chamber into second chamber accompanied with creation new automata in the second chamber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Those two consequent processes are accompanied by effective slowdown in diffusion that is seen in left part of Figure 11c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' At the same time we observe slowdown in entropy increase as in an accordance to the right part of Figure 11c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Once mass saturation was obtained in the simulation we observe maximization and small drop in entropy that later stabilizes and saturates, as in the right part of Figure 11c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Entropy unlike mass is characterized by a large variation of values in the middle of the population and at the edges of the cellular automata population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' In the situation with barrier (case of Mass Entropy 25000 1200 1000 20000 800 15000 600 10000 400 5000 200 0 0 0 25 50 75 100 125 150 175 200 0 25 50 75 100 125 150 175 200 Cycle Cycle(a) Mass at t=5 (b) Mass at t=70 (c) Mass at t=100 (d) Entropy at t=5 (e) Entropy at t=70 (f) Entropy at t=100 (g) T(x,y,t=5) (h) T(x,y,t=70) (i) T(x,y,t=100) (j) dm dt with time (k) dS dt with time (l) Temperature with time Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' 12: Dynamics of thermodynamical parameters with 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='175 80vass derivative with resoect to time 15 1 dm/dt 0 25 54 S 14D 125 150 175 CycleEntropy derivative with respect to time 500 310 p/Sp 240 10 0- 100 200 0 25 50 75 140 125 150 175 CycleTemperature 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='D 25 2D 15 LD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='0 0 25 50 75 140 125 150 175 CycleFigure 12e) this is particularly evident after the cells pass through the gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' This particular process is due to the logarithm function dependence of Von Neumann entropy, which tends to negative infinity for arguments going to zero from the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Observed criteria of an equilibrium is fact that mass and entropy have steady values and that temperature have negative value in thermodynamical equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Almost always before the equilibrium moment is achieved, time derivatives of mass and entropy have positive values, and still the temperature is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' From the moment the equilibrium is achieved, we are dealing with small fluctuations of entropy and temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' We observe the correlation stating that slight increase in mass results in slight decrease in entropy and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' In an analogical way to the system with one barrier, simulations were conducted on the system with two barriers and the initial structure as depicted in Figure 13a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' We consider a single cellular automata (a) (b) (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' 13: Dynamics of diffusion process for a system SCCGoLp20L100b100 with two barriers with two small holes in each barrier (generalization of situation from Figure 12) creating three weakly interconnected chambers perturbed by mutual interactions mediated by holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Monotonicity in in- crease of entropy is twice shortly interrupted by small decline, what is associated with automata cells ”colliding” with barriers and experiencing short lasting slowing down in its propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Details on space depended thermodynamical parameter evolution with time are given by Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' placed in empty chamber with impenetrable walls with two small holes that link it to the second empty chamber, which is connected to a third empty chamber by impenetrable walls with two small holes as depicted in the Figure 13a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' We observe a diffusion of cellular automata with simulation time that consists of three main processes: creation and diffusion of cellular automata in the first chamber, diffusion of cellular automata from the first chamber into second chamber accompanied with creation new automata in the second chamber and diffusion of cellular automata from the second chamber into third chamber accompanied with creation new automata in the third chamber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Twice cells try Mass Entropy 1200 25000 1000 20000 800 15000 600 10000 400 5000 200 0 0 25 50 75 100 125 150 175 200 0 25 50 75 100 125 150 175 200 Cycle Cycleto reach impenetrable barriers, we observe small drops in entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' After the second time entropy stabilizes and saturates, as in the right part of Figure 13c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' In a system with two barriers, it lasts longer for mass and entropy to reach equilibrium than in the case of a system with only one barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' As depicted in Figure 14k, due to the cells approaching the barriers and losing the extra entropy at the edges of the population, we observe significant fluctuations in the time derivative of the entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' This results in large peaks seen in the Figure 14l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' (a) Mass at t=30 (b) Mass at t=70 (c) Mass at t=110 (d) Entropy at t=30 (e) Entropy at t=70 (f) Entropy at t=110 (g) T(x,y,t=30) (h) T(x,y,t=70) (i) T(x,y,t=110) (j) dm dt with time (k) dS dt with time (l) Temperature with time Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' 14: Space depended dynamics of thermodynamical parameters with simulation time in SCC- GoLp20L100b100 with three weekly interconnected chambers by four small holes also depicted in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='150 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='125 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} 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CycleTemperature t0 Temperature 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='0 0 25 50 75 140 125 150 175 Cycle7 Numerical study of two species cellular automata in perturbative interaction by narrow constriction Further simulations for the case of two species cellular automata were carried out for a system divided into two reservoirs separated by two impenetrable barriers with one small hole (case of Figure 15a) that implies perturbative interaction between tribes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' In the left upper corner of the first part of the system there have been located cells of the first cellular automata tribe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' At a closer distance to the hole in impenetrable wall, but in the second right reservoir there have been located cells of the second cellular automata tribe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' As depicted in Figure 15c, we observe similar final masses and their dynamics in case of both tribes, but in accordance to Figure 15d, different entropy dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Very last is due to the distance of the cellular automata tribes from the small hole in impenetrable wall: a tribe located further from that hole needs more time to propagate and occupy its natural neighborhood and first left chamber of the system, and thus this tribe has a lower probability of taking over the territory of the other tribe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' However the noticeable fact is that mass and entropy of both tribes achieves equilibrium and finally tribes end up in bit same geometrical and dynamical situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' As depicted in Figure 15f, the right tribe closer to the small hole occupies its nearest neighborhood territory more quickly, resulting in an attempt to occupy a rival tribe territory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' As depicted in Figure 16 we observe large oscillations in the time derivatives of mass and entropy of both cellular automata tribes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' In contrast to previously conducted simulations, the temperature of the system after reaching equilibrium is not only characterized by negative values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Tribes existential competition is the reason of occurrence of both positive and negative temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' (a) (b) (c) (d) (e) Mass at t=3 (f) Mass at t=34 (g) Mass at t=200 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' 15: Diffusion process in case of system with two cellular automata tribes weekly interacting with each other via a small hole in double barrier as depicted in (a), where initial configuration is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' After long thermodynamic equilibrium is achieved as given by (b) so two cellular automata tribes coexist in two different geometrical domains effectively geographically separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' In case of both tribes mass and entropy saturates having tendency to oscillate in thermodynamical equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='Mass of the first tribe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='Mass of the second tribe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='140 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='140 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='120 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='120 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='80 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='dt with time for first and second automata tribe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='(b) dS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='dt with time for first and second automata tribe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='(c) Temperature with time for first and second automata tribe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' 16: Dynamics of thermodynamical variables for the case of two competing cellular automata tribes (first tribe is placed on the left and second tribe is placed on the right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='Derivative of mass with resoect to time of the first tribe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='Derivative of mass with resoect to time of the second tribe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='3 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='175 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='Cycle ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='CycleTemperature of the first tribe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='Temperature of the second tribe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='4 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='2 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='0 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='8 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='6 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='4 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='2 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='0 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='2 14 0 25 50 75 100 125 150 175 200 0 25 50 75 100 125 150 175 200 Cycle Cycle8 Conclusions and future perspectives Cellular automata can simulate many complex physical phenomena using the power of simple rules as it was shown in the case of cellular automata diffusion dynamics confirmed by various simulations for one and many automata species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Certain type of automata Darwinism was spotted by studying 4 automata species dynamics as given by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' The SCCGoL dynamics study provides strong evidence that despite the fact that the principle of conservation of mass is not fulfilled, since we have creation- ism and annihilation of automata, the entropy and temperature comes to equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' In conducted various simulations of Stochastic Conway’s Game of Life dynamics we report transition from positive to negative values of temperatures and we are aware that there is maximum level of mass and energy density allowed for cellular automata, since otherwise they would die due to overpopulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' The fact that the temperature can be negative is known in condensed matter physics, but with assumption that the energy is top-bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' In most ”normal” situations this is impossible, but in a rare cases in solid state physics approximately it can be achieved by inverting the population state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Obviously there is a such limitation on top-bounded energy value (mass density value) in Stochastic Conway’s Game of Life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Therefore, it is still consistent with thermodynamics methodology, as it was pointed by professor Adam Bednorz (Faculty of Physics, University of Warsaw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Following conclusions were derived basing on conducted simulations and described methodology: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Identification of thermodynamically defined temperature as proper measure of system evolution with ’-’ sign (case of Figures 10, 12, 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Identification of mass as effective energy of system (in first approximation) (case of Figures 8, 11, 13, 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Identification of Shannon Entropy as effective system entropy (in first approximation) (case of Figures 8, 11, 13, 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Generalization of Stochastic Conway Game of Life of N tribes (approximated analogy to N-body Quantum Physics can be conceptionally drawn) as depicted in Figures 7, 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Confirmation validity of second law of thermodynamics in SCCGoL (entropy maximises and saturates, case of Figures 8, 11, 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Identification of short lasting Shannon entropy peak that later minimizes and saturates in SGoL (case of Figures 8, 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Monotonicity in increase of entropy is twice shortly interrupted by small decline, what is associated with automata cells ”colliding” with barriers and experiencing short lasting slowing down in its propagation (case of Figure 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Conducted analysis of Stochastic Game of Life allows to treat such system as mathematical object well described by methodology of classical statistical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Obtained numerical results by various simulations suggest that we shall introduce another definition of temperature in Stochastic Conway’s Game of Life system by adding ’minus’ sign to temperature known in statistical physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' so we obtain the following formula: TemperatureConway−P omorski−Kotula = −dE dS (6) TemperatureStatisticalP hysics = +dE dS Having such a definition of Conway-Pomorski-Kotula temperature we can use tools of statistical physics in Stochastic Game of Life preserving most classical physics thermodynamical intuition about various situations we can come across.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' There are well-known inherent analogous between classical statistical physics [12][7][2][5][4] and quantum mechanics [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Therefore further research perspectives in study of Stochastic Classical Conway’s Game of Life assume the usage of quantum mechanics being able to simulate classical statistical physics (as expressed by epidemic model or stochastic finite-state machine) as explicitly represented by tight-binding model [9][10] or Schroedinger model directly proposing structures implemented in semiconductor single-electron devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Noticeable var- ious obtained map of probability of Stochastic Conway’s Game of Life can be parameterized by non-linear Schrodinger equation and especially by Ginzburg-Landau model [6], what can be the base for quantization of Stochastic Classical Conway’s Game of Life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' 9 Acknowledgment We would like to express our acknowledgment to professor Adam Bednorz (University of Warsaw), Adam Chochla (Cracow University of Technology) and to doctor Lukasz Stepien (The Pedagogical University in Cracow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' The consultations with them on manuscript has allowed to improve it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' The Authors have no conflict of interests and equally contributed to this work with 50 percent of contri- bution on each side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' First Author proposed methodological and conceptual framework for this work, while Second Author conducted all numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' The interpretations of obtained results is equally assigned to each Author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content=' Baez, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} +page_content='C.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfdgQp/content/2301.03195v1.pdf'} diff --git a/cdE5T4oBgHgl3EQfEw5P/content/tmp_files/2301.05416v1.pdf.txt b/cdE5T4oBgHgl3EQfEw5P/content/tmp_files/2301.05416v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f51d023f006e9140411c71f026f16879d0a3d43e --- /dev/null +++ b/cdE5T4oBgHgl3EQfEw5P/content/tmp_files/2301.05416v1.pdf.txt @@ -0,0 +1,1596 @@ +Two spectral extremal results for graphs with +given order and rank +Xiuqing Li, Xian’an Jin1, Chao Shi, Ruiling Zheng +School of Mathematical Sciences, Xiamen University, +Xiamen, Fujian 361005, P. R. China +E-mails: xiuqingli2021@163.com; xajin@xmu.edu.cn; +cshi@aliyun.com; rlzheng2017@163.com +Abstract +The spectral radius and rank of a graph are defined to be the spectral radius +and rank of its adjacency matrix, respectively. It is an important problem +in spectral extremal graph theory to determine the extremal graph that has +the maximum or minimum spectral radius over certain families of graphs. +Monsalve and Rada [Extremal spectral radius of graphs with rank 4, Linear +Algebra Appl. 609 (2021) 1–11] obtained the extremal graphs with maximum +and minimum spectral radii among all graphs with order n and rank 4. In +this paper, we first determine the extremal graph which attains the maximum +spectral radius among all graphs with any given order n and rank r, and +further determine the extremal graph which attains the minimum spectral +radius among all graphs with order n and rank 5. +Keywords: Rank of graphs; Extremal graphs; Maximum spectral radius; +Minimum spectral radius +1. Introduction +Graphs considered in the paper are all simple, connected and undirected. +Let G = (V (G), E(G)) be a graph. For v ∈ V (G), the degree d(v) is the +cardinality of the neighborhood NG(v) (or N(v) for short) of v in G. Let +A(G) be the adjacency matrix of G. +The characteristic polynomial of a +1Corresponding author +Preprint submitted to Linear Algebra and its Applications +arXiv:2301.05416v1 [math.CO] 13 Jan 2023 + +graph G is the determinantal expansion of xI − A(G), denoted by φ(G, x). +According to the famous Perron-Frobenius theorem, the largest eigenvalue +ρ(G) of A(G) is exactly the spectral radius of G and there is a unique positive +unit eigenvector corresponding to ρ(G), called the principal eigenvector of G. +Let G be a graph with vertex set V (G) = {v1, v2, . . . , vk} and m = +(n1, n2, . . . , nk) be a vector of positive integers. Denote by G ◦ m, the graph +obtained from G by replacing each vertex vi with an independent set Vi with +ni vertices v1 +i , v2 +i , . . . , vni +i +and joining each vertex in Vi with each vertex in +Vj if and only if vivj ∈ E(G). +The resulting graph G ◦ m is said to be +obtained from G by multiplication of vertices by Chang, Huang and Yeh in +[1]. Further, let G be a graph of order k, we define Mn(G) to be the set of +all graphs G ◦ (n1, n2, . . . , nk) with �k +i=1 ni = n. Moreover, for a given set of +graphs {H1, . . . , Hl}, we denote the set �l +i=1 Mn(Hi) by Mn(H1, . . . , Hl). +Let G be a connected graph of order n and R(G) be its rank. Sciriha [4] +proved that R(G) = i if and only if G ∈ Mn(Ki) for i = 2, 3, where Ki is the +complete graph of order i. Chang, Huang and Yeh [1, 5] characterized the +set of all connected graphs with rank 4 and 5, respectively. They obtained +the set of connected graphs of order n and rank 5 is +Mn(G1, G2, . . . , G24), +where the graphs G1, G2, . . . , G24 are shown in Figure 1. +2 + +Figure 1: Reduced graphs of rank 5. +3 + +V1 +5 +15 +V2 +V2 +V2 +G, +G +G3 +V3 +V4 +V4 +V3 +Vs +V4 +V3 +V5 +V +V2 +V +V +G4 +G, +G。 +V +V +V4 +V1 +V3 +V5 +G, +G: +G。 +V3 +V4 +V6 +V5 +V3 +V5 +V1 +V1 +V4 +V6 +V5 +V6 +V +V2 +G10 +V4 +V6 +V3 +VA +V +V2 +V5 +V. +V. +G +G +V +V +V +V. +V. +V. +V +V +G18 +G +V +V3 +V1 +V +V +V +G21 +V5 +V +V6 +V4 +V6 +V5 +V3 +V6 +V3 +V5 +V3 +V, +V8 +V7 +V- +G, +G2 +G,4For a given class of graphs G , there are many results on character- +izing the extramal graphs with maximum and minimum spectral radius +among Mn(G ). For example, in [6], Stevanovi´c, Gutman and Rehman de- +termined the extremal graphs with the maximum and minimum spectral +radii in Mn(Kp). Monsalve and Rada [7] obtained the extremal graphs with +maximum and minimum spectral radii among all connected graphs of or- +der n and rank 4. In the same article, they conjectured that in Mn(Pk), +Pk ◦(1, . . . , 1, ⌊ n−k+2 +2 +⌋, ⌈ n−k+2 +2 +⌉, 1, . . . , 1) and Pk ◦(⌊ n−k+2 +2 +⌋, 1, . . . , 1, ⌈ n−k+2 +2 +⌉) +attain the maximum and minimum spectral radius, respectively, and Ck ◦ +(⌊ n−k+2 +2 +⌋, ⌈ n−k+2 +2 +⌉, 1, . . . , 1) attains the maximum spectral radius in Mn(Ck). +Recently, Lou, Zhai [2] and Sun, Das [3] independently proved the above +conjectures on the extremal graphs with the maximum spectral radius in +Mn(Pk) and Mn(Ck) by using different techniques, and they independently +constructed a class of graphs disproving the conjecture on the minimum spec- +tral radius in Mn(Pk). +The Tur´an graph T(n, r) is the complete r-partite graph on n vertices +where its part sizes are as equal as possible. In this paper, we first determine +the extremal graph that attains the maximum spectral radius with any given +order and rank, and obtain: +Theorem 1.1. T(n, r) is the unique extremal graph that attains the maxi- +mum spectral radius among all graphs of order n and rank r. +However, it seems that it is a difficult task to find the extremal graph +that attains the minimum spectral radius with given order and rank. In this +paper, we focus on graphs with order n and rank 5, and obtain: +Theorem 1.2. The extremal graph that attains the minimum spectral radius +among all connected graphs of order n and rank 5 is: +• G7 = C5, for n = 5; +• G1 ◦ (1, 1, 1, 1, n − 4), for 6 ≤ n ≤ 10; +• G10 ◦ (1, 1, 1, 1, 1, n − 5), for n = 11; +• G10◦(1, 1, 1, 1, k, n−k−4), where k = ⌊ 6n−37−√24n+1 +18 +⌋ or ⌈ 6n−37−√24n+1 +18 +⌉, +for n ≥ 12. +4 + +2. The proof of Theorem 1.1 +We will use the following results to prove Theorem 1.1. +Theorem 2.1. [1] Suppose that G and H are two graphs. If H ∈ Mn(G), +then R(H) = R(G). +Theorem 2.2. [8] Let T(n, r) be the r-partite Tur´an graph of order n. If G +is a Kr+1-free graph of order n, then ρ(G) < ρ(T(n, r)) unless G = T(n, r). +Proof of Theorem 1.1. Let G be a graph of order n and rank r. +We +claim that G is a Kr+1-free graph. Otherwise, since Kr+1 is a subgraph of +G, selecting the rows and columns corresponding to the vertices in Kr+1 can +obtain a nonzero minor of order r + 1 of A(G), i.e., +det +� +� +� +� +� +0 +1 +· · · +1 +1 +0 +· · · +1 +... +... +... +... +1 +1 +· · · +0 +� +� +� +� +� +(r+1)×(r+1) += (−1)r · r ̸= 0. +Therefore, we have R(G) ≥ r + 1, a contradiction. Since T(n, r) = Kr ◦ +(⌈ n +r ⌉, . . . , ⌈ n +r ⌉, ⌊ n +r ⌋, . . . , ⌊ n +r ⌋) ∈ Mn(Kr), by Theorem 2.1, we have R(T(n, r)) = +R(Kr) = r. +By Theorem 2.2, we obtain ρ(G) < ρ(T(n, r)) unless G = +T(n, r). +3. The proof of Theorem 1.2 +In this section, we focus on the extremal graph that has the minimum +spectral radius among all connected graphs of order n and rank 5. We firstly +outline our proof for Theorem 1.2. +Step 1. We first apply a result of Monsalve and Rada in [7] to prove +that the extremal graph with minimum spectral radius belongs to Mn(G1, G7, +G10). +Step 2. Then, using the method of comparing characteristic polynomials, +we characterize the extremal graph with minimum spectral radius in Mn(G1), +Mn(G7) and Mn(G10), respectively. +Step 3. +Next, for n ≥ 12, we compare the spectral radii of these +three types of extremal graphs by some well-known results and obtain that +the extremal graph of order n and rank 5 with minimum spectral radius +5 + +is G10 ◦ (1, 1, 1, 1, k, n − 4 − k) for some integer k. Further, we determine +k ∈ {⌊6n−37−√24n+1 +18 +⌋, ⌈ 6n−37−√24n+1 +18 +⌉}. +Step 4. +Finally, for 5 ≤ n ≤ 11, we obtain the extremal graphs by +calculating directly the spectral radii of the extremal graphs in Mn(G1), +Mn(G7) and Mn(G10), respectively. +3.1. Step 1 +We begin with recalling a well-known result. +Theorem 3.1. [9] If H is a proper subgraph of a connected graph G, then +ρ(H) < ρ(G). +In [7], Theorem 3.1 is used to prove the following results. +Theorem 3.2. [7] Let G be a connected graph with k vertices and m = +(n1, n2, . . . , nk) a vector of positive integers. If v1v2 ∈ E(G), then +ρ((G − v1v2) ◦ m) < ρ(G ◦ m). +Theorem 3.3. [7] Let G be a connected graph with k vertices and m = +(n1, n2, . . . , nk) a vector of positive integers. If vivj /∈ E(G) and N(vi) ⊊ +N(vj), then +ρ(G◦(n1, . . . , ni, . . . , nj, . . . , nk)) < ρ(G◦(n1, . . . , ni −1, . . . , nj +1, . . . , nk)). +By Theorem 3.2, we obtain the following proposition. +Proposition 3.4. Let G be the extremal graph with minimum spectral radius +among all connected graphs of order n and rank 5. Then G ∈ Mn(G1, G7, G10). +Proof. Let m1 = (n1, n2, n3, n4, n5), m2 = (n1, n2, n3, n4, n5, n6), m3 = (n1, n2, +n3, n4, n5, n6, n7) and m4 = (n1, n2, n3, n4, n5, n6, n7, n8) be arbitrary vectors +of positive integers with � +i=1 ni = n. As a consequence of Theorem 3.2, we +have +ρ(G1 ◦ m1) < ρ(Gi ◦ m1), i = 2, 3, 4, 5, 6, 8, +ρ(G10 ◦ m2) < ρ(Gj ◦ m2), j = 11, 12, 13, 14, 15. +6 + +Thus, +G ∈ Mn(G1, G7, G9, G10, G16, G17, G18, G19, G20, G21, G22, G23, G24). +Let H1 = G1 ◦ (1, 1, 1, 1, 2), H2 = G10 ◦ (1, 1, 1, 1, 1, 2), H3 = G10 ◦ +(1, 1, 1, 1, 2, 1) and H4 = G10 ◦ (1, 1, 1, 1, 1, 3), as shown in Figure 2. +Obiviously, +• H1 is the spanning proper subgraph of G9; +• H2 is the spanning proper subgraph of Gi, i ∈ {16, 17, 18, 19, 21, 22}; +• H3 is the spanning proper subgraph of G20; +• H4 is the spanning proper subgraph of Gj, j ∈ {23, 24}. +Therefore, it follows from Theorem 3.2 that +ρ(G1 ◦ m′ +2) = ρ(H1 ◦ m2) < ρ(G9 ◦ m2), +ρ(G10 ◦ m′ +3) = ρ(H2 ◦ m3) < ρ(Gi ◦ m3), i = 16, 17, 18, 19, 21, 22, +ρ(G10 ◦ m′′ +3) = ρ(H3 ◦ m3) < ρ(G20 ◦ m3), +ρ(G10 ◦ m′ +4) = ρ(H4 ◦ m4) < ρ(Gj ◦ m4), j = 23, 24, +where m′ +2 = (n1, n2, n3, n4, n5 + n6), m′ +3 = (n1, n2, n3, n4, n5, n6 + n7), m′′ +3 = +(n1, n2, n3, n4, n5 + n7, n6) and m′ +4 = (n1, n2, n3, n4, n5, n6 + n7 + n8). +Hence, G ∈ Mn(G1, G7, G10). +Figure 2: The graphs Hi, i = 1, 2, 3, 4. +7 + +V +V1 +V +4 +5 +3 +H, +V5 +V3V1 V4V6 +V3 +V1 +V4 +V +V +V +V +2 +H, +H3.2. Step 2 +In this subsection we characterize the extremal graphs with minimum +spectral radii in Mn(G1), Mn(G7) and Mn(G10), respectively. To accomplish +this, let’s introduce some classic results in spectral graph theory. +Definition 3.5. [10] Let A be an n×n real matrix whose rows and columns +are indexed by X = {1, 2, . . . , n}. We partition X into X1, X2, . . . , Xk in +order and rewrite A according to the partition X1, X2, . . . , Xk as follows: +A = +� +� +� +A1,1 +· · · +A1,k +... +... +... +Ak,1 +· · · +Ak,k +� +� +� , +where Ai,j is the block of A formed by rows in Xi and the columns in Xj. Let +bi,j denote the average row sum of Ai,j. Then the matrix B = [bi,j] will be +called the quotient matrix of the partition of A. In particular, when the +row sum of each block Ai,j is constant, the partition is called an equitable +partition. +Theorem 3.6. [10] Let A ≥ 0 be an irreducible square matrix, B be the quo- +tient matrix of an equitable partition of A. Then the spectrum of A contains +the spectrum of B and ρ(A) = ρ(B). +Theorem 3.7. [11] Let G and H be two connected graphs such that φ(H, x) > +φ(G, x) for x ≥ ρ(G). Then ρ(H) < ρ(G). +Theorem 3.8. [9] Let Kn1,n2,...,nk be the complete multipartite graph of order +n. Then +φ(Kn1,n2,...,nk, x) = xn−k(1 − +k +� +i=1 +ni +x + ni +) +k +� +i=1 +(x + ni). +The following Propositions 3.9, 3.10 and 3.11 give the extremal graph +which attains the minimum spectral radius in Mn(G1), Mn(G10) and Mn(G7), +respectively. +Proposition 3.9. The extremal graph in Mn(G1) which attains minimum +spectral radius is of the form +G1 ◦ (1, 1, 1, k, n − k − 3), +8 + +where 1 ≤ k ≤ n−3 +2 . +Proof. Since N(v5) = {v3} ⊊ N(v1) and v1v5 /∈ E(G1), then by Theorem 3.3 +we have +ρ(G1 ◦ (1, n2, n3, n4, n5 + n1 − 1)) ≤ ρ(G1 ◦ (n1, n2, n3, n4, n5)), +with equality if and only if n1 = 1. It follows that the extremal graph in +Mn(G1) which attains minimum spectral radius is of the form F = G1 ◦ +(1, n2, n3, n4, n5). +Then V (F) can be naturally partitioned into 5 parts: +{V1, V2, V3, V4, V5}, +where Vi = {v1 +i , . . . , vni +i }, i = 1, 2, 3, 4, 5. Obviously, this partition of A(F) is +equitable and the corresponding quotient matrix B is +B = +� +� +� +� +� +� +0 +n2 +n3 +n4 +0 +1 +0 +0 +n4 +0 +1 +0 +0 +0 +n5 +1 +n2 +0 +0 +0 +0 +0 +n3 +0 +0 +� +� +� +� +� +� +. +Then the characteristic polynomial of the quotient matrix B is: +φ(B, x) = x5 − (n2 + n3 + n4 + n2n4 + n3n5)x3 − 2n2n4x2+ +(n2n3n4 + n2n3n5 + n3n4n5 + n2n3n4n5)x + 2n2n3n4n5. +Since R(A(F)) = 5, by Theorem 3.6 we have φ(F, x) = xn−5φ(B, x) and +ρ(F) = ρ(A(F)) = ρ(B). +Note that G1 ◦(1, n2, n3, n4, n5) ∼= G1 ◦(1, n4, n3, n2, n5). Therefore, with- +out loss of generality, we suppose that n4 ≥ n2. +Claim 1. n2 = 1. +Assume n2 ≥ 2, let F1 = G1 ◦ (1, n2 − 1, n3, n4 + 1, n5) then +r(x) = φ(F1, x) − φ(F, x) += xn−5(n4 − n2 + 1)(x3 + 2x2 − (n3 + n3n5)x − 2n3n5) += xn−5(n4 − n2 + 1) +� +x(x2 − n3(n5 + 1)) + 2(x2 − n3n5) +� +. +Since n4 ≥ n2, we have n4 − n2 + 1 > 0. It is clear that Kn3,n5+1 is a +9 + +proper subgraph of F, we obtain ρ(F) > ρ(Kn3,n5+1) = +� +n3(n5 + 1), then +r(x) > 0 for x ≥ ρ(F). +Thus, by Theorem 3.7, we have ρ(F1) < ρ(F) which contradicts to the +extremality of F. +Claim 2. n3 = 1. +Now F = G1 ◦ (1, 1, n3, n4, n5), we claim that n5 ≥ n3. If not, let F2 = +G1 ◦ (1, 1, n5, n4, n3), then +r(x) = φ(F2, x) − φ(F, x) = xn−4(x2 − n4)(n3 − n5). +Since n3 > n5, we have n3 − n5 > 0. +It can be seen that Kn4,2 is a +proper subgraph of F, we obtain ρ(F) > ρ(Kn4,2) = √2n4, then r(x) > 0 for +x ≥ ρ(F). +Thus, by Theorem 3.7, we have ρ(F2) < ρ(F), a contradiction. Therefore +n5 ≥ n3. +Next, we assume n3 ≥ 2, let F3 = G1 ◦ (1, 1, n3 − 1, n4, n5 + 1) then +r(x) = φ(F3, x) − φ(F, x) += xn−5 � +(n5 − n3 + 1)(x3 − (2n4 + 1)x − 2n4) + x(x2 − n4) +� +. +Since n5 ≥ n3, we have n5 − n3 + 1 > 0. +It is clear that Kn4,1,1 is +a proper subgraph of F, by Theorem 3.8, we obtain ρ(F) > ρ(Kn4,1,1) = +(√8n4 + 1 + 1)/2, then r(x) > 0 for x ≥ ρ(F). +Thus, by Theorem 3.7, we have ρ(F3) < ρ(F), which contradicts to the +extremality of F. +Claim 3. n5 ≥ n4. +Now F = G1 ◦ (1, 1, 1, n4, n5). Otherwise, let F4 = G1 ◦ (1, 1, 1, n5, n4) +then +r(x) = φ(F4, x) − φ(F, x) = xn−3(x + 2)(n4 − n5). +Since n4 > n5 and ρ(F) > 0, then r(x) > 0 for x ≥ ρ(F). By Theorem +3.7, we have ρ(F4) < ρ(F) which contradicts to the extremality of F, thus +n5 ≥ n4. +From above three claims, we conclude that the extremal graph with min- +imum spectral radius in Mn(G1) is of the form G1 ◦ (1, 1, 1, k, n − k − 3), +where 1 ≤ k ≤ (n − 3)/2. +10 + +Similarly, we characterize the extremal graph with minimum spectral ra- +dius in Mn(G10). +Proposition 3.10. The extremal graph in Mn(G10) which attains minimum +spectral radius is of the form +G10 ◦ (1, 1, 1, 1, k, n − k − 4), +where 1 ≤ k ≤ n−4 +2 . +Proof. By Theorem 3.3, we have +ρ(G10 ◦ (1, n2, n3, n4, n5, n6 + n1 − 1)) ≤ ρ(G10 ◦ (n1, n2, n3, n4, n5, n6)), +with equality if and only if n1 = 1. Thus, we may suppose that the extremal +graph in Mn(G10) which attains minimum spectral radius is of the form +F = G10 ◦ (1, n2, n3, n4, n5, n6). +Similarly, we obtain +B = +� +� +� +� +� +� +� +� +0 +n2 +n3 +n4 +0 +0 +1 +0 +n3 +0 +n5 +0 +1 +n2 +0 +0 +n5 +0 +1 +0 +0 +0 +0 +n6 +0 +n2 +n3 +0 +0 +0 +0 +0 +0 +n4 +0 +0 +� +� +� +� +� +� +� +� +, +is the quotient matrix of an equitable partition of A(F). The characteristic +polynomial of the quotient matrix B is: +φ(B, x) = x(x5 − (n2 + n3 + n4 + n2n3 + n2n5 + n3n5 + n4n6)x3 +− (2n2n3 + 2n2n3n5)x2 + (n2n3n4 + n2n4n5 + n2n4n6 ++ n3n4n5 + n3n4n6 + n2n3n4n6 + n2n4n5n6 + n3n4n5n6)x ++ 2n2n3n4n5 + 2n2n3n4n6 + 2n2n3n4n5n6). +Since R(A(F)) = 5, by Theorem 3.6 we have φ(F, x) = xn−6φ(B, x) and +ρ(F) = ρ(A(F)) = ρ(B). +Note that G10 ◦ (1, n2, n3, n4, n5, n6) ∼= G10 ◦ (1, n3, n2, n4, n5, n6). There- +fore, without loss of generality, we suppose that n3 ≥ n2. +Claim 1. n2 = 1. +11 + +Assume n2 ≥ 2, let F1 = G10 ◦ (1, n2 − 1, n3 + 1, n4, n5, n6) then +r(x) = φ(F1, x) − φ(F, x) += xn−5(n3 − n2 + 1)(x3 + 2(1 + n5)x2 − (n4 + n4n6)x − 2n4n5 +− 2n4n6 − 2n4n5n6) += xn−5(n3 − n2 + 1)(x(x2 − n4(n6 + 1)) + 2n5(x2 − n4(n6 + 1)) ++ 2(x2 − n4n6)). +Since n3 ≥ n2, we have n3 − n2 + 1 > 0. It is clear that Kn4,n6+1 is a +proper subgraph of F, we obtain ρ(F) > ρ(Kn4,n6+1) = +� +n4(n6 + 1), then +r(x) > 0 for x ≥ ρ(F). +Thus, by Theorem 3.7, we have ρ(F1) < ρ(F) which contradicts to the +extremality of F. +Claim 2. n3 = 1. +Now F = G10 ◦ (1, 1, n3, n4, n5, n6), we claim that n5 ≥ n3. If not, let +F2 = G10 ◦ (1, 1, n5, n4, n3, n6), then +r(x) = φ(F2, x) − φ(F, x) = xn−5(n3 − n5)(x2 − n4n6)(x + 2). +Since n3 > n5, we have n3 − n5 > 0. It can be seen that Kn4,n6 is a +proper subgraph of F, we obtain ρ(F) > ρ(Kn4,n6) = √n4n6, then r(x) > 0 +for x ≥ ρ(F). +Thus, by Theorem 3.7, we have ρ(F2) < ρ(F), a contradiction. Therefore +n5 ≥ n3. +Next, we assume n3 ≥ 2, let F3 = G10 ◦ (1, 1, n3 − 1, n4, n5 + 1, n6) then +r(x) = φ(F3, x) − φ(F, x) += xn−5(x + 2) +� +(n5 − n3 + 2)(x2 − n4(n6 + 1)) + n4 +� +. +Since n5 ≥ n3, we have n5 − n3 + 2 > 0. It is clear that Kn4,n6+1 is a +proper subgraph of F, we obtain ρ(F) > ρ(Kn4,n6+1) = +� +n4(n6 + 1), then +r(x) > 0 for x ≥ ρ(F). +Thus, by Theorem 3.7, we have ρ(F3) < ρ(F) which contradicts to the +extremality of F. +Claim 3. n4 = 1. +Now F = G10 ◦ (1, 1, 1, n4, n5, n6), we claim that n6 ≥ n4. If not, let +12 + +F4 = G10 ◦ (1, 1, 1, n6, n5, n4), then +r(x) = φ(F4, x) − φ(F, x) = xn−5(n4 − n6)(x2 − x − 2n5)(x + 1). +Since n4 > n6, we have n4 − n6 > 0. It can be seen that Kn5,1,1 is a +proper subgraph of F, we obtain ρ(F) > ρ(Kn5,1,1) = (√8n5 + 1+1)/2, then +r(x) > 0 for x ≥ ρ(F). +Thus, by Theorem 3.7, we have ρ(F4) < ρ(F), a contradiction. Therefore +n6 ≥ n4. +Next, we assume n4 ≥ 2, let F5 = G10 ◦ (1, 1, 1, n4 − 1, n5, n6 + 1) then +r(x) = φ(F5, x) − φ(F, x) += xn−5(x + 1) +� +(n6 − n4 + 2)(x2 − x − 2n5 − 2) + 2 +� +. +Since n6 ≥ n4, we have n6 − n4 + 2 > 0. It is clear that H ◦ (n5, 1, 1, 1) +is a proper subgraph of F, where H is shown in Figure 3, we obtain ρ(F) > +ρ(H ◦ (n5, 1, 1, 1)) = (√8n5 + 9 + 1)/2, then r(x) > 0 for x ≥ ρ(F). +Thus, by Theorem 3.7, we have ρ(F5) < ρ(F), which contradicts to the +extremality of F. +Figure 3: The graph H. +Claim 4. n6 ≥ n5. +Now F = G10◦(1, 1, 1, 1, n5, n6). Otherwise, let F6 = G10◦(1, 1, 1, 1, n6, n5) +then +r(x) = φ(F6, x) − φ(F, x) = xn−4(x + 1)2(n5 − n6). +Since n5 > n6 and ρ(F) > 0, then r(x) > 0 for x ≥ ρ(F), by Theorem +3.7, we have ρ(F6) < ρ(F) which contradicts to the extremality of F, thus +n6 ≥ n5. +It follows from above four claims that the extremal graph with minimum +spectral radius in Mn(G10) is of the form G10 ◦ (1, 1, 1, 1, k, n − k − 4), where +13 + +H1 ≤ k ≤ (n − 4)/2. +Next we determine the extremal graph with minimum spectral radius in +Mn(G7). +Proposition 3.11. The extremal graph in Mn(G7) which attains minimum +spectral radius is +G7 ◦ (⌈n − 3 +2 +⌉, 1, ⌊n − 3 +2 +⌋, 1, 1). +Proof. Suppose that the extremal graph in Mn(G7) which attains minimum +spectral radius is of the form F = G7 ◦ (n1, n2, n3, n4, n5). Similarlly, we +obtain +B = +� +� +� +� +� +� +0 +n2 +0 +0 +n5 +n1 +0 +n3 +0 +0 +0 +n2 +0 +n4 +0 +0 +0 +n3 +0 +n5 +n1 +0 +0 +n4 +0 +� +� +� +� +� +� +, +is the quotient matrix of an equitable partition of A(F) and the characteristic +polynomial of B is: +φ(B, x) = x5 − (n1n2 + n2n3 + n1n5 + n3n4 + n4n5)x3 + (n1n2n3n4+ +n1n2n3n5 + n1n2n4n5 + n1n3n4n5 + n2n3n4n5)x − 2n1n2n3n4n5. +Since R(A(F)) = 5, by Theorem 3.6 we have φ(F, x) = xn−5φ(B, x) and +ρ(F) = ρ(A(F)) = ρ(B). +Without loss of generality, we may suppose that n1 = max{ni, i = +1, 2, 3, 4, 5} and n2 ≤ n5, then we have the following claims. +Claim 1. n2 ≤ n3 and n5 ≤ n4. +Suppose that n2 > n3. Let F1 = G7 ◦ (n1, n3, n2, n4, n5), then +r(x) = φ(F1, x) − φ(F, x) = xn−2(n1 − n4)(n2 − n3). +Since n1 = max{ni, i = 1, 2, 3, 4, 5}, we have n1 ≥ n4. And if n1 = n4, +then F1 ∼= F. Thus, without loss of generality, we may suppose that n1 > n4. +Since n2 > n3, n1 > n4 and ρ(F) > 0, then r(x) > 0 for x ≥ ρ(F). By +Theorem 3.7, we have ρ(F1) < ρ(F) which contradicts to the extremality of +F. +14 + +Similarly, we obtain n5 ≤ n4. +Claim 2. n4 ≤ n3. +Suppose to the contrary that n4 > n3. Let F2 = G7 ◦ (n1, n2, n4, n3, n5), +then +r(x) = φ(F2, x) − φ(F, x) = xn−2(n2 − n5)(n3 − n4). +Since n5 ≥ n2 and if n5 = n2, then F2 ∼= F. +Thus without loss of +generality we may suppose that n5 > n2. +Since n4 > n3, n5 > n2 and ρ(F) > 0. Then r(x) > 0 for x ≥ ρ(F). By +Theorem 3.7, we have ρ(F2) < ρ(F) which contradicts to the extremality of +F. +From above two claims, we have n1 ≥ n3 ≥ n4 ≥ n5 ≥ n2. Next, we will +prove n2 = n4 = n5 = 1 and n1 − n3 ≤ 1. +Claim 3. n2 = n5 +Assume n2 < n5, let F3 = G7 ◦ (n1 + n5 − n2, n2, n3, n4, n2) then +r(x) = φ(F3, x) − φ(F, x) += xn−5(n5 − n2)((n1 − 2n2 + n4)x3 − (n1 − n2)((n3n4 + n2n3 + n2n4)x +− 2n2n3n4)) +≥ xn−5(n5 − n2)(n1 − n2) +� +x3 − (n3n4 + n2n3 + n2n4)x + 2n2n3n4 +� += xn−5(n5 − n2)(n1 − n2)g(x). +Since n1 ≥ n3 ≥ n4 ≥ n5 > n2 ≥ 1, we have n1 − n2 > 0 and n5 − +n2 > 0. +It is clear that Kn3,n2+n4 is a proper subgraph of F, we obtain +ρ(F) > ρ(Kn3,n2+n4) = +� +n3(n2 + n4). +Since g( +� +n3(n2 + n4)) > 0 and +� +n3(n2 + n4) > +� +(n3(n2 + n4) + n2n4)/3, +where +� +(n3(n2 + n4) + n2n4) /3 is the largest root of g′(x), we have r(x) > 0 +for x ≥ ρ(F). +Thus, by Theorem 3.7, we have ρ(F3) < ρ(F) which contradicts to the +extremality of F. +Note that G7 ◦ (n1, n2, n3, n4, n5) ∼= G7 ◦ (n1, n2, n4, n3, n5) when n2 = n5, +therefore without loss of generality we may suppose that n3 ≥ n4. +Claim 4. n4 = 1. +Assume n4 ≥ 2, let F4 = G7 ◦ (n1, n2, n3 + 1, n4 − 1, n5) then +r(x) = φ(F4, x) − φ(F, x) += xn−5(x − n5)(n3 − n4 + 1)(x2 + n5x − 2n1n5). +15 + +Since n1 ≥ n3 ≥ n4 ≥ n5 = n2, we have n3 − n4 + 1 > 0. It can be +seen that Kn1,2n5 is a proper subgraph of F, we obtain ρ(F) > ρ(Kn1,2n5) = +√2n1n5 > n5, then r(x) > 0 for x ≥ ρ(F). +Thus, by Theorem 3.7, we have ρ(F4) < ρ(F) which contradicts to the +extremality of F, therefore n4 = 1 and hence n2 = n5 = 1. +Claim 5. n1 − n3 ≤ 1. +Now F = G7 ◦ (n1, 1, n3, 1, 1). Assume n1 ≥ n3 + 2, let F5 = G7 ◦ (n1 − +1, 1, n3 + 1, 1, 1) then +r(x) = φ(F5, x) − φ(F, x) = xn−5(3x − 2)(n1 − n3 − 1). +Since n1 ≥ n3 + 2, we have n1 − n3 − 1 > 0. It is clear that Kn1,2 is a +proper subgraph of F, we obtain ρ(F) > ρ(Kn1,2) = √2n1 > 1, then r(x) > 0 +for x ≥ ρ(F). +Thus, by Theorem 3.7, we have ρ(F5) < ρ(F) which contradicts to the +extremality of F, therefore n1 − n3 ≤ 1 and hence n1 = ⌈(n − 3)/2⌉, n3 = +⌊(n − 3)/2⌋. +From above five claims, we obtain G7 ◦ (⌈ n−3 +2 ⌉, 1, ⌊ n−3 +2 ⌋, 1, 1) attains the +minimum spectral radius in Mn(G7). +3.3. Step 3 +We first prove that the extremal graph with minimum spectral radius in +Mn(G1, G7) must be in Mn(G1) by the following lemma. +Lemma 3.12. For n ≥ 8, we have ρ(G1 ◦ (1, 1, 1, ⌊ n−3 +2 ⌋, ⌈ n−3 +2 ⌉)) < ρ(G7 ◦ +(⌈ n−3 +2 ⌉, 1, ⌊ n−3 +2 ⌋, 1, 1)). +Proof. Let F1 = G1◦(1, 1, 1, ⌊ n−3 +2 ⌋, ⌈ n−3 +2 ⌉) and F2 = G7◦(⌈ n−3 +2 ⌉, 1, ⌊ n−3 +2 ⌋, 1, 1). +For 8 ≤ n ≤ 12, we use the MATLAB software to calculate the spectral +radii of Fi for i = 1, 2, as shown in the Table 1. +Table 1: ρ(Fi). +n +ρ(F1) +ρ(F2) +8 +2.7676 +2.9764 +9 +3.1474 +3.2176 +10 +3.1713 +3.4630 +11 +3.5047 +3.6737 +12 +3.5223 +3.8879 +16 + +So let us assume that n ≥ 13. +Case 1. n − 3 = 2k is even. +In this case, F1 = G1 ◦ (1, 1, 1, k, k) and F2 = G7 ◦ (k, 1, k, 1, 1), then +r(x) = φ(F1, x) − φ(F2, x) = xn−5 � +(k − 1)x3 − 2kx2 − k2x + 4k2� +. +It can be seen that K2k,1 is a proper subgraph of F2, we obtain ρ(F2) > +ρ(K2k,1) = +√ +2k. Since n ≥ 13, we have r( +√ +2k) > 0 and +√ +2k > (2k + +k +√ +3k + 1)/3(k − 1). Since (2k + k +√ +3k + 1)/3(k − 1) is the largest root of +r′(x), we obtain r(x) > 0 for x ≥ ρ(F2). +Thus by Theorem 3.7, we have ρ(F1) < ρ(F2). +Case 2. n − 3 = 2k + 1 is odd. +In this case, F1 = G1 ◦ (1, 1, 1, k, k + 1) and F2 = G7 ◦ (k + 1, 1, k, 1, 1), +then +r(x) = φ(F1, x) − φ(F2, x) = xn−5k +� +x3 − 2x2 − (k + 1)x + 4k+ +� +. +It is clear that K2k+1,1 is a proper subgraph of F2, we obtain ρ(F2) > +ρ(K2k+1,1) = +√ +2k + 1. Since n ≥ 13, we have r( +√ +2k + 1) > 0 and +√ +2k + 1 > +(2+ +√ +3k + 7)/3. Since (2+ +√ +3k + 7)/3 is the largest root of r′(x), we obtain +r(x) > 0 for x ≥ ρ(F2). +Thus by Theorem 3.7, we have ρ(F1) < ρ(F2). +Next we prove the extremal graph with minimum spectral radius in +Mn(G1, G10) must be in Mn(G10). We need the following theorem. +Theorem 3.13. [12] Let G be a graph with m edges and n vertices. Then +ρ(G) ≤ √2m − n + 1, with equality if and only if G is isomorphic to the star +K1,n−1 or the complete graph Kn. +Lemma 3.14. Let G1 ◦ (1, 1, 1, k, n − k − 3) be the extremal graph with min- +imum spectral radius in Mn(G1) for n ≥ 12. Then 2 ≤ k ≤ n−3 +2 . +Proof. We denote Fk = G1 ◦ (1, 1, 1, k, n − k − 3) for convenience. By Propo- +sition 3.9, we have 1 ≤ k ≤ (n − 3)/2. +For 12 ≤ n ≤ 18, we use the MATLAB software to calculate the spectral +radii of Fk, as shown in the Table 2, where the minimum spectral radius is +bolded. +17 + +Table 2: ρ(Fk). +n +ρ(Fk) +k +1 +2 +3 +4 +5 +6 +7 +12 +3.0751 +3.0649 +3.2427 +3.5223 +\ +\ +\ +13 +3.2229 +3.1791 +3.2951 +3.5443 +3.8231 +\ +\ +14 +3.3668 +3.3013 +3.3616 +3.5722 +3.8368 +\ +\ +15 +3.5064 +3.4274 +3.4422 +3.6076 +3.8535 +4.1131 +\ +16 +3.6418 +3.5544 +3.5353 +3.6526 +3.8742 +4.1243 +\ +17 +3.7731 +3.6807 +3.6377 +3.7088 +3.8998 +4.1376 +4.3813 +18 +3.9006 +3.8053 +3.7459 +3.7767 +3.9318 +4.1536 +4.3906 +For n ≥ 19, note that F1 = G1◦(1, 1, 1, 1, n−4), F2 = G1◦(1, 1, 1, 2, n−5), +let x be the principal eigenvector of F2 and xi correspond to vertices in Vi +for i = 1, 2, 3, 4, 5. By ρ(F2)x = A(F2)x, we have +ρ(F2)x1 = x2 + x3 + 2x4, +(1) +ρ(F2)x2 = x1 + 2x4, +(2) +ρ(F2)x3 = x1 + (n − 5)x5, +(3) +ρ(F2)x4 = x1 + x2, +(4) +ρ(F2)x5 = x3, +(5) +From (1)-(3), we have +ρ(F2)(x3 − x1 − x2) = x1 + (n − 5)x5 − x2 − x3 − 2x4 − x1 − 2x4, +multiplying ρ(F2) on both sides, by (4) and (5), yields +ρ(F2)2(x3 − x1 − x2) = (n − 5)x3 − ρ(F2)x3 − ρ(F2)x2 − 4(x1 + x2), +then +(ρ(F2)2 − ρ(F2) − 4)(x3 − x1 − x2) = (n − 9 − 2ρ(F2))x3 + ρ(F2)x1. +(6) +Since n ≥ 19 and Kn−5,1 is a proper subgraph of F2, we have ρ(F2) > +ρ(Kn−5,1) = √n − 5 > 3, thus ρ(F2)2 − ρ(F2) − 4 > 0. By Theorem 3.13 +and n ≥ 19, we obtain ρ(F2) < +� +2m(F2) − n + 1 = +� +2(n + 1) − n + 1 = +√n + 3 < (n − 9)/2, therefore n − 9 − 2ρ(F2) > 0. Since x is the principal +eigenvector of F2, we have xi > 0. +Thus, it follows from (6) that x3 − x1 − x2 > 0. +18 + +Now we have +ρ(F1) − ρ(F2) ≥ xTA(F2)x − xTA(F1)x += 2x4x3 − 2x4(x1 + x2) += 2x4(x3 − x1 − x2) > 0. +Therefore, ρ(F1) > ρ(F2), which means k ≥ 2. +Now we prove that ρ(G10◦(1, 1, 1, 1, k−1, n−k−3)) < ρ(G1◦(1, 1, 1, k, n− +k − 3)) for k ≥ 2 and n ≥ 12 by using a well-known operation. +Theorem 3.15. [8] Let v1, v2 be two vertices of a connected graph G and +let {u1, u2, . . . , ut} ⊆ N(v1) \ N(v2). Let G′ be the graph obtained from G by +rotating the edge v1ui to v2ui for i = 1, 2, . . . , t. If xv1 ≤ xv2, where x is the +principal eigenvector of G, then ρ(G′) > ρ(G). +Lemma 3.16. For k ≥ 2 and n ≥ 12, we have ρ(G10 ◦ (1, 1, 1, 1, k − 1, n − +k − 3)) < ρ(G1 ◦ (1, 1, 1, k, n − k − 3)). +Proof. Let F1 = G1 ◦ (1, 1, 1, k, n − k − 3) and F2 = G10 ◦ (1, 1, 1, 1, k − 1, n − +k −3). Let x be the principal eigenvector of F2 and xi correspond to vertices +in Vi for i = 1, 2, 3, 4, 5, 6. +Let us first suppose that x3 ≥ x1, then by Theorem 3.15 we have ρ(F2) < +ρ(F ′), where F ′ is obtained from F2 by rotating the edge v1v4 to v3v4. Since +F ′ ∼= F1, we obtain ρ(F2) < ρ(F1). +Now, suppose that x3 < x1. Since F ′′ ∼= F1, where F ′′ is obtained from +F2 by rotating the edge v3vi +5 to v1vi +5 for i = 1, 2, . . . , k − 1, we have ρ(F2) < +ρ(F ′′) = ρ(F1). +Thus, we complete the proof of the Lemma. +Now we know that the extremal graph of order n and rank 5 with mini- +mum spectral radius is G10 ◦ (1, 1, 1, 1, k, n − 4 − k) for some integer k with +1 ≤ k ≤ n−4 +2 +when n ≥ 12. +For convenience, we set Fn(i) = G10 ◦ (1, 1, 1, 1, i, n − 4 − i) and F = +{Fn(i) : 1 ≤ i ≤ n−4 +2 }. It is only remained to find the extremal graph with +minimum spectral radius in F. +19 + +Theorem 3.17. [10] Let A be an n×n nonnegative matrix. Then the largest +eigenvalue ρ(A) ≥ xTAx for any unit vector x, with equality if and only if +Ax = ρ(A)x. +Lemma 3.18. Let α = 6n−37−√24n+1 +18 +and n ≥ 12. Then for 1 ≤ i ≤ n−4 +2 , we +have +ρ(Fn(i)) > min{ρ(Fn(⌊α⌋)), ρ(Fn(⌈α⌉))} +unless i = ⌊α⌋ or ⌈α⌉. +Proof. Let ρi = ρ(Fn(i)). Our aim is to prove that ρi < ρi−1 if 2 ≤ i ≤ ⌊α⌋ +and ρi < ρi+1 if ⌈α⌉ ≤ i ≤ n−6 +2 . +Let xi be the principal eigenvector of Fn(i) and xi +j correspond to vertices +in Vj for j = 1, 2, 3, 4, 5, 6. Then by ρixi = A(Fn(i))xi we have +ρixi +1 = xi +2 + xi +3 + xi +4, +(7) +ρixi +2 = xi +1 + xi +3 + ixi +5, +(8) +ρixi +3 = xi +1 + xi +2 + ixi +5, +(9) +ρixi +4 = xi +1 + (n − i − 4)xi +6, +(10) +ρixi +5 = xi +2 + xi +3, +(11) +ρixi +6 = xi +4, +(12) +By (8) and (9), we have +ρi(xi +2 − xi +3) = xi +3 − xi +2, i.e., +(ρi + 1)(xi +2 − xi +3) = 0, +which implies that +xi +2 = xi +3. +(13) +20 + +Therefore, by (7) and (10)-(13), we have +xi +5 = 2xi +2 +ρi += xi +1 − xi +4 +ρi += xi +1 − xi +6 += ρixi +4 − (n − i − 4)xi +6 − xi +6 += ρ2 +i xi +6 − (n − i − 4)xi +6 − xi +6 += (ρ2 +i − n + i + 3)xi +6, +and from (7)-(8) and (11)-(13), we have +xi +6 = xi +4 +ρi += xi +1 − 2xi +2 +ρi += xi +1 − xi +5 += (ρi − 1)xi +2 − ixi +5 − xi +5 += ρi(ρi − 1) +2 +xi +5 − ixi +5 − xi +5 += 1 +2(ρ2 +i − ρi − 2i − 2)xi +5. +Hence, we obtain that +� +� +� +� +� +(ρ2 +i − n + i + 3)(ρ2 +i − ρi − 2i − 2) = 2, +ρ2 +i − n + i + 3 > 0, +ρ2 +i − ρi − 2i − 2 > 0. +(14) +Note that, if we let +� +ρ2 +i − n + i + 3 = 1, +ρ2 +i − ρi − 2i − 2 = 2, +then we have +� +ρi = √n − i − 2, +ρi = 1+√8i+17 +2 +. +By calculation, we can find that i = α = (6n−37−√24n + 1)/18 is the only +21 + +solution of √n − i − 2 = (1 + √8i + 17)/2. Since i ∈ N, we will complete +the proof by classifying the value of i. +Case 1. If 2 ≤ i ≤ ⌊α⌋. +We have √n − i − 2 ≥ (1+√8i + 17)/2. We claim that (1+√8i + 17)/2 ≤ +ρi ≤ √n − i − 2. Suopose that ρi < (1 + √8i + 17)/2. By (14), we have +0 < ρ2 +i − n + i + 3 < 1 and 0 < ρ2 +i − ρi − 2i − 2 < 2. Then (ρ2 +i − n + i + +3)(ρ2 +i − ρi − 2i − 2) < 2, a contradiction. Suopose that ρi > √n − i − 2. By +(14), we obtain that ρ2 +i − n + i + 3 > 1 and ρ2 +i − ρi − 2i − 2 > 2. Then +(ρ2 +i − n + i + 3)(ρ2 +i − ρi − 2i − 2) > 2, a contradiction. +Thus we have (1 + √8i + 17)/2 ≤ ρi ≤ √n − i − 2. This induces that +ρ2 +i − n + i + 3 ≤ 1 and ρ2 +i − ρi − 2i − 2 ≥ 2, which lead to xi +6 ≥ xi +5. Therefore +ρi−1 − ρi +≥xT +i A(Fn(i − 1))xi − xT +i A(Fn(i))xi +=2xi +5(xi +4 − xi +2 − xi +3) +=2ρixi +5(xi +6 − xi +5) +≥0. +(15) +Now we only need to prove ρi−1 ̸= ρi. +Suppose that ρi−1 = ρi, then +ρi−1 = xT +i A(Fn(i − 1))xi. By Theorem 3.17, we have +ρi−1xi +4 = xi +1 + (n − i − 4)xi +6 + xi +5, +and since +ρixi +4 = xi +1 + (n − i − 4)xi +6, +we obtain 0 = (ρi−1 − ρi)xi +4 = xi +5, which contradicts to the definition of the +principal eigenvector. +Therefore, from (15) we have ρi−1 > ρi for 2 ≤ i ≤ ⌊α⌋. +Case 2. If ⌈α⌉ ≤ i ≤ n−6 +2 . +We have √n − i − 2 ≤ (1 + √8i + 17)/2. Similarly, by (14), we conclude +that √n − i − 2 ≤ ρi ≤ (1+√8i + 17)/2. This induces that ρ2 +i −n+i+3 ≥ 1 +22 + +and ρ2 +i − ρi − 2i − 2 ≤ 2, which lead to xi +5 ≥ xi +6, therefore +ρi+1 − ρi +≥xT +i A(Fn(i + 1))xi − xT +i A(Fn(i))xi +=2xi +6(xi +2 + xi +3 − xi +4) +=2ρixi +6(xi +5 − xi +6) +≥0. +(16) +Similarly, we have ρi+1 ̸= ρi. Using this, from (16), we obtain ρi+1 > ρi +for ⌈α⌉ ≤ i ≤ n−6 +2 . +Therefore, the proof of Lemma is completed. +3.4. Step 4 +It only remains for the case that 5 ≤ n ≤ 11. Applying Proposition 3.9, +3.10 and 3.11, we obtain the extremal graphs with minimum spectral radius +in Mn(G1), Mn(G7) and Mn(G10), respectively. +And then calculate their +spectral radii by using MATLAB, as shown in Table 3, where the extremal +graphs and the minimum spectral radii are bolded. +Table 3: The extremal graph with minimum spectral radius in Mn(G1), Mn(G7), Mn(G10). +n +Mn(G1) +Mn(G7) +Mn(G10) +Extremal graph +Spectral radius +Extremal graph +Spectral radius +Extremal graph +Spectral radius +5 +G1 ◦ (1, 1, 1, 1, 1) +2.2143 +G7 ◦ (1, 1, 1, 1, 1) +2.0000 +\ +\ +6 G1 ◦ (1, 1, 1, 1, 2) +2.2784 +G7 ◦ (2, 1, 1, 1, 1) +2.3912 +G10 ◦ (1, 1, 1, 1, 1, 1) +2.6544 +7 G1 ◦ (1, 1, 1, 1, 3) +2.3686 +G7 ◦ (2, 1, 2, 1, 1) +2.6813 +G10 ◦ (1, 1, 1, 1, 1, 2) +2.6751 +8 G1 ◦ (1, 1, 1, 1, 4) +2.4860 +G7 ◦ (3, 1, 2, 1, 1) +2.9764 +G10 ◦ (1, 1, 1, 1, 1, 3) +2.7033 +9 G1 ◦ (1, 1, 1, 1, 5) +2.6239 +G7 ◦ (3, 1, 3, 1, 1) +3.2176 +G10 ◦ (1, 1, 1, 1, 1, 4) +2.7448 +10 G1 ◦ (1, 1, 1, 1, 6) +2.7724 +G7 ◦ (4, 1, 3, 1, 1) +3.4630 +G10 ◦ (1, 1, 1, 1, 1, 5) +2.8060 +11 +G1 ◦ (1, 1, 1, 1, 7) +2.9243 +G7 ◦ (4, 1, 4, 1, 1) +3.6737 +G10 ◦ (1, 1, 1, 1, 1, 6) +2.8915 +By Table 4, we obtain that when 5 ≤ n ≤ 11, the extremal graph with +minimum spectral radius of rank 5 is: +• G7 = C5, for n = 5; +• G1 ◦ (1, 1, 1, 1, n − 4), for 6 ≤ n ≤ 10; +• G10 ◦ (1, 1, 1, 1, 1, n − 5), for n = 11. +23 + +4. Concluding remarks +In the last case of Theorem 1.2, we obtain that k ∈ {⌊ 6n−37−√24n+1 +18 +⌋ +, ⌈ 6n−37−√24n+1 +18 +⌉}. When 12 ≤ n ≤ 23, we use the MATLAB software to +calculate the spectral radii of the graphs in F = {Fn(i) : 1 ≤ i ≤ +n−4 +2 }, +as shown in the Table 4, where the minimum spectral radius is bolded. It +demonstrates that k = ⌊ 6n−37−√24n+1 +18 +⌋ or ⌈ 6n−37−√24n+1 +18 +⌉ depends on n. +Table 4: ρ(Fn(i)). +n +ρ(Fn(i)) i +1 +2 +3 +4 +5 +6 +7 +8 +9 +6n−37−√24n+1 +18 +12 +3 +3.1370 +3.4319 +3.7362 +\ +\ +\ +\ +\ +1 +13 +3.1239 +3.1818 +3.4431 +3.7404 +\ +\ +\ +\ +\ +1.2949 +14 +3.255 +3.2470 +3.4588 +3.7457 +4.0278 +\ +\ +\ +\ +1.5912 +15 +3.3894 +3.3347 +3.4817 +3.7525 +4.0308 +\ +\ +\ +\ +1.8889 +16 +3.5227 +3.4402 +3.5160 +3.7616 +4.0344 +4.2979 +\ +\ +\ +2.1877 +17 +3.6539 +3.5563 +3.5674 +3.7743 +4.0389 +4.3001 +\ +\ +\ +2.4876 +18 +3.7824 +3.6770 +3.6394 +3.7926 +4.0446 +4.3027 +4.5506 +\ +\ +2.7884 +19 +3.9079 +3.7889 +3.7303 +3.8199 +4.0523 +4.3058 +4.5522 +\ +\ +3.0901 +20 +4.0303 +3.9201 +3.8338 +3.8612 +4.0628 +4.3097 +4.5542 +4.7888 +\ +3.3927 +21 +4.1498 +4.0396 +3.9439 +3.9211 +4.0779 +4.3147 +4.5565 +4.7900 +\ +3.6960 +22 +4.2663 +4.1570 +4.0564 +4 +4.1002 +4.3213 +4.5593 +4.7915 +5.0146 +4 +23 +4.3801 +4.2721 +4.1694 +4.0929 +4.1341 +4.3303 +4.5627 +4.7933 +5.0157 +4.3047 +It is a natural problem to determine the extramal spectral radii of the +graphs of order n and rank r. By Theorem 1.1, we know that the maximum +spectral radius of all connected graphs of order n and rank r is ρ(T(n, r)). +Feng et al. gave the spectral radius of T(n, r) in [13]. +Theorem 4.1. [13] Let T(n, r) be a Tur´an graph. Then +ρ(T(n, r)) = 1 +2 +� +n − 2⌊n +r ⌋ − 1 + +� +(n + 1)2 − 4(n − r⌊n +r ⌋)⌈n +r ⌉ +� +≤ n − ⌊n +r ⌋ +with the last equality if and only if T(n, r) is regular. +Further, we obtain a sharp upper and lower bound for the spectral radius +of the extremal graph G which attains the minimum spectral radius among +all connected graphs of order n ≥ 12 and rank 5. By Theorem 1.2, we know +that +ρ(G) = min {ρ(Fn(⌊α⌋)), ρ(Fn(⌈α⌉))}, +24 + +where α = 6n−37−√24n+1 +18 +. +From the proof of Lemma 3.18, we have +1 + +� +8⌊α⌋ + 17 +2 +≤ ρ(Fn(⌊α⌋) ≤ +� +n − ⌊α⌋ − 2, +� +n − ⌈α⌉ − 2 ≤ ρ(Fn(⌈α⌉) ≤ 1 + +� +8⌈α⌉ + 17 +2 +. +Therefore, we obtain that +ρ(G) ≥ min{1 + +� +8⌊α⌋ + 17 +2 +, +� +n − ⌈α⌉ − 2}, +and +ρ(G) ≤ min{ +� +n − ⌊α⌋ − 2, 1 + +� +8⌈α⌉ + 17 +2 +}. +In general, the problem of determining the minimum spectral radius of all +connected graphs with order n and rank r deserves further study. +Declaration of compting interest +There is no competing interest. +Acknowledgement +This research is supported by the National Natural Science Foundation +of China [Grant number, 12171402]. +References +[1] G. Chang, L. Huang, H.G. Yeh, A characterization of graphs with rank +4, Linear Algebra Appl. 434 (2011) 1793–1798. +[2] Z.Z. Lou, M.Q. Zhai, Proof of a conjecture on extremal spectral radii of +blow-up graphs, Linear Algebra Appl. 617 (2021) 168–178. +[3] S.W. Sun, K.C. Das, Proof and disproof of conjectures on spectral radii +of coclique extension of cycles and paths, Linear Algebra Appl. 618 +(2021) 1–11 +25 + +[4] I. Sciriha, On the rank of graphs, in: Y. Alavi, D.R. Lick, A. Schwenk +(Eds.), Combinatorics, Graph Theory, and Algorithms, vol. II, New Is- +sue Press, Western Michigan University, Kalamazoo, Michigan, 1999, +pp. 769–778. +[5] G.J. Chang, L.H. Huang, H.G. Yeh, A characterization of graphs with +rank 5, Linear Algebra Appl. 436 (2012) 4241–4250. +[6] D. Stevanovi´c, I. Gutman, M.U. Rehman, On spectral radius and energy +of complete multipartite graphs, Ars Math. Contemp. 9 (2015) 109–113. +[7] J. Monsalve, J. Rada, Extremal spectral radius of graphs with rank 4, +Linear Algebra Appl. 609 (2021) 1–11. +[8] V. Nikiforov, Bounds on graph eigenvalues II, Linear Algebra Appl. 427 +(2007) 183–189. +[9] D. Stevanovi´c, Spectral Radius of Graphs, Academic Press, Amsterdam, +2015. +[10] A. Brouwer, W. Haemers, Spectra of Graphs, Springer, New York, 2012. +[11] Q. Li, K.Q. Feng, On the largest eigenvalue of graphs, Acta Math. Appl. +Sinica. 2 (1979) 167–175 (in Chinese). +[12] Y. Hong, A bound on the spectral radius of graphs, Linear Algebra +Appl. 108 (1988) 135–139. +[13] L.H. Feng, Q. Li, X.D. Zhang, Spectral radii of graphs with given chro- +matic number, Appl. Math. Lett. 20 (2007) 158–162. +26 + diff --git a/cdE5T4oBgHgl3EQfEw5P/content/tmp_files/load_file.txt b/cdE5T4oBgHgl3EQfEw5P/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3a4fe18a0ca754eac3169837a53ae4b4ddeeb6eb --- /dev/null +++ b/cdE5T4oBgHgl3EQfEw5P/content/tmp_files/load_file.txt @@ -0,0 +1,809 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf,len=808 +page_content='Two spectral extremal results for graphs with given order and rank Xiuqing Li, Xian’an Jin1, Chao Shi, Ruiling Zheng School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' China E-mails: xiuqingli2021@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' xajin@xmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' cshi@aliyun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' rlzheng2017@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='com Abstract The spectral radius and rank of a graph are defined to be the spectral radius and rank of its adjacency matrix, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' It is an important problem in spectral extremal graph theory to determine the extremal graph that has the maximum or minimum spectral radius over certain families of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Monsalve and Rada [Extremal spectral radius of graphs with rank 4, Linear Algebra Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' 609 (2021) 1–11] obtained the extremal graphs with maximum and minimum spectral radii among all graphs with order n and rank 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' In this paper, we first determine the extremal graph which attains the maximum spectral radius among all graphs with any given order n and rank r, and further determine the extremal graph which attains the minimum spectral radius among all graphs with order n and rank 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Keywords: Rank of graphs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Extremal graphs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Maximum spectral radius;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Minimum spectral radius 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Introduction Graphs considered in the paper are all simple, connected and undirected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Let G = (V (G), E(G)) be a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' For v ∈ V (G), the degree d(v) is the cardinality of the neighborhood NG(v) (or N(v) for short) of v in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Let A(G) be the adjacency matrix of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' The characteristic polynomial of a 1Corresponding author Preprint submitted to Linear Algebra and its Applications arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='05416v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='CO] 13 Jan 2023 graph G is the determinantal expansion of xI − A(G), denoted by φ(G, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' According to the famous Perron-Frobenius theorem, the largest eigenvalue ρ(G) of A(G) is exactly the spectral radius of G and there is a unique positive unit eigenvector corresponding to ρ(G), called the principal eigenvector of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Let G be a graph with vertex set V (G) = {v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' , vk} and m = (n1, n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' , nk) be a vector of positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Denote by G ◦ m, the graph obtained from G by replacing each vertex vi with an independent set Vi with ni vertices v1 i , v2 i , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' , vni i and joining each vertex in Vi with each vertex in Vj if and only if vivj ∈ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' The resulting graph G ◦ m is said to be obtained from G by multiplication of vertices by Chang, Huang and Yeh in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Further, let G be a graph of order k, we define Mn(G) to be the set of all graphs G ◦ (n1, n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' , nk) with �k i=1 ni = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Moreover, for a given set of graphs {H1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' , Hl}, we denote the set �l i=1 Mn(Hi) by Mn(H1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' , Hl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Let G be a connected graph of order n and R(G) be its rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Sciriha [4] proved that R(G) = i if and only if G ∈ Mn(Ki) for i = 2, 3, where Ki is the complete graph of order i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Chang, Huang and Yeh [1, 5] characterized the set of all connected graphs with rank 4 and 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' They obtained the set of connected graphs of order n and rank 5 is Mn(G1, G2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' , G24), where the graphs G1, G2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' , G24 are shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' 2 Figure 1: Reduced graphs of rank 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' 3 V1 5 15 V2 V2 V2 G, G G3 V3 V4 V4 V3 Vs V4 V3 V5 V V2 V V G4 G, G。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' V V V4 V1 V3 V5 G, G: G。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' V3 V4 V6 V5 V3 V5 V1 V1 V4 V6 V5 V6 V V2 G10 V4 V6 V3 VA V V2 V5 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' G G V V V V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' V V G18 G V V3 V1 V V V G21 V5 V V6 V4 V6 V5 V3 V6 V3 V5 V3 V, V8 V7 V- G, G2 G,4For a given class of graphs G , there are many results on character- izing the extramal graphs with maximum and minimum spectral radius among Mn(G ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' For example, in [6], Stevanovi´c, Gutman and Rehman de- termined the extremal graphs with the maximum and minimum spectral radii in Mn(Kp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Monsalve and Rada [7] obtained the extremal graphs with maximum and minimum spectral radii among all connected graphs of or- der n and rank 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' In the same article, they conjectured that in Mn(Pk), Pk ◦(1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' , 1, ⌊ n−k+2 2 ⌋, ⌈ n−k+2 2 ⌉, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' , 1) and Pk ◦(⌊ n−k+2 2 ⌋, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' , 1, ⌈ n−k+2 2 ⌉) attain the maximum and minimum spectral radius, respectively, and Ck ◦ (⌊ n−k+2 2 ⌋, ⌈ n−k+2 2 ⌉, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' , 1) attains the maximum spectral radius in Mn(Ck).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Recently, Lou, Zhai [2] and Sun, Das [3] independently proved the above conjectures on the extremal graphs with the maximum spectral radius in Mn(Pk) and Mn(Ck) by using different techniques, and they independently constructed a class of graphs disproving the conjecture on the minimum spec- tral radius in Mn(Pk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' The Tur´an graph T(n, r) is the complete r-partite graph on n vertices where its part sizes are as equal as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' In this paper, we first determine the extremal graph that attains the maximum spectral radius with any given order and rank, and obtain: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' T(n, r) is the unique extremal graph that attains the maxi- mum spectral radius among all graphs of order n and rank r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' However, it seems that it is a difficult task to find the extremal graph that attains the minimum spectral radius with given order and rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' In this paper, we focus on graphs with order n and rank 5, and obtain: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' The extremal graph that attains the minimum spectral radius among all connected graphs of order n and rank 5 is: G7 = C5, for n = 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' G1 ◦ (1, 1, 1, 1, n − 4), for 6 ≤ n ≤ 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' G10 ◦ (1, 1, 1, 1, 1, n − 5), for n = 11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' G10◦(1, 1, 1, 1, k, n−k−4), where k = ⌊ 6n−37−√24n+1 18 ⌋ or ⌈ 6n−37−√24n+1 18 ⌉, for n ≥ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='1 We will use the following results to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' [1] Suppose that G and H are two graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' If H ∈ Mn(G), then R(H) = R(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' [8] Let T(n, r) be the r-partite Tur´an graph of order n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' If G is a Kr+1-free graph of order n, then ρ(G) < ρ(T(n, r)) unless G = T(n, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Let G be a graph of order n and rank r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' We claim that G is a Kr+1-free graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Otherwise, since Kr+1 is a subgraph of G, selecting the rows and columns corresponding to the vertices in Kr+1 can obtain a nonzero minor of order r + 1 of A(G), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=', det � � � � � 0 1 · · 1 1 0 · · 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' 1 1 · · 0 � � � � � (r+1)×(r+1) = (−1)r · r ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Therefore, we have R(G) ≥ r + 1, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Since T(n, r) = Kr ◦ (⌈ n r ⌉, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' , ⌈ n r ⌉, ⌊ n r ⌋, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' , ⌊ n r ⌋) ∈ Mn(Kr), by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='1, we have R(T(n, r)) = R(Kr) = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='2, we obtain ρ(G) < ρ(T(n, r)) unless G = T(n, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='2 In this section, we focus on the extremal graph that has the minimum spectral radius among all connected graphs of order n and rank 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' We firstly outline our proof for Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' We first apply a result of Monsalve and Rada in [7] to prove that the extremal graph with minimum spectral radius belongs to Mn(G1, G7, G10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Then, using the method of comparing characteristic polynomials, we characterize the extremal graph with minimum spectral radius in Mn(G1), Mn(G7) and Mn(G10), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Next, for n ≥ 12, we compare the spectral radii of these three types of extremal graphs by some well-known results and obtain that the extremal graph of order n and rank 5 with minimum spectral radius 5 is G10 ◦ (1, 1, 1, 1, k, n − 4 − k) for some integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Further, we determine k ∈ {⌊6n−37−√24n+1 18 ⌋, ⌈ 6n−37−√24n+1 18 ⌉}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Finally, for 5 ≤ n ≤ 11, we obtain the extremal graphs by calculating directly the spectral radii of the extremal graphs in Mn(G1), Mn(G7) and Mn(G10), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Step 1 We begin with recalling a well-known result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' [9] If H is a proper subgraph of a connected graph G, then ρ(H) < ρ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' In [7], Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='1 is used to prove the following results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' [7] Let G be a connected graph with k vertices and m = (n1, n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' , nk) a vector of positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' If v1v2 ∈ E(G), then ρ((G − v1v2) ◦ m) < ρ(G ◦ m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' [7] Let G be a connected graph with k vertices and m = (n1, n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' , nk) a vector of positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' If vivj /∈ E(G) and N(vi) ⊊ N(vj), then ρ(G◦(n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' , ni, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' , nj, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' , nk)) < ρ(G◦(n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' , ni −1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' , nj +1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' , nk)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='2, we obtain the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Let G be the extremal graph with minimum spectral radius among all connected graphs of order n and rank 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Then G ∈ Mn(G1, G7, G10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Let m1 = (n1, n2, n3, n4, n5), m2 = (n1, n2, n3, n4, n5, n6), m3 = (n1, n2, n3, n4, n5, n6, n7) and m4 = (n1, n2, n3, n4, n5, n6, n7, n8) be arbitrary vectors of positive integers with � i=1 ni = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' As a consequence of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='2, we have ρ(G1 ◦ m1) < ρ(Gi ◦ m1), i = 2, 3, 4, 5, 6, 8, ρ(G10 ◦ m2) < ρ(Gj ◦ m2), j = 11, 12, 13, 14, 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' 6 Thus, G ∈ Mn(G1, G7, G9, G10, G16, G17, G18, G19, G20, G21, G22, G23, G24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Let H1 = G1 ◦ (1, 1, 1, 1, 2), H2 = G10 ◦ (1, 1, 1, 1, 1, 2), H3 = G10 ◦ (1, 1, 1, 1, 2, 1) and H4 = G10 ◦ (1, 1, 1, 1, 1, 3), as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Obiviously, H1 is the spanning proper subgraph of G9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' H2 is the spanning proper subgraph of Gi, i ∈ {16, 17, 18, 19, 21, 22};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' H3 is the spanning proper subgraph of G20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' H4 is the spanning proper subgraph of Gj, j ∈ {23, 24}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Therefore, it follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='2 that ρ(G1 ◦ m′ 2) = ρ(H1 ◦ m2) < ρ(G9 ◦ m2), ρ(G10 ◦ m′ 3) = ρ(H2 ◦ m3) < ρ(Gi ◦ m3), i = 16, 17, 18, 19, 21, 22, ρ(G10 ◦ m′′ 3) = ρ(H3 ◦ m3) < ρ(G20 ◦ m3), ρ(G10 ◦ m′ 4) = ρ(H4 ◦ m4) < ρ(Gj ◦ m4), j = 23, 24, where m′ 2 = (n1, n2, n3, n4, n5 + n6), m′ 3 = (n1, n2, n3, n4, n5, n6 + n7), m′′ 3 = (n1, n2, n3, n4, n5 + n7, n6) and m′ 4 = (n1, n2, n3, n4, n5, n6 + n7 + n8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Hence, G ∈ Mn(G1, G7, G10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Figure 2: The graphs Hi, i = 1, 2, 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' 7 V V1 V 4 5 3 H, V5 V3V1 V4V6 V3 V1 V4 V V V V 2 H, H3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Step 2 In this subsection we characterize the extremal graphs with minimum spectral radii in Mn(G1), Mn(G7) and Mn(G10), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' To accomplish this, let’s introduce some classic results in spectral graph theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' [10] Let A be an n×n real matrix whose rows and columns are indexed by X = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' We partition X into X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' , Xk in order and rewrite A according to the partition X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' , Xk as follows: A = � � � A1,1 · · A1,k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Ak,1 · · Ak,k � � � , where Ai,j is the block of A formed by rows in Xi and the columns in Xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Let bi,j denote the average row sum of Ai,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Then the matrix B = [bi,j] will be called the quotient matrix of the partition of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' In particular, when the row sum of each block Ai,j is constant, the partition is called an equitable partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' [10] Let A ≥ 0 be an irreducible square matrix, B be the quo- tient matrix of an equitable partition of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Then the spectrum of A contains the spectrum of B and ρ(A) = ρ(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' [11] Let G and H be two connected graphs such that φ(H, x) > φ(G, x) for x ≥ ρ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Then ρ(H) < ρ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' [9] Let Kn1,n2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=',nk be the complete multipartite graph of order n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Then φ(Kn1,n2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=',nk, x) = xn−k(1 − k � i=1 ni x + ni ) k � i=1 (x + ni).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' The following Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='9, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='10 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='11 give the extremal graph which attains the minimum spectral radius in Mn(G1), Mn(G10) and Mn(G7), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' The extremal graph in Mn(G1) which attains minimum spectral radius is of the form G1 ◦ (1, 1, 1, k, n − k − 3), 8 where 1 ≤ k ≤ n−3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Since N(v5) = {v3} ⊊ N(v1) and v1v5 /∈ E(G1), then by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='3 we have ρ(G1 ◦ (1, n2, n3, n4, n5 + n1 − 1)) ≤ ρ(G1 ◦ (n1, n2, n3, n4, n5)), with equality if and only if n1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' It follows that the extremal graph in Mn(G1) which attains minimum spectral radius is of the form F = G1 ◦ (1, n2, n3, n4, n5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Then V (F) can be naturally partitioned into 5 parts: {V1, V2, V3, V4, V5}, where Vi = {v1 i , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' , vni i }, i = 1, 2, 3, 4, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Obviously, this partition of A(F) is equitable and the corresponding quotient matrix B is B = � � � � � � 0 n2 n3 n4 0 1 0 0 n4 0 1 0 0 0 n5 1 n2 0 0 0 0 0 n3 0 0 � � � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Then the characteristic polynomial of the quotient matrix B is: φ(B, x) = x5 − (n2 + n3 + n4 + n2n4 + n3n5)x3 − 2n2n4x2+ (n2n3n4 + n2n3n5 + n3n4n5 + n2n3n4n5)x + 2n2n3n4n5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Since R(A(F)) = 5, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='6 we have φ(F, x) = xn−5φ(B, x) and ρ(F) = ρ(A(F)) = ρ(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Note that G1 ◦(1, n2, n3, n4, n5) ∼= G1 ◦(1, n4, n3, n2, n5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Therefore, with- out loss of generality, we suppose that n4 ≥ n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' n2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Assume n2 ≥ 2, let F1 = G1 ◦ (1, n2 − 1, n3, n4 + 1, n5) then r(x) = φ(F1, x) − φ(F, x) = xn−5(n4 − n2 + 1)(x3 + 2x2 − (n3 + n3n5)x − 2n3n5) = xn−5(n4 − n2 + 1) � x(x2 − n3(n5 + 1)) + 2(x2 − n3n5) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Since n4 ≥ n2, we have n4 − n2 + 1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' It is clear that Kn3,n5+1 is a 9 proper subgraph of F, we obtain ρ(F) > ρ(Kn3,n5+1) = � n3(n5 + 1), then r(x) > 0 for x ≥ ρ(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Thus, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='7, we have ρ(F1) < ρ(F) which contradicts to the extremality of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' n3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Now F = G1 ◦ (1, 1, n3, n4, n5), we claim that n5 ≥ n3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' If not, let F2 = G1 ◦ (1, 1, n5, n4, n3), then r(x) = φ(F2, x) − φ(F, x) = xn−4(x2 − n4)(n3 − n5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Since n3 > n5, we have n3 − n5 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' It can be seen that Kn4,2 is a proper subgraph of F, we obtain ρ(F) > ρ(Kn4,2) = √2n4, then r(x) > 0 for x ≥ ρ(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Thus, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='7, we have ρ(F2) < ρ(F), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Therefore n5 ≥ n3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Next, we assume n3 ≥ 2, let F3 = G1 ◦ (1, 1, n3 − 1, n4, n5 + 1) then r(x) = φ(F3, x) − φ(F, x) = xn−5 � (n5 − n3 + 1)(x3 − (2n4 + 1)x − 2n4) + x(x2 − n4) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Since n5 ≥ n3, we have n5 − n3 + 1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' It is clear that Kn4,1,1 is a proper subgraph of F, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='8, we obtain ρ(F) > ρ(Kn4,1,1) = (√8n4 + 1 + 1)/2, then r(x) > 0 for x ≥ ρ(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Thus, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='7, we have ρ(F3) < ρ(F), which contradicts to the extremality of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' n5 ≥ n4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Now F = G1 ◦ (1, 1, 1, n4, n5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Otherwise, let F4 = G1 ◦ (1, 1, 1, n5, n4) then r(x) = φ(F4, x) − φ(F, x) = xn−3(x + 2)(n4 − n5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Since n4 > n5 and ρ(F) > 0, then r(x) > 0 for x ≥ ρ(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='7, we have ρ(F4) < ρ(F) which contradicts to the extremality of F, thus n5 ≥ n4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' From above three claims, we conclude that the extremal graph with min- imum spectral radius in Mn(G1) is of the form G1 ◦ (1, 1, 1, k, n − k − 3), where 1 ≤ k ≤ (n − 3)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' 10 Similarly, we characterize the extremal graph with minimum spectral ra- dius in Mn(G10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' The extremal graph in Mn(G10) which attains minimum spectral radius is of the form G10 ◦ (1, 1, 1, 1, k, n − k − 4), where 1 ≤ k ≤ n−4 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='3, we have ρ(G10 ◦ (1, n2, n3, n4, n5, n6 + n1 − 1)) ≤ ρ(G10 ◦ (n1, n2, n3, n4, n5, n6)), with equality if and only if n1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Thus, we may suppose that the extremal graph in Mn(G10) which attains minimum spectral radius is of the form F = G10 ◦ (1, n2, n3, n4, n5, n6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Similarly, we obtain B = � � � � � � � � 0 n2 n3 n4 0 0 1 0 n3 0 n5 0 1 n2 0 0 n5 0 1 0 0 0 0 n6 0 n2 n3 0 0 0 0 0 0 n4 0 0 � � � � � � � � , is the quotient matrix of an equitable partition of A(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' The characteristic polynomial of the quotient matrix B is: φ(B, x) = x(x5 − (n2 + n3 + n4 + n2n3 + n2n5 + n3n5 + n4n6)x3 − (2n2n3 + 2n2n3n5)x2 + (n2n3n4 + n2n4n5 + n2n4n6 + n3n4n5 + n3n4n6 + n2n3n4n6 + n2n4n5n6 + n3n4n5n6)x + 2n2n3n4n5 + 2n2n3n4n6 + 2n2n3n4n5n6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Since R(A(F)) = 5, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='6 we have φ(F, x) = xn−6φ(B, x) and ρ(F) = ρ(A(F)) = ρ(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Note that G10 ◦ (1, n2, n3, n4, n5, n6) ∼= G10 ◦ (1, n3, n2, n4, n5, n6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' There- fore, without loss of generality, we suppose that n3 ≥ n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' n2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' 11 Assume n2 ≥ 2, let F1 = G10 ◦ (1, n2 − 1, n3 + 1, n4, n5, n6) then r(x) = φ(F1, x) − φ(F, x) = xn−5(n3 − n2 + 1)(x3 + 2(1 + n5)x2 − (n4 + n4n6)x − 2n4n5 − 2n4n6 − 2n4n5n6) = xn−5(n3 − n2 + 1)(x(x2 − n4(n6 + 1)) + 2n5(x2 − n4(n6 + 1)) + 2(x2 − n4n6)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Since n3 ≥ n2, we have n3 − n2 + 1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' It is clear that Kn4,n6+1 is a proper subgraph of F, we obtain ρ(F) > ρ(Kn4,n6+1) = � n4(n6 + 1), then r(x) > 0 for x ≥ ρ(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Thus, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='7, we have ρ(F1) < ρ(F) which contradicts to the extremality of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' n3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Now F = G10 ◦ (1, 1, n3, n4, n5, n6), we claim that n5 ≥ n3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' If not, let F2 = G10 ◦ (1, 1, n5, n4, n3, n6), then r(x) = φ(F2, x) − φ(F, x) = xn−5(n3 − n5)(x2 − n4n6)(x + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Since n3 > n5, we have n3 − n5 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' It can be seen that Kn4,n6 is a proper subgraph of F, we obtain ρ(F) > ρ(Kn4,n6) = √n4n6, then r(x) > 0 for x ≥ ρ(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Thus, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='7, we have ρ(F2) < ρ(F), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Therefore n5 ≥ n3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Next, we assume n3 ≥ 2, let F3 = G10 ◦ (1, 1, n3 − 1, n4, n5 + 1, n6) then r(x) = φ(F3, x) − φ(F, x) = xn−5(x + 2) � (n5 − n3 + 2)(x2 − n4(n6 + 1)) + n4 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Since n5 ≥ n3, we have n5 − n3 + 2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' It is clear that Kn4,n6+1 is a proper subgraph of F, we obtain ρ(F) > ρ(Kn4,n6+1) = � n4(n6 + 1), then r(x) > 0 for x ≥ ρ(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Thus, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='7, we have ρ(F3) < ρ(F) which contradicts to the extremality of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' n4 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Now F = G10 ◦ (1, 1, 1, n4, n5, n6), we claim that n6 ≥ n4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' If not, let 12 F4 = G10 ◦ (1, 1, 1, n6, n5, n4), then r(x) = φ(F4, x) − φ(F, x) = xn−5(n4 − n6)(x2 − x − 2n5)(x + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Since n4 > n6, we have n4 − n6 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' It can be seen that Kn5,1,1 is a proper subgraph of F, we obtain ρ(F) > ρ(Kn5,1,1) = (√8n5 + 1+1)/2, then r(x) > 0 for x ≥ ρ(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Thus, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='7, we have ρ(F4) < ρ(F), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Therefore n6 ≥ n4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Next, we assume n4 ≥ 2, let F5 = G10 ◦ (1, 1, 1, n4 − 1, n5, n6 + 1) then r(x) = φ(F5, x) − φ(F, x) = xn−5(x + 1) � (n6 − n4 + 2)(x2 − x − 2n5 − 2) + 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Since n6 ≥ n4, we have n6 − n4 + 2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' It is clear that H ◦ (n5, 1, 1, 1) is a proper subgraph of F, where H is shown in Figure 3, we obtain ρ(F) > ρ(H ◦ (n5, 1, 1, 1)) = (√8n5 + 9 + 1)/2, then r(x) > 0 for x ≥ ρ(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Thus, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='7, we have ρ(F5) < ρ(F), which contradicts to the extremality of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Figure 3: The graph H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' n6 ≥ n5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Now F = G10◦(1, 1, 1, 1, n5, n6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Otherwise, let F6 = G10◦(1, 1, 1, 1, n6, n5) then r(x) = φ(F6, x) − φ(F, x) = xn−4(x + 1)2(n5 − n6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Since n5 > n6 and ρ(F) > 0, then r(x) > 0 for x ≥ ρ(F), by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='7, we have ρ(F6) < ρ(F) which contradicts to the extremality of F, thus n6 ≥ n5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' It follows from above four claims that the extremal graph with minimum spectral radius in Mn(G10) is of the form G10 ◦ (1, 1, 1, 1, k, n − k − 4), where 13 H1 ≤ k ≤ (n − 4)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Next we determine the extremal graph with minimum spectral radius in Mn(G7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' The extremal graph in Mn(G7) which attains minimum spectral radius is G7 ◦ (⌈n − 3 2 ⌉, 1, ⌊n − 3 2 ⌋, 1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Suppose that the extremal graph in Mn(G7) which attains minimum spectral radius is of the form F = G7 ◦ (n1, n2, n3, n4, n5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Similarlly, we obtain B = � � � � � � 0 n2 0 0 n5 n1 0 n3 0 0 0 n2 0 n4 0 0 0 n3 0 n5 n1 0 0 n4 0 � � � � � � , is the quotient matrix of an equitable partition of A(F) and the characteristic polynomial of B is: φ(B, x) = x5 − (n1n2 + n2n3 + n1n5 + n3n4 + n4n5)x3 + (n1n2n3n4+ n1n2n3n5 + n1n2n4n5 + n1n3n4n5 + n2n3n4n5)x − 2n1n2n3n4n5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Since R(A(F)) = 5, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='6 we have φ(F, x) = xn−5φ(B, x) and ρ(F) = ρ(A(F)) = ρ(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Without loss of generality, we may suppose that n1 = max{ni, i = 1, 2, 3, 4, 5} and n2 ≤ n5, then we have the following claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' n2 ≤ n3 and n5 ≤ n4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Suppose that n2 > n3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Let F1 = G7 ◦ (n1, n3, n2, n4, n5), then r(x) = φ(F1, x) − φ(F, x) = xn−2(n1 − n4)(n2 − n3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Since n1 = max{ni, i = 1, 2, 3, 4, 5}, we have n1 ≥ n4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' And if n1 = n4, then F1 ∼= F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Thus, without loss of generality, we may suppose that n1 > n4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Since n2 > n3, n1 > n4 and ρ(F) > 0, then r(x) > 0 for x ≥ ρ(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='7, we have ρ(F1) < ρ(F) which contradicts to the extremality of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' 14 Similarly, we obtain n5 ≤ n4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' n4 ≤ n3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Suppose to the contrary that n4 > n3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Let F2 = G7 ◦ (n1, n2, n4, n3, n5), then r(x) = φ(F2, x) − φ(F, x) = xn−2(n2 − n5)(n3 − n4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Since n5 ≥ n2 and if n5 = n2, then F2 ∼= F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Thus without loss of generality we may suppose that n5 > n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Since n4 > n3, n5 > n2 and ρ(F) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Then r(x) > 0 for x ≥ ρ(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='7, we have ρ(F2) < ρ(F) which contradicts to the extremality of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' From above two claims, we have n1 ≥ n3 ≥ n4 ≥ n5 ≥ n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Next, we will prove n2 = n4 = n5 = 1 and n1 − n3 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' n2 = n5 Assume n2 < n5, let F3 = G7 ◦ (n1 + n5 − n2, n2, n3, n4, n2) then r(x) = φ(F3, x) − φ(F, x) = xn−5(n5 − n2)((n1 − 2n2 + n4)x3 − (n1 − n2)((n3n4 + n2n3 + n2n4)x − 2n2n3n4)) ≥ xn−5(n5 − n2)(n1 − n2) � x3 − (n3n4 + n2n3 + n2n4)x + 2n2n3n4 � = xn−5(n5 − n2)(n1 − n2)g(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Since n1 ≥ n3 ≥ n4 ≥ n5 > n2 ≥ 1, we have n1 − n2 > 0 and n5 − n2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' It is clear that Kn3,n2+n4 is a proper subgraph of F, we obtain ρ(F) > ρ(Kn3,n2+n4) = � n3(n2 + n4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Since g( � n3(n2 + n4)) > 0 and � n3(n2 + n4) > � (n3(n2 + n4) + n2n4)/3, where � (n3(n2 + n4) + n2n4) /3 is the largest root of g′(x), we have r(x) > 0 for x ≥ ρ(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Thus, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='7, we have ρ(F3) < ρ(F) which contradicts to the extremality of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Note that G7 ◦ (n1, n2, n3, n4, n5) ∼= G7 ◦ (n1, n2, n4, n3, n5) when n2 = n5, therefore without loss of generality we may suppose that n3 ≥ n4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' n4 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Assume n4 ≥ 2, let F4 = G7 ◦ (n1, n2, n3 + 1, n4 − 1, n5) then r(x) = φ(F4, x) − φ(F, x) = xn−5(x − n5)(n3 − n4 + 1)(x2 + n5x − 2n1n5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' 15 Since n1 ≥ n3 ≥ n4 ≥ n5 = n2, we have n3 − n4 + 1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' It can be seen that Kn1,2n5 is a proper subgraph of F, we obtain ρ(F) > ρ(Kn1,2n5) = √2n1n5 > n5, then r(x) > 0 for x ≥ ρ(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Thus, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='7, we have ρ(F4) < ρ(F) which contradicts to the extremality of F, therefore n4 = 1 and hence n2 = n5 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' n1 − n3 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Now F = G7 ◦ (n1, 1, n3, 1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Assume n1 ≥ n3 + 2, let F5 = G7 ◦ (n1 − 1, 1, n3 + 1, 1, 1) then r(x) = φ(F5, x) − φ(F, x) = xn−5(3x − 2)(n1 − n3 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Since n1 ≥ n3 + 2, we have n1 − n3 − 1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' It is clear that Kn1,2 is a proper subgraph of F, we obtain ρ(F) > ρ(Kn1,2) = √2n1 > 1, then r(x) > 0 for x ≥ ρ(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Thus, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='7, we have ρ(F5) < ρ(F) which contradicts to the extremality of F, therefore n1 − n3 ≤ 1 and hence n1 = ⌈(n − 3)/2⌉, n3 = ⌊(n − 3)/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' From above five claims, we obtain G7 ◦ (⌈ n−3 2 ⌉, 1, ⌊ n−3 2 ⌋, 1, 1) attains the minimum spectral radius in Mn(G7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Step 3 We first prove that the extremal graph with minimum spectral radius in Mn(G1, G7) must be in Mn(G1) by the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' For n ≥ 8, we have ρ(G1 ◦ (1, 1, 1, ⌊ n−3 2 ⌋, ⌈ n−3 2 ⌉)) < ρ(G7 ◦ (⌈ n−3 2 ⌉, 1, ⌊ n−3 2 ⌋, 1, 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Let F1 = G1◦(1, 1, 1, ⌊ n−3 2 ⌋, ⌈ n−3 2 ⌉) and F2 = G7◦(⌈ n−3 2 ⌉, 1, ⌊ n−3 2 ⌋, 1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' For 8 ≤ n ≤ 12, we use the MATLAB software to calculate the spectral radii of Fi for i = 1, 2, as shown in the Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Table 1: ρ(Fi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' n ρ(F1) ρ(F2) 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='7676 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='9764 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='1474 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='2176 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='1713 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='4630 11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='5047 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='6737 12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='5223 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='8879 16 So let us assume that n ≥ 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' n − 3 = 2k is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' In this case, F1 = G1 ◦ (1, 1, 1, k, k) and F2 = G7 ◦ (k, 1, k, 1, 1), then r(x) = φ(F1, x) − φ(F2, x) = xn−5 � (k − 1)x3 − 2kx2 − k2x + 4k2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' It can be seen that K2k,1 is a proper subgraph of F2, we obtain ρ(F2) > ρ(K2k,1) = √ 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Since n ≥ 13, we have r( √ 2k) > 0 and √ 2k > (2k + k √ 3k + 1)/3(k − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Since (2k + k √ 3k + 1)/3(k − 1) is the largest root of r′(x), we obtain r(x) > 0 for x ≥ ρ(F2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Thus by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='7, we have ρ(F1) < ρ(F2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' n − 3 = 2k + 1 is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' In this case, F1 = G1 ◦ (1, 1, 1, k, k + 1) and F2 = G7 ◦ (k + 1, 1, k, 1, 1), then r(x) = φ(F1, x) − φ(F2, x) = xn−5k � x3 − 2x2 − (k + 1)x + 4k+ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' It is clear that K2k+1,1 is a proper subgraph of F2, we obtain ρ(F2) > ρ(K2k+1,1) = √ 2k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Since n ≥ 13, we have r( √ 2k + 1) > 0 and √ 2k + 1 > (2+ √ 3k + 7)/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Since (2+ √ 3k + 7)/3 is the largest root of r′(x), we obtain r(x) > 0 for x ≥ ρ(F2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Thus by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='7, we have ρ(F1) < ρ(F2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Next we prove the extremal graph with minimum spectral radius in Mn(G1, G10) must be in Mn(G10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' We need the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' [12] Let G be a graph with m edges and n vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Then ρ(G) ≤ √2m − n + 1, with equality if and only if G is isomorphic to the star K1,n−1 or the complete graph Kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Let G1 ◦ (1, 1, 1, k, n − k − 3) be the extremal graph with min- imum spectral radius in Mn(G1) for n ≥ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Then 2 ≤ k ≤ n−3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' We denote Fk = G1 ◦ (1, 1, 1, k, n − k − 3) for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' By Propo- sition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='9, we have 1 ≤ k ≤ (n − 3)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' For 12 ≤ n ≤ 18, we use the MATLAB software to calculate the spectral radii of Fk, as shown in the Table 2, where the minimum spectral radius is bolded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' 17 Table 2: ρ(Fk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' n ρ(Fk) k 1 2 3 4 5 6 7 12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='0751 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='0649 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='2427 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='5223 \\ \\ \\ 13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='2229 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='1791 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='2951 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='5443 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='8231 \\ \\ 14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='3668 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='3013 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='3616 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='5722 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='8368 \\ \\ 15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='5064 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='4274 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='4422 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='6076 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='8535 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='1131 \\ 16 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='6418 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='5544 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='5353 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='6526 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='8742 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='1243 \\ 17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='7731 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='6807 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='6377 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='7088 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='8998 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='1376 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='3813 18 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='9006 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='8053 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='7459 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='7767 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='9318 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='1536 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='3906 For n ≥ 19, note that F1 = G1◦(1, 1, 1, 1, n−4), F2 = G1◦(1, 1, 1, 2, n−5), let x be the principal eigenvector of F2 and xi correspond to vertices in Vi for i = 1, 2, 3, 4, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' By ρ(F2)x = A(F2)x, we have ρ(F2)x1 = x2 + x3 + 2x4, (1) ρ(F2)x2 = x1 + 2x4, (2) ρ(F2)x3 = x1 + (n − 5)x5, (3) ρ(F2)x4 = x1 + x2, (4) ρ(F2)x5 = x3, (5) From (1)-(3), we have ρ(F2)(x3 − x1 − x2) = x1 + (n − 5)x5 − x2 − x3 − 2x4 − x1 − 2x4, multiplying ρ(F2) on both sides, by (4) and (5), yields ρ(F2)2(x3 − x1 − x2) = (n − 5)x3 − ρ(F2)x3 − ρ(F2)x2 − 4(x1 + x2), then (ρ(F2)2 − ρ(F2) − 4)(x3 − x1 − x2) = (n − 9 − 2ρ(F2))x3 + ρ(F2)x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' (6) Since n ≥ 19 and Kn−5,1 is a proper subgraph of F2, we have ρ(F2) > ρ(Kn−5,1) = √n − 5 > 3, thus ρ(F2)2 − ρ(F2) − 4 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='13 and n ≥ 19, we obtain ρ(F2) < � 2m(F2) − n + 1 = � 2(n + 1) − n + 1 = √n + 3 < (n − 9)/2, therefore n − 9 − 2ρ(F2) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Since x is the principal eigenvector of F2, we have xi > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Thus, it follows from (6) that x3 − x1 − x2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' 18 Now we have ρ(F1) − ρ(F2) ≥ xTA(F2)x − xTA(F1)x = 2x4x3 − 2x4(x1 + x2) = 2x4(x3 − x1 − x2) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Therefore, ρ(F1) > ρ(F2), which means k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Now we prove that ρ(G10◦(1, 1, 1, 1, k−1, n−k−3)) < ρ(G1◦(1, 1, 1, k, n− k − 3)) for k ≥ 2 and n ≥ 12 by using a well-known operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' [8] Let v1, v2 be two vertices of a connected graph G and let {u1, u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' , ut} ⊆ N(v1) \\ N(v2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Let G′ be the graph obtained from G by rotating the edge v1ui to v2ui for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' , t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' If xv1 ≤ xv2, where x is the principal eigenvector of G, then ρ(G′) > ρ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' For k ≥ 2 and n ≥ 12, we have ρ(G10 ◦ (1, 1, 1, 1, k − 1, n − k − 3)) < ρ(G1 ◦ (1, 1, 1, k, n − k − 3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Let F1 = G1 ◦ (1, 1, 1, k, n − k − 3) and F2 = G10 ◦ (1, 1, 1, 1, k − 1, n − k −3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Let x be the principal eigenvector of F2 and xi correspond to vertices in Vi for i = 1, 2, 3, 4, 5, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Let us first suppose that x3 ≥ x1, then by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='15 we have ρ(F2) < ρ(F ′), where F ′ is obtained from F2 by rotating the edge v1v4 to v3v4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Since F ′ ∼= F1, we obtain ρ(F2) < ρ(F1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Now, suppose that x3 < x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Since F ′′ ∼= F1, where F ′′ is obtained from F2 by rotating the edge v3vi 5 to v1vi 5 for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' , k − 1, we have ρ(F2) < ρ(F ′′) = ρ(F1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Thus, we complete the proof of the Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Now we know that the extremal graph of order n and rank 5 with mini- mum spectral radius is G10 ◦ (1, 1, 1, 1, k, n − 4 − k) for some integer k with 1 ≤ k ≤ n−4 2 when n ≥ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' For convenience, we set Fn(i) = G10 ◦ (1, 1, 1, 1, i, n − 4 − i) and F = {Fn(i) : 1 ≤ i ≤ n−4 2 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' It is only remained to find the extremal graph with minimum spectral radius in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' 19 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' [10] Let A be an n×n nonnegative matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Then the largest eigenvalue ρ(A) ≥ xTAx for any unit vector x, with equality if and only if Ax = ρ(A)x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Let α = 6n−37−√24n+1 18 and n ≥ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Then for 1 ≤ i ≤ n−4 2 , we have ρ(Fn(i)) > min{ρ(Fn(⌊α⌋)), ρ(Fn(⌈α⌉))} unless i = ⌊α⌋ or ⌈α⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Let ρi = ρ(Fn(i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Our aim is to prove that ρi < ρi−1 if 2 ≤ i ≤ ⌊α⌋ and ρi < ρi+1 if ⌈α⌉ ≤ i ≤ n−6 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Let xi be the principal eigenvector of Fn(i) and xi j correspond to vertices in Vj for j = 1, 2, 3, 4, 5, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Then by ρixi = A(Fn(i))xi we have ρixi 1 = xi 2 + xi 3 + xi 4, (7) ρixi 2 = xi 1 + xi 3 + ixi 5, (8) ρixi 3 = xi 1 + xi 2 + ixi 5, (9) ρixi 4 = xi 1 + (n − i − 4)xi 6, (10) ρixi 5 = xi 2 + xi 3, (11) ρixi 6 = xi 4, (12) By (8) and (9), we have ρi(xi 2 − xi 3) = xi 3 − xi 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=', (ρi + 1)(xi 2 − xi 3) = 0, which implies that xi 2 = xi 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' (13) 20 Therefore, by (7) and (10)-(13), we have xi 5 = 2xi 2 ρi = xi 1 − xi 4 ρi = xi 1 − xi 6 = ρixi 4 − (n − i − 4)xi 6 − xi 6 = ρ2 i xi 6 − (n − i − 4)xi 6 − xi 6 = (ρ2 i − n + i + 3)xi 6, and from (7)-(8) and (11)-(13), we have xi 6 = xi 4 ρi = xi 1 − 2xi 2 ρi = xi 1 − xi 5 = (ρi − 1)xi 2 − ixi 5 − xi 5 = ρi(ρi − 1) 2 xi 5 − ixi 5 − xi 5 = 1 2(ρ2 i − ρi − 2i − 2)xi 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Hence, we obtain that � � � � � (ρ2 i − n + i + 3)(ρ2 i − ρi − 2i − 2) = 2, ρ2 i − n + i + 3 > 0, ρ2 i − ρi − 2i − 2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' (14) Note that, if we let � ρ2 i − n + i + 3 = 1, ρ2 i − ρi − 2i − 2 = 2, then we have � ρi = √n − i − 2, ρi = 1+√8i+17 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' By calculation, we can find that i = α = (6n−37−√24n + 1)/18 is the only 21 solution of √n − i − 2 = (1 + √8i + 17)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Since i ∈ N, we will complete the proof by classifying the value of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' If 2 ≤ i ≤ ⌊α⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' We have √n − i − 2 ≥ (1+√8i + 17)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' We claim that (1+√8i + 17)/2 ≤ ρi ≤ √n − i − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Suopose that ρi < (1 + √8i + 17)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' By (14), we have 0 < ρ2 i − n + i + 3 < 1 and 0 < ρ2 i − ρi − 2i − 2 < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Then (ρ2 i − n + i + 3)(ρ2 i − ρi − 2i − 2) < 2, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Suopose that ρi > √n − i − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' By (14), we obtain that ρ2 i − n + i + 3 > 1 and ρ2 i − ρi − 2i − 2 > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Then (ρ2 i − n + i + 3)(ρ2 i − ρi − 2i − 2) > 2, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Thus we have (1 + √8i + 17)/2 ≤ ρi ≤ √n − i − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' This induces that ρ2 i − n + i + 3 ≤ 1 and ρ2 i − ρi − 2i − 2 ≥ 2, which lead to xi 6 ≥ xi 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Therefore ρi−1 − ρi ≥xT i A(Fn(i − 1))xi − xT i A(Fn(i))xi =2xi 5(xi 4 − xi 2 − xi 3) =2ρixi 5(xi 6 − xi 5) ≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' (15) Now we only need to prove ρi−1 ̸= ρi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Suppose that ρi−1 = ρi, then ρi−1 = xT i A(Fn(i − 1))xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='17, we have ρi−1xi 4 = xi 1 + (n − i − 4)xi 6 + xi 5, and since ρixi 4 = xi 1 + (n − i − 4)xi 6, we obtain 0 = (ρi−1 − ρi)xi 4 = xi 5, which contradicts to the definition of the principal eigenvector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Therefore, from (15) we have ρi−1 > ρi for 2 ≤ i ≤ ⌊α⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' If ⌈α⌉ ≤ i ≤ n−6 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' We have √n − i − 2 ≤ (1 + √8i + 17)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Similarly, by (14), we conclude that √n − i − 2 ≤ ρi ≤ (1+√8i + 17)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' This induces that ρ2 i −n+i+3 ≥ 1 22 and ρ2 i − ρi − 2i − 2 ≤ 2, which lead to xi 5 ≥ xi 6, therefore ρi+1 − ρi ≥xT i A(Fn(i + 1))xi − xT i A(Fn(i))xi =2xi 6(xi 2 + xi 3 − xi 4) =2ρixi 6(xi 5 − xi 6) ≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' (16) Similarly, we have ρi+1 ̸= ρi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Using this, from (16), we obtain ρi+1 > ρi for ⌈α⌉ ≤ i ≤ n−6 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Therefore, the proof of Lemma is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Step 4 It only remains for the case that 5 ≤ n ≤ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Applying Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='9, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='10 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='11, we obtain the extremal graphs with minimum spectral radius in Mn(G1), Mn(G7) and Mn(G10), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' And then calculate their spectral radii by using MATLAB, as shown in Table 3, where the extremal graphs and the minimum spectral radii are bolded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Table 3: The extremal graph with minimum spectral radius in Mn(G1), Mn(G7), Mn(G10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' n Mn(G1) Mn(G7) Mn(G10) Extremal graph Spectral radius Extremal graph Spectral radius Extremal graph Spectral radius 5 G1 ◦ (1, 1, 1, 1, 1) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='2143 G7 ◦ (1, 1, 1, 1, 1) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='0000 \\ \\ 6 G1 ◦ (1, 1, 1, 1, 2) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='2784 G7 ◦ (2, 1, 1, 1, 1) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='3912 G10 ◦ (1, 1, 1, 1, 1, 1) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='6544 7 G1 ◦ (1, 1, 1, 1, 3) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='3686 G7 ◦ (2, 1, 2, 1, 1) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='6813 G10 ◦ (1, 1, 1, 1, 1, 2) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='6751 8 G1 ◦ (1, 1, 1, 1, 4) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='4860 G7 ◦ (3, 1, 2, 1, 1) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='9764 G10 ◦ (1, 1, 1, 1, 1, 3) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='7033 9 G1 ◦ (1, 1, 1, 1, 5) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='6239 G7 ◦ (3, 1, 3, 1, 1) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='2176 G10 ◦ (1, 1, 1, 1, 1, 4) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='7448 10 G1 ◦ (1, 1, 1, 1, 6) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='7724 G7 ◦ (4, 1, 3, 1, 1) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='4630 G10 ◦ (1, 1, 1, 1, 1, 5) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='8060 11 G1 ◦ (1, 1, 1, 1, 7) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='9243 G7 ◦ (4, 1, 4, 1, 1) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='6737 G10 ◦ (1, 1, 1, 1, 1, 6) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='8915 By Table 4, we obtain that when 5 ≤ n ≤ 11, the extremal graph with minimum spectral radius of rank 5 is: G7 = C5, for n = 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' G1 ◦ (1, 1, 1, 1, n − 4), for 6 ≤ n ≤ 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' G10 ◦ (1, 1, 1, 1, 1, n − 5), for n = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' 23 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Concluding remarks In the last case of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='2, we obtain that k ∈ {⌊ 6n−37−√24n+1 18 ⌋ , ⌈ 6n−37−√24n+1 18 ⌉}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' When 12 ≤ n ≤ 23, we use the MATLAB software to calculate the spectral radii of the graphs in F = {Fn(i) : 1 ≤ i ≤ n−4 2 }, as shown in the Table 4, where the minimum spectral radius is bolded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' It demonstrates that k = ⌊ 6n−37−√24n+1 18 ⌋ or ⌈ 6n−37−√24n+1 18 ⌉ depends on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Table 4: ρ(Fn(i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' n ρ(Fn(i)) i 1 2 3 4 5 6 7 8 9 6n−37−√24n+1 18 12 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='1370 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='4319 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='7362 \\ \\ \\ \\ \\ 1 13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='1239 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='1818 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='4431 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='7404 \\ \\ \\ \\ \\ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='2949 14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='255 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='2470 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='4588 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='7457 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='0278 \\ \\ \\ \\ 1.' 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3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='5227 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='4402 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='5160 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='7616 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='0344 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='2979 \\ \\ \\ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='1877 17 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='5522 \\ \\ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='0901 20 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='0303 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='9201 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='8338 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='8612 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='0628 4.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='9211 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='0779 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='3147 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='5565 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='7900 \\ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='6960 22 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='2663 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='1570 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='0564 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='1002 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='3213 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='5593 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='7915 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='0146 4 23 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='3801 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='2721 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='1694 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='0929 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='1341 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='3303 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='5627 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='7933 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='0157 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='3047 It is a natural problem to determine the extramal spectral radii of the graphs of order n and rank r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' By Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='1, we know that the maximum spectral radius of all connected graphs of order n and rank r is ρ(T(n, r)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' gave the spectral radius of T(n, r) in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' [13] Let T(n, r) be a Tur´an graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Then ρ(T(n, r)) = 1 2 � n − 2⌊n r ⌋ − 1 + � (n + 1)2 − 4(n − r⌊n r ⌋)⌈n r ⌉ � ≤ n − ⌊n r ⌋ with the last equality if and only if T(n, r) is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Further, we obtain a sharp upper and lower bound for the spectral radius of the extremal graph G which attains the minimum spectral radius among all connected graphs of order n ≥ 12 and rank 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' By Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='2, we know that ρ(G) = min {ρ(Fn(⌊α⌋)), ρ(Fn(⌈α⌉))}, 24 where α = 6n−37−√24n+1 18 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' From the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='18, we have 1 + � 8⌊α⌋ + 17 2 ≤ ρ(Fn(⌊α⌋) ≤ � n − ⌊α⌋ − 2, � n − ⌈α⌉ − 2 ≤ ρ(Fn(⌈α⌉) ≤ 1 + � 8⌈α⌉ + 17 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Therefore, we obtain that ρ(G) ≥ min{1 + � 8⌊α⌋ + 17 2 , � n − ⌈α⌉ − 2}, and ρ(G) ≤ min{ � n − ⌊α⌋ − 2, 1 + � 8⌈α⌉ + 17 2 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' In general, the problem of determining the minimum spectral radius of all connected graphs with order n and rank r deserves further study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Declaration of compting interest There is no competing interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Acknowledgement This research is supported by the National Natural Science Foundation of China [Grant number, 12171402].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' References [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Chang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Huang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' Yeh, A characterization of graphs with rank 4, Linear 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} +page_content=' 26' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE5T4oBgHgl3EQfEw5P/content/2301.05416v1.pdf'} diff --git a/d9AzT4oBgHgl3EQfaPwm/content/tmp_files/2301.01364v1.pdf.txt b/d9AzT4oBgHgl3EQfaPwm/content/tmp_files/2301.01364v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0e996a33e3c3ca8007006d4473cf2b2afbb0781c --- /dev/null +++ b/d9AzT4oBgHgl3EQfaPwm/content/tmp_files/2301.01364v1.pdf.txt @@ -0,0 +1,1074 @@ +arXiv:2301.01364v1 [stat.AP] 3 Jan 2023 +Notes on Correspondence Analysis of Power +Transformed Data Sets +Choulakian V., Universit´e de Moncton, Canada +vartan.choulakian@umoncton.ca +January 2023 +R´esum´e +We prospect for a clear simple picture on CA of power transformed +or the Box-Cox transformed data initiated since 2009 by Grenacre. We +distinguish two types of data sets : strictly positive and with zeros ; +we concentrate on the latter. +Key words : Contingency tables ; compositional data ; double cen- +tering ; power transformation ; interactions ; zeros ; taxicab correspon- +dence analysis. +AMS 2010 subject classifications : 62H25, 62H30 +1 +Introduction +Correspondence analysis (CA) and logratio analysis (LRA) are two po- +pular methods for the analysis and visualization of a two-way contingency +table or a compositional data set N = (nij) of size I × J for i = 1, ..., I +and j = 1, ..., J. LRA is considered an ideal desirable method for analysis- +visualization of N, because it is row and column scales invariant ; but it can +be applied only when nij > 0. While CA can be applied to data when nij ≥ 0, +but it is NOT row and column scale invariant because it is dependent on its +row and column marginals. +The central topic of this paper is Greenacre’s (2010, Result 1), that we re- +name it as theorem, and its application to N. Greenacre’s Theorem concerns +the convergence of CA of N(α) = (nα +ij) for α ∈ (0, 1] as α → 0 to uniformly +weighted LRA of N when nij > 0. However, Greenacre (2011, 2022) attempts +1 + +to show its applicability - usefulness when nij ≥ 0 ; that is, when the data +set N contains zero valued cells. We aim to shed further light on this latter +situation. +This paper is organized as follows : Section 2 presents preliminaries ; sec- +tion 3 presents Greenacre’s Theorem concerning CA of strictly positive data +sets ; section 4 presents CA of nonnegative data sets having zero valued cells ; +section 5 discusses three examples of real data sets with zero cells ; finally we +conclude in section 6. +1.1 +Some references +CA and LRA are based on three different principles : CA on Benz´ecri’s +distributional equivalence principle, RC association models on Yule’s scale +invariance principle, and CoDA on Aitchison’s subcompositional coherence +principle. A recent discussion of these three principles can be found in Chou- +lakian et al. (2023). +Benz´ecri (1973) is the reference book on CA. Beh and Lombardo (2014) +present a panoramic review of CA and its variants. +LRA includes two independently well developed methods : RC association +models for the analysis of contingency tables by Goodman (1979, 1981a, +1981b, 1991, 1996) and a set of compositional vectors (CoDA) by Aitchison +(1986), see also among others Greenacre (2018). +The relationship between CA and LRA has been discussed in many pa- +pers ; see for example, among others, Goodman (1996), Cuadras et al. (2006), +Cuadras and Cuadras (2015 ), Greenacre (2009, 2010), Beh and Lombardo +(2022) and Choulakian (2022). +2 +Preliminaries on analysis of contingency tables +We consider a two-way contingency table N = (nij) for i = 1, ..., I and +j = 1, ..., J, and P = N/n = (pij) of size I ×J the associated correspondence +matrix (probability table) of the contingency table N. We define as usual +pi+ = �J +j=1 pij , p+j = �I +i=1 pij, the vector r = (pi+) ∈ RI, the vector +c = (p+j) ∈ RJ, and MI = Diag(r) the diagonal matrix having diagonal +elements pi+, and similarly MJ = Diag(c). We suppose that MI and MJ +are positive definite metric matrices of size I × I and J × J, respectively ; +this means that the diagonal elements of MI and MJ are strictly positive. +2 + +2.1 +Independence of the row and column categories +a) The I row categories and the J column categories are mutually inde- +pendent, then +σij = pij − pi+p+j = 0 +(1) +for i = 1, ..., I and j = 1, ..., J, and, where σij is the residual matrix of pij +with respect to the independence model pi+p+j. +Remark 1 : The contingency table N = (nij) can also be represented +(coded) as an indicator matrix Z = [ZI ZJ] = [(zαi) (zαj)] of size n×(I+J), +where zαi = 0 if individual α does not have level i of the row variable, zαi = 1 +if individual α has level i of the row variable ; zαj = 0 if individual α does +not have level j of the column variable, zαj = 1 if individual α has level j +of the column variable. Note that N = ZI ′ZJ and σij = pij − pi+p+j is the +covariance between the i-th column of ZI and the j-th column of ZJ. +b) The independence assumption σij = 0 can also be interpreted in ano- +ther way as +∆ij = ( +pij +pi+p+j +− 1) = 0 +(2) += 1 +pi+ +( pij +p+j +− pi+) += +1 +p+j +( pij +pi+ +− p+j); +this is the column and row homogeneity model. Benz´ecri (1973, p.31) named +the conditional probability vector ( pij +p+j for i = 1, ..., I and j fixed) the profile +of the j-th column. He also referred to the element of +pij +pi+p+j as the density +function of the probability measure (pij) with respect to the product measure +pi+p+j. The element +pij +pi+p+j is named Pearson ratio in Goodman (1996) and +Beh and Lombardo (2014, p.123). +c) A third way to represent the independence assumption σij = 0 and the +row and column homogeneity models ∆ij = 0 is via the (wR +i , wC +j ) weighted +loglinear formulation, equation (3), assuming pij > 0 and defining Gij = +log(pij), +λ(pij, wR +i , wC +j ) = Gij − Gi+ − G+j + G++ = 0, +(3) +3 + +where Gi+ = �J +j=1 GijwC +j , G+j = �I +i=1 GijwR +i and G++ = �J +j=1 +�I +i=1 GijwC +j wR +i ; +wC +j > 0 and wR +i > 0, satisfying �J +j=1 wC +j = �I +i=1 wR +i += 1, are a priori fixed +or data dependent probability weights. Two popular weights are marginal +(wR +i = pi+, wC +j = p+j) and uniform (wR +i = 1/I, wC +j = 1/J). This is implicit +in equation 7 in Goodman (1996) or equation 2.2.6 in Goodman (1991) ; and +explicit in Egozcue et al. (2015). +Equation (3) is equivalent to the logratios +log(pijpi1j1 +pij1pi1j +) = 0 +for i ̸= i1 and j ̸= j1, +which Goodman (1979, equation 2.2) refers to as the ”null association” mo- +del. +Equation (3) is also equivalent to +pij = exp(Gi+) exp(G+j) +exp(G++) +, +from which we deduce that : under the independence assumption the mar- +ginal row probability vector (pi+) is proportional to the vector of weighted +geometric means (exp(Gi+)); and a similar property is true also for the co- +lumns ; see for instance Egozcue et al. (2015). +2.2 +Interaction factorization +Suppose the independence-homogeneity-null association models are not +true, then each of the three equivalent model formulations (1), (2), (3) can +be generalized to explain the nonindependence-nonhomogeneity-association, +named interaction, among the I rows and the J columns by adding k bilinear +terms, where k = rank(N)−1. +We designate any one of the interaction indices (1), (2), (3) by τij. +Benz´ecri (1973, Vol.2, p. 31-32) emphasized the importance of row and +column weights or metrics in multidimensional data analysis ; this is the +reason in the French data analysis circles any study starts with a triplet +(X, MI, MJ), where X represents the data set, MI = (Diag(mr +i)) is the +metric matrix defined on the rows and MJ = (Diag(mc +j)) the metric ma- +trix defined on the columns. We follow the same procedure but X is the +preprocessed data, where : +a) In covariance analysis, X = (τij) = (IJσij) and +4 + +(MI, MJ) = (Diag(1/I), Diag(1/J)). +b) In CA, X = (τij) = (∆ij) and (MI, MJ) = (Diag(pi+), Diag(p+j)); +c) In LRA, X = (τij) = (λ(pij, wR +i , wC +j )) and (MI, MJ) = (Diag(wR +i ), Diag(wC +j )) +with �J +j=1 wC +j = �I +i=1 wR +i = 1. +Note that in a) we have multiplied σij by IJ to unify theoretically the +covariance analysis with the CA analysis, see the subsection on double cen- +tering. +We factorize the interactions in (1), (2) and (3) by singular value decom- +position (SVD) or taxicab SVD (TSVD) as +τij = +k +� +α=1 +fα(i)gα(j)/δα, +(4) +where fα(i) is the ith row principal coordinate and gα(j) is the jth co- +lumn principal coordinate along the αth principal dimension. Also δα is the +dispersion measure of the αth principal axis. +Remark 2 : +a) In the SVD case the parameters (fα(i), gα(j), δα) satisfy the conditions : +for α, β = 1, ..., k +δ2 +α = +I +� +i=1 +f +2 +α (i)mr +i = +J +� +j=1 +g2 +α(j)mc +j +0 = +I +� +i=1 +fα(i)mr +i = +J +� +j=1 +gα(j)mc +j +0 = +I +� +i=1 +fα(i)fβ(i)mr +i = +J +� +j=1 +gα(j)gβ(j)mc +j for α ̸= β. +b) In the TSVD case the parameters (fα(i), gα(j), δα) satisfy the conditions : +for α, β = 1, ..., k +δα = +I +� +i=1 +|fα|(i)mr +i = +J +� +j=1 +|gα(j)|mc +j +0 = +I +� +i=1 +fα(i)mr +i = +J +� +j=1 +gα(j)mc +j +5 + +0 = +I +� +i=1 +fα(i) sign(fβ(i))mr +i = +J +� +j=1 +gα(j) sign(gβ(j))mc +j for α > β. +A description of TSVD can be found in Choulakian (2006, 2016). +Remark 3 +a) In the case (τij) = (IJσij), the bilinear decomposition (4) is also named +interbattery analysis first proposed by Tucker (1958) ; later on, Tenenhaus +and Augendre (1996) reintroduced it within correspondence analysis circles, +where they showed that the Tucker decomposition by SVD produced on some +correspondence tables more interesting (interpretable) structure than CA. +b) In the case (τij) = (∆ij), the CA decomposition has many interpreta- +tions. Essentially, for data analysis, purposes Benz´ecri (1973) interpreted it +as weighted principal components analysis of row and column profiles. Ano- +ther useful interpretation of CA, comparable to Tucker interbattery analysis, +is Hotelling(1936)’s canonical correlation analysis, see Lancaster (1958) and +Goodman (1991, 1996). +c) In the case (τij) = (λ(pij, wR +i , wC +j )) and (MI, MJ) = (Diag(wR +i ), Diag(wC +j )), +where (wR +i , wC +j ) are prespecified ; we note this case by TLRA or LRA ; for an +example of general prespecified weights see Egozcue and Pawlowsky-Glahn +(2016). For the important particular case λ(pij, wR +i , wC +j ) = λ(pij, 1/I, 1/J), +we get uniformly-weighted (or taxicab) logratio analysis uwLRA (or uwTLRA). +d) We are concerned with the property of scale dependence or indepen- +dence of the three interaction indices (1), (2) and (3). The three indices +depend on pij where pij = nij/ � +i,j nij. To emphasize this dependence, we ex- +press any one of of the three interaction indices by τij(nij) = τ(pij, mR +i , mC +j ). +Following Yule (1912), we state the following +Definition 1 : An interaction index τij(nij) is scale invariant if τij(nij) = +τij(ainijbj) for arbitrary scales ai > 0 and bj > 0. +It is evident that neither of the three indices with data dependent mar- +ginal weights (pi+, p+j) are scale invariant. +Concerning the association index (3) we have the following lemma, see +Choulakian (2022) or Choulakian et al. (2023). +Lemma 1 : a) The association index (3) with prespecified weights (wR +i , wC +j ) +6 + +is scale invariant. That is, +λ(nij, wR +i , wC +j ) = λ(ainijbj/n∗, wR +i , wC +j ), +for arbitrary scales ai > 0 and bj > 0 and n∗ = � +i,j ainijbj. +b) To a first-order approximation, λ(nij, wR +i , wC +j ) ≈ ( +pij +wC +j wR +i − pi+ +wR +i − p+j +wC +j +1). +2.3 +Double centering +The interaction matrix (τij) is double centered, that is : +0 = +I +� +i=1 +mr +i mc +jτij += +J +� +j=1 +mr +i mc +jτij. +(5) +According to Tukey (1977, chapter 10), there are two kinds of double cen- +tering row-PLUS-column and row-TIMES-column ; we name them additive +and multiplicative, that we describe. +We consider the triplet (Y, MI, MJ), where Yij = h(pij) where h is a +function and (MI, MJ) = (Diag(mr +i), Diag(mc +j)). We compute the three +means, two marginals and the total : Yi+ = �J +j=1 Yijmc +j, Y+j = �I +i=1 Yijmr +i +and Y++ = �J +j=1 +�I +i=1 Yijmr +imc +j. +The multiplicative double centering is +τij = Yij − Yi+Y+j +Y++ +, +(6) +where rank(τij) = rank(Yij) − 1. +The additive double centering is +τij = Yij − Yi+ − Y+j + Y++, +(7) +where rank(τij) = rank(Yij) − 1 or rank(Yij) − 2. +An evident difference between the two types of double centering is that +(7) is invariant to a row or column additive constants ; Definition 1 and +Lemma1a become a corollary to this fact. +We consider two functional forms of Yij = h(pij). +7 + +a) Yij = h(pij) = +pij +mr +i mc +j is the density function of the joint probability +measure pij with respect to the product measure mr +i mc +j. First, the cova- +riance interaction (τij) = (IJσij) is obtained from (6), when mr +i = 1/I and +mc +j = 1/J. So the rank(IJσij) = rank(IJpij) −1. Second, the CA interaction +(τij) = (∆ij) is obtained either from (6) or from (7), when mr +i = pi+ and +mc +j = p+j. So the rank(∆ij) = rank( +pij +pi+p+j ) − 1. Third, the right-hand side of +Lemma 1b is obtained from (7) when mr +i = wR +i and mc +j = wC +j . +b) Yij = h(pij) = log pij for pij > 0. First, the log interaction (τij) = (λ(pij, wR +i , wC +j )) +is obtained from (7), when mr +i = wR +i and mc +j = wC +j . So the rank (λ(pij, wR +i , wC +j )) = +rank(log pij) − 1, as in Goodman (1991, Table 11) ; or rank(log pij) − 2, as +in Goodman (1991, Table 10). Second, to our knowledge the log interaction +obtained from (6) has not been applied yet, most probably due to the fact +that it is not row and column scale invariant ; see also subsection 3.1. +Choulakian et al. (2023) used the quality of the sign of the residuals index, +QSR, to choose an optimal case among the different cases mentioned above. +3 +CA of power transformed strictly positive +data +Here, we state Greenacre’s Theorem and provide a mathematical proof +in the appendix. +Let (p(α) +ij += +nα +ij +� +i,j nα +ij ) for i = 1, ..., I and +j = 1, ..., J be the correspon- +dence table of the power transformed strictly positive data, and (∆(α) +ij += +p(α) +ij −p(α) +i+ p(α) ++j +p(α) +i+ p(α) ++j +) the CA interaction matrix. +Theorem (Greenacre (2010, Result 1) +Under the assumption nij > 0, for α → 0 and α > 0 +∆(α) +ij += p(α) +ij − p(α) +i+ p(α) ++j +p(α) +i+ p(α) ++j +≈ αλ(pij, wR +i = 1/I, wC +j = 1/J) += α(log pij + 1 +IJ +� +i,j +log pij − 1 +I +� +i +log pij − 1 +J +� +j +log pij) +Furthermore, p(α) +i+ ≈ 1/I for i = 1, ..., I and p(α) ++j ≈ 1/J for j = 1, ..., J. +8 + +A way to see the theorem is to look at the following sequence of approxi- +mations using Lemma1b, and, the fact that p(α) +i+ ≈ 1/I for i = 1, ..., I and +p(α) ++j ≈ 1/J for j = 1, ..., J as α → 0. +λ(p(α) +ij , wR +i = 1/I, wC +j = 1/J) = αλ(pij, wR +i = 1/I, wC +j = 1/J) +≈ (IJp(α) +ij + 1 − Ip(α) +i+ − Jp(α) ++j ) +≈ (IJp(α) +ij − 1) +≈ ∆(α) +ij . +3.1 +Remarks +a) The Theorem is interesting, but not useful in empirical contexts, be- +cause one can directly compute the log interactions λ(pij, wR +i = 1/I, wC +j = +1/J). +b) In the above Theorem, the assumption nij > 0 is fundamental : if +α = 0, then nα +ij = n0 +ij = 1 for all (i, j); that is, the rank of the matrix +(nα +ij = n0 +ij = 1) is 1 ; thus both sides of the equation in the Theorem are zero. +c) In the proof of the above Theorem in the appendix, one sees that +lim +α→0 +∆(α) +ij +α += λ(pij, wR +i = 1/I, wC +j = 1/J). +d) Assume nij ≥ 1; then another power transformation is the logarithmic +transformation, also known as the Box-Cox transformation, L = (lognij = +limα→0 +1 +α(nα +ij − 1)). We have not seen any application of CA to L, see the +subsection 2.3 on double centering. Note that the assumption in Greenacre’s +Theorem nij > 0 is weaker than the assumption nij ≥ 1 for the use of the log +transformation. Goodman’s RC model, based on the log transformation, has +been developed for the analysis of contingency tables (tables with strictly +positive counts), and not for compositional data where often data in % have +strictly positive values less than 1, besides having positive values larger than +1 ; see for instance, the archeological compositional CUPS data in Greenacre +and Lewi (2008). +e) There is some kind of kinship between Greenacre’s Theorem and Good- +man’s marginal free CA (mfCA), see Goodman (1996, equation (46)). In +mfCA of a probability table P = (pij) with row and column marginals (pi+) +9 + +and (p+j) respectively, CA is applied to a matrix Q = (qij = aipijbj) related +to P in two steps : +Step 1 : by the scale invariance property of the log interaction index, +see Lemma1a, under the assumption nij > 0 there exists a unique proba- +bility matrix Q =(qij) which is related to P = (pij) via the strictly positive +scales (ai, bj), that keeps Yule’s association between the i-th row and the j-th +column unchanged ; that is, +λ(pij, wR +i = 1/I, wC +j = 1/J) = λ(qij = aipijbj, wR +i = qi+ = 1/I, wC +j = q+j = 1/J). +Note that Q has uniform row and column marginal weights similar to (p(α) +ij ). +Step 2, mfCA of P is CA representation of Q +IJqij − 1 = +k +� +α=1 +fα(i)gα(j)/δα. +For further details, see Choulakian (2022). +4 +CA of power transformed data set having +zero valued cells +It is quite common that large compositional data sets or contingency +tables have zero valued cells. For this case, let m be the number of zero valued +cells ; so IJ −m is the number of strictly positive valued cells in the table. We +see that for such tables the log transformation discussed in Remark b is not +valid, while the simple power transformation nα +ij for α ∈ [0, 1] is valid. That +is, the matrix (lim nα +ij) as α → 0 converges to an indicator matrix Z = (zij), +where zij = 1 if nij > 0 and zij = 0 if nij = 0. A comparison of this case with +the observation in Remark a) under the assumption nij > 0 is insightful, +where the indicator matrix becomes Z = (zij = 1) of rank 1. Here, it is +insightful to consider two distinct cases : Sparse tables containing multiple +zeros in at least two nonproportional rows or nonproportional columns, and +tables with exactly one column (or row) having at least one zero valued +cell. In the latter case the zero entries exhibit a dominating influence very +similar to the cases of a heavyweight cell and a heavyweight column (or row) +discussed by Benz´ecri (1979) and Lebart (1979) in CA, and by Choulakian +(2008) in TCA. +10 + +4.1 +Tables with one column having m zero-valued cells +Here we discuss the case where there are m for m = 1, ..., I −1 zero valued +cells in one column of the indicator matrix Z of size I × J. By Benz´ecri’s +principle of distributional equivalence property which states that in CA (or +TCA) proportional rows or columns can be merged, CA (or TCA) of Z is +identical to CA (or TCA) of the 2 × 2 contingency table +R = +� 0 +m(J − 1) +(I − m) +(I − m)(J − 1) +� +, +(8) +which has only one CA (or TCA) principal dimension. +Lemma 2 +a) In CA of R the dispersion, named inertia, ρ2 +1 = +m +IJ.. +b) In TCA of R the taxicab dispersion δ1 = 4m(J−1)(I−m) +(IJ−m)2 +. +Proof of a) : By equation 2.1.4 in Goodman (1996), for a 2×2 contingency +table the ”ninindependence” based on the correlation coefficient is +ρ2 +1 = (p11p22 − p12p21)2 +p1+p2+p+1p+2 +. +(9) +By replacing the probability values of the elements of R in (9), we get +ρ2 +1 = +m2(J − 1)2(I − m)2 +(I − m)m(J − 1)(J − 1)I(I − m)J += m +IJ . +Proof of b) : We have to calculate the cross-covariance values σij = pij − +pi+p+j for i = 1, 2 and j = 1, 2 of R +Σ = +� σ11 +σ12 +σ21 +σ22 +� += +� σ11 +−σ11 +−σ11 +σ11 +� +, +(10) +because (σij = pij − pi+p+j) is row and column centered. +11 + +There is one principal axis u′ +1 = (1 − 1). So, +δ1 = ||Σu1||1 += 4|σ11| += 4m(J − 1)(I − m) +(IJ − m)2 +. +4.2 +Sparse tables +Suppose Z = (zij) is an incidence matrix, a presence-absence data set, +where zij = 0 means level j is absent in the i-th individual, zij = 1 means +level j is present in the i-th individual. CA (or TCA) is a popular method for +the analysis of such tables, see for an example Choulakian and Abou-Samra +(2020). Putting pij = zij/ � +i,j zij and supposing that the marginals p+j > 0 +and pi+ > 0, CA (or TCA) data reconstruction formula is +pij = p+jpi+(1 + +k +� +α=1 +fα(i)gα(j)/δα). +(11) +Now suppose we apply uniformly weighted LRA (or TLRA) ; observing log2(zij+ +1) = zij, then the data reconstruction formula becomes +pij = p+j/I + pi+/J − 1/(IJ) + +k +� +α=1 +fα(i)gα(j)/δα; +(12) +a familiar one known as a FANOVA ( factor analysis and analysis of variance), +see Mandel (1971). +To choose the ”best” between (11) and (12) we use the quality of the +signs of the residuals index, (QSR), within Taxicab framework, see Choula- +kian(2021) and Choulakian et al. (2023). +5 +Examples +Here we analyze three publicly available contingency tables and provide +summary results. For the computations we use the two R packages ca and +TaxicabCA. +12 + +5.1 +Example 1 : Author data set +Greenacre and Lewi (2009) discussed the author contingency table that +has one zero valued count. This data set is of size 12 × 26 and is included in +the correspondence analysis ca in R package by Greenacre et al. (2022). +We consider the power transformed data set N(α) = (nα +ij) for α = 10ˆ(−4); +this value of α is used by Greenacre (2022). By the ca in R package the first +two dispersion-inertia values of N(α) are : +0.003204, +0; +while by Lemma2a, via the merged indicator matrix R of size 2 × 2, with +m = 1, I = 12 and J = 26, we get the value of the first principal inertia : +1/(12 ∗ 26) = 0.003205. +By the TaxicabCA in R package the first two dispersion values of N(α) +are : +0.011369, +9e − 06; +while by Lemma2b, via the R matrix of size 2 × 2, with m = 1, I = 12 and +J = 26, we get the value of the first principal taxicab dispersion : 0.011373. +This example shows the dominant influence of one zero-valued cell in +N(α) = (nα +ij) for α = 10ˆ(−4). The next example shows that similar result is +also obtained for multiple zero-valued cells in one column. +5.2 +Example 2 : RBGlass1 data set +RBGlass1 is a compositional data set of size 105 × 11 that has m = 26 +zeros in the variable Sb column. It is found in the R package archdata by +Carlson and Roth (2022). +We consider the power transformed data set N(α) = (nα +ij) for α = 10ˆ(−4). By +the ca in R package the first two dispersion-inertia values of N(α) are : +0.0225, 0; +while by Lemma2a, via the merged indicator matrix R of size 2 × 2, with +m = 26, I = 105 and J = 11, we get the value of the first principal inertia : +26/(11 ∗ 105) = 0.022511. +By the TaxicabCA in R package the first two dispersion values of N(α) +are : +0.0645, +7e − 06; +13 + +while by Lemma2b, via the R matrix of size 2×2, with m = 26, I = 105 and +J = 11, we get the value of the first principal taxicab dispersion : 0.0645. +5.3 +Example 3 : Rodent abundance data +We consider the rodent data set of size 28 by 9 found in the R package +TaxicabCA. This is an abundance data set of 9 species of rats in 28 cities in +California. Choulakian (2017) analyzed it by comparing the CA and TCA +maps ; Choulakian (2021) showed that it has quasi-2-blocks diagonal struc- +ture ; furthermore Choulakian (2022) analyzed it by Goodman’s marginal-free +CA and marginal-free TCA methods. +Let N be the original data set of size 28 × 9 ; the apparent percentage of +zero counts in N is 66.27%. The function ”CombineCollinearRowsCols” in +the package TaxicabCA in R merges the rows and the columns of N, which +are proportional ; we see that the size of the Nmerged is 21 × 9. So within +the CA framework the real percentage of zero counts in N is the percentage +of zero counts in Nmerged, which is 58.73%. Similarly the real size of the +indicator matrix Z is the size of the Zmerged, which is 14 × 9, whose % of +zeros is 60.32%. +We consider the power transformed data set N(α) = (nα +ij) for α = 10ˆ(−4). +The ca in R package produced exactly the same maps applied to N(α) and +Zmerged. Similarly, the TaxicabCA in R package produced exactly the same +maps applied to N(α) and Zmerged. +The four maps can be seen by applying the R code in the appendix. +6 +Conclusion +We attempted to have a simple clear picture on CA of power transfor- +med or the Box-Cox transformed data initiated since 2009 by Grenacre. We +distinguished two types of data sets : strictly positive and with zeros. In par- +ticular, we showed the dominant influence of the zero entries in the CA of +power transformed data when the power goes to zero ; in this case the power +transformed data set becomes almost a 0-1 indicator matrix. +An alternative approach is to add a positive constant to any zero-valued +cell, then use the log transformation as in the development of (11) ; see, +among others, Lubbe et al. (2021) and Choulakian et al. (2023). +14 + +Acknowledgements. +Choulakian’s research has been supported by NSERC of Canada. +Appendix 1: The R code +The execution of the R code will produce the numerical results and the +maps discussed in the paper. The R code uses the following three packages. +a) The ca package, by Greenacre et al. (2022), does CA and produces the +CA map. +b) The TaxicabCA package, by Allard and Choulakian (2019), does TCA +and produces the TCA map. +c) The archdata package, by Carlson and Roth (2022) for the RBGlasse1 +dataset. +# install packages +install.packages(c(”ca”, ”TaxicabCA”, ”archdata”)) +library(TaxicabCA) +library(ca) +library(archdata) +#Choose a data set +dataMatrix = as.matrix(rodent) +dataMatrix <- t(author) +data(RBGlass1) +dataMatrix <- RBGlass1[, -1] +dim(dataMatrix) +#Compute dataMerged and IndicatorM +dataMerged <- CombineCollinearRowsCols(dataMatrix, rows = T, cols += T) +dim(dataMerged) +IndicatorM = 1-(dataMatrix == 0) +IndicMerged <- CombineCollinearRowsCols(IndicatorM, rows = T, cols += T) +dim(IndicMerged) +#Compute dataPowered and its marginals +alpha <- 10ˆ(-4) +dataPowered <- dataMatrixˆalpha +sum(dataPowered) +apply(dataPowered, 2, function(x) sum(x)) +15 + +apply(dataPowered, 1, function(x) sum(x)) +#CA map of rodent dataset +plot(ca(dataMatrix)) +plot(ca(dataPowered)) +plot(ca(IndicatorM)) +# TCA maps +tca.Data <- tca(dataMatrix, nAxes=2,algorithm = ”exhaustive”) +plot(tca.Data, axes = c(1, 2),labels.rc = c(2, 2)) +tca.Data <- tca(dataPowered, nAxes=2,algorithm = ”exhaustive”) +plot(tca.Data, axes = c(1, 2),labels.rc = c(2, 2)) +tca.Data <- tca(IndicatorM, nAxes=2,algorithm = ”exhaustive”) +plot(tca.Data, axes = c(1, 2),labels.rc = c(2, 2)) +Appendix 2: +Theorem (Greenacre (2010, Result 1) +Under the assumption nij > 0, +∆(α) +ij += p(α) +ij − p(α) +i+ p(α) ++j +p(α) +i+ p(α) ++j +≈ αλ(pij, wR +i = 1/I, wC +j = 1/J) += α(log pij + 1 +IJ +� +i,j +log pij − 1 +I +� +i +log pij − 1 +J +� +j +log pij) +for α → 0 and α > 0. +Proof : By Maclaurin-Taylor series expansion, in the neighborhood of +α = 0, nα +ij = exp(α log nij) = 1 + α log x + O(α2), where r(α) = O(h(α)) +16 + +means limα→0 | r(α) +h(α)| = constant > 0. So for i = 1, ..., I and j = 1, ..., J +p(α) +ij = +nα +ij +� +i,j nα +ij += +1 + α log nij + O(α2) +IJ + � +i,j α log nij + O(α2) +(A1) += +[1 + α log nij + O(α2)] +� +IJ + α � +i,j log nij + O(α2) +� +� +IJ + α � +i,j log nij + O(α2) +�2 += +IJ + αIJ log nij + α � +i,j log nij + O(α2) +[IJ + O(α)]2 +. +(A2) +By (5a) we have +p(α) +i+ = +� +j +p(α) +ij += +J + α � +j log nij + O(α2) +IJ + O(α) +; +(A3) +p(α) ++j = I + α � +i log nij + O(α2) +IJ + O(α) +; +(A4) +p(α) +i+ p(α) ++j = +IJ + αJ � +i log nij + αI � +j log nij + O(α2) +[IJ + O(α)]2 +. +By (A1) and (A4), we have +∆(α) +ij = p(α) +ij − p(α) +i+ p(α) ++j +p(α) +i+ p(α) ++j += +� +IJ + αIJ log nij + α � +i,j log nij + O(α2) +� +− +� +IJ + αJ � +i log nij + αI � +j log nij + O(α2) +� +IJ + αJ � +i log nij + αI � +j log nij + O(α2) += (αIJ +IJ ) +log nij + +1 +IJ +� +i,j log nij − 1 +I +� +i log nij − 1 +J +� +j log nij + O(α) +1 + α 1 +I +� +i log nij + α 1 +J +� +j log nij + O(α2) += αλ(pij, wR +i = 1/I, wC +j = 1/J) + O(α) +1 + O(α) +; +17 + +which is the required result by Lemma 1. +Corollary +By (A3) and (A4), we see that p(α) +i+ = 1/I + O(α) for i = 1, ..., I and +p(α) ++j = 1/J + O(α) for j = 1, ..., J. +References +Aitchison J (1986) The Statistical Analysis of Compositional Data. Lon- +don : Chapman and Hall +Allard J, Choulakian V (2019) Package TaxicabCA in R +Beh E, Lombardo R (2014) Correspondence Analysis : Theory, Practice +and New Strategies. N.Y : Wiley +Beh E, Lombardo R (2022) Correspondence Analysis and the Cressie- +Read Divergence Statistic. National Institute for Applied Statistics Research +, University of Wollongong, Australia, Working Paper 06-22 +Benz´ecri JP (1973) L’Analyse des Donn´ees : Vol. 2 : L’Analyse des Cor- +respondances. Paris : Dunod +Carlson D.L, Roth G (2022) Package archdata in R +Choulakian V (2006) Taxicab correspondence analysis. Psychometrika, +71, 333-345 +Choulakian V (2008) Taxicab correspondence analysis of contingency +tables with one heavyweight column. Psychometrika, 73(2), 309-319 +Choulakian V (2016) Matrix factorizations based on induced norms. Sta- +tistics, Optimization and Information Computing, 4, 1-14 +Choulakian V (2017) Taxicab correspondence analysis of sparse contin- +gency tables. Italian Journal of Applied Statistics, 29 (2-3), 153-179 +Choulakian V (2021) Quantification of intrinsic quality of a principal +dimension in correspondence analysis and taxicab correspondence analysis. +Available on arXiv :2108.10685 +Choulakian V (2022) Some notes on Goodman’s marginal-free correspon- +dence analysis. https ://arxiv.org/pdf/2202.01620.pdf +Choulakian V, Allard J, Mahdi S (2023) Taxicab correspondence analysis +and Taxicab logratio analysis : A comparison on contingency tables and +compositional data. To appear in Austrian Journal of Statistics. +Cuadras CM, Cuadras D, Greenacre M (2006) A comparison of different +methods for representing categorical data. Communications in Statistics- +Simul. and Comp, 35(2), 447-459 +Cuadras CM, Cuadras D (2015 ) A unified approach for the multivariate +analysis of contingency tables. Open Journal of Statistics, 5, 223-232 +18 + +Egozcue JJ, Pawlowsky-Glahn V (2016) Changing the reference measure +in the simplex and its weighting effects. Austrian Journal of Statistics, 45(4), +25-44 +Egozcue JJ, Pawlowsky-Glahn V, Templ M, Hron K (2015) Independence +in contingency tables using simplicial geometry. Communications in Statistics +- Theory and Methods, 44 :18, 3978-3996 +Goodman LA (1979) Simple models for the analysis of association in +cross-classifications having ordered categories. Journal of the American Sta- +tistical Association, 74,537-55 +Goodman LA (1981a) Association models and the bivariate normal for +contingency tables with ordered categories. Biometrika, 68, 347-355 +Goodman LA (1981b) Association models and canonical correlation in +the analysis of cross-classifications having ordered categories. 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Mathema- +tical Geosciences, 43, 681–93 +Greenacre M (2022) The chi-square standardization, combined with Box- +Cox transformation, is a valid alternative to transforming to logratios in +compositional data analysis. Available at https ://arxiv.org/abs/2211.06755 +Greenacre M, Lewi P (2009) Distributional equivalence and subcompo- +sitional coherence in the analysis of compositional data, contingency tables +and ratio-scale measurements. Journal of Classification, 26, 29-54 +Greenacre M, Nenadic O, Friendly M (2022) Package ca in R +Hotelling H (1936) Relations between two sets of variables. Biometrika +28, 321-377 +19 + +Lancaster HO (1958). The structure of bivariate distributions. Ann. Math. +Statist. 29, 719–736 +Mandel J (1971) A new analysis of variance model for non-additive data. +Technometrics, 13(1), 1-18 +Lubbe S, Filzmoser P, Templ M (2021) Comparison of zero replacement +strategies for compositional data with large numbers of zeros. Chem Intell +Lab Syst, 210, 104248. +Tenenhaus M, Augendre H (1996) Analyse factorielle inter-batteries de +Tucker et analyse canonique aux moindres carr´es partiels. In Recueil des +r´esum´es des communications des 28‘eme Journ´ees de statistique, 693-697 +Tucker LR (1958) An inter-battery method of factor analysis. Psychome- +trika, 23, 111-136 +Tukey JW (1977) Exploratory Data Analysis. Addison-Wesley : Reading, +Massachusetts +Yule GU (1912) On the methods of measuring association between two +attributes. JRSS, 75, 579-642 +20 + diff --git a/d9AzT4oBgHgl3EQfaPwm/content/tmp_files/load_file.txt b/d9AzT4oBgHgl3EQfaPwm/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f9df231b82d43cb97e26c615df1f2a69e4a0ab58 --- /dev/null +++ b/d9AzT4oBgHgl3EQfaPwm/content/tmp_files/load_file.txt @@ -0,0 +1,373 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf,len=372 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='01364v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='AP] 3 Jan 2023 Notes on Correspondence Analysis of Power Transformed Data Sets Choulakian V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=', Universit´e de Moncton, Canada vartan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='choulakian@umoncton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='ca January 2023 R´esum´e We prospect for a clear simple picture on CA of power transformed or the Box-Cox transformed data initiated since 2009 by Grenacre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' We distinguish two types of data sets : strictly positive and with zeros ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' we concentrate on the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' Key words : Contingency tables ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' compositional data ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' double cen- tering ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' power transformation ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' interactions ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' zeros ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' taxicab correspon- dence analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' AMS 2010 subject classifications : 62H25, 62H30 1 Introduction Correspondence analysis (CA) and logratio analysis (LRA) are two po- pular methods for the analysis and visualization of a two-way contingency table or a compositional data set N = (nij) of size I × J for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=', I and j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' LRA is considered an ideal desirable method for analysis- visualization of N, because it is row and column scales invariant ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' but it can be applied only when nij > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' While CA can be applied to data when nij ≥ 0, but it is NOT row and column scale invariant because it is dependent on its row and column marginals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' The central topic of this paper is Greenacre’s (2010, Result 1), that we re- name it as theorem, and its application to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' Greenacre’s Theorem concerns the convergence of CA of N(α) = (nα ij) for α ∈ (0, 1] as α → 0 to uniformly weighted LRA of N when nij > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' However, Greenacre (2011, 2022) attempts 1 to show its applicability - usefulness when nij ≥ 0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' that is, when the data set N contains zero valued cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' We aim to shed further light on this latter situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' This paper is organized as follows : Section 2 presents preliminaries ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' sec- tion 3 presents Greenacre’s Theorem concerning CA of strictly positive data sets ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' section 4 presents CA of nonnegative data sets having zero valued cells ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' section 5 discusses three examples of real data sets with zero cells ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' finally we conclude in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='1 Some references CA and LRA are based on three different principles : CA on Benz´ecri’s distributional equivalence principle, RC association models on Yule’s scale invariance principle, and CoDA on Aitchison’s subcompositional coherence principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' A recent discussion of these three principles can be found in Chou- lakian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' Benz´ecri (1973) is the reference book on CA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' Beh and Lombardo (2014) present a panoramic review of CA and its variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' LRA includes two independently well developed methods : RC association models for the analysis of contingency tables by Goodman (1979, 1981a, 1981b, 1991, 1996) and a set of compositional vectors (CoDA) by Aitchison (1986), see also among others Greenacre (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' The relationship between CA and LRA has been discussed in many pa- pers ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' see for example, among others, Goodman (1996), Cuadras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' (2006), Cuadras and Cuadras (2015 ), Greenacre (2009, 2010), Beh and Lombardo (2022) and Choulakian (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' 2 Preliminaries on analysis of contingency tables We consider a two-way contingency table N = (nij) for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=', I and j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=', J, and P = N/n = (pij) of size I ×J the associated correspondence matrix (probability table) of the contingency table N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' We define as usual pi+ = �J j=1 pij , p+j = �I i=1 pij, the vector r = (pi+) ∈ RI, the vector c = (p+j) ∈ RJ, and MI = Diag(r) the diagonal matrix having diagonal elements pi+, and similarly MJ = Diag(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' We suppose that MI and MJ are positive definite metric matrices of size I × I and J × J, respectively ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' this means that the diagonal elements of MI and MJ are strictly positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='1 Independence of the row and column categories a) The I row categories and the J column categories are mutually inde- pendent, then σij = pij − pi+p+j = 0 (1) for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=', I and j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=', J, and, where σij is the residual matrix of pij with respect to the independence model pi+p+j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' Remark 1 : The contingency table N = (nij) can also be represented (coded) as an indicator matrix Z = [ZI ZJ] = [(zαi) (zαj)] of size n×(I+J), where zαi = 0 if individual α does not have level i of the row variable, zαi = 1 if individual α has level i of the row variable ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' zαj = 0 if individual α does not have level j of the column variable, zαj = 1 if individual α has level j of the column variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' Note that N = ZI ′ZJ and σij = pij − pi+p+j is the covariance between the i-th column of ZI and the j-th column of ZJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' b) The independence assumption σij = 0 can also be interpreted in ano- ther way as ∆ij = ( pij pi+p+j − 1) = 0 (2) = 1 pi+ ( pij p+j − pi+) = 1 p+j ( pij pi+ − p+j);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' this is the column and row homogeneity model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' Benz´ecri (1973, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='31) named the conditional probability vector ( pij p+j for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=', I and j fixed) the profile of the j-th column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' He also referred to the element of pij pi+p+j as the density function of the probability measure (pij) with respect to the product measure pi+p+j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' The element pij pi+p+j is named Pearson ratio in Goodman (1996) and Beh and Lombardo (2014, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='123).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' c) A third way to represent the independence assumption σij = 0 and the row and column homogeneity models ∆ij = 0 is via the (wR i , wC j ) weighted loglinear formulation, equation (3), assuming pij > 0 and defining Gij = log(pij), λ(pij, wR i , wC j ) = Gij − Gi+ − G+j + G++ = 0, (3) 3 where Gi+ = �J j=1 GijwC j , G+j = �I i=1 GijwR i and G++ = �J j=1 �I i=1 GijwC j wR i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' wC j > 0 and wR i > 0, satisfying �J j=1 wC j = �I i=1 wR i = 1, are a priori fixed or data dependent probability weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' Two popular weights are marginal (wR i = pi+, wC j = p+j) and uniform (wR i = 1/I, wC j = 1/J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' This is implicit in equation 7 in Goodman (1996) or equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='6 in Goodman (1991) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' and explicit in Egozcue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' Equation (3) is equivalent to the logratios log(pijpi1j1 pij1pi1j ) = 0 for i ̸= i1 and j ̸= j1, which Goodman (1979, equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='2) refers to as the ”null association” mo- del.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' Equation (3) is also equivalent to pij = exp(Gi+) exp(G+j) exp(G++) , from which we deduce that : under the independence assumption the mar- ginal row probability vector (pi+) is proportional to the vector of weighted geometric means (exp(Gi+));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' and a similar property is true also for the co- lumns ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' see for instance Egozcue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='2 Interaction factorization Suppose the independence-homogeneity-null association models are not true, then each of the three equivalent model formulations (1), (2), (3) can be generalized to explain the nonindependence-nonhomogeneity-association, named interaction, among the I rows and the J columns by adding k bilinear terms, where k = rank(N)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' We designate any one of the interaction indices (1), (2), (3) by τij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' Benz´ecri (1973, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' 31-32) emphasized the importance of row and column weights or metrics in multidimensional data analysis ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' this is the reason in the French data analysis circles any study starts with a triplet (X, MI, MJ), where X represents the data set, MI = (Diag(mr i)) is the metric matrix defined on the rows and MJ = (Diag(mc j)) the metric ma- trix defined on the columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' We follow the same procedure but X is the preprocessed data, where : a) In covariance analysis, X = (τij) = (IJσij) and 4 (MI, MJ) = (Diag(1/I), Diag(1/J)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' b) In CA, X = (τij) = (∆ij) and (MI, MJ) = (Diag(pi+), Diag(p+j));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' c) In LRA, X = (τij) = (λ(pij, wR i , wC j )) and (MI, MJ) = (Diag(wR i ), Diag(wC j )) with �J j=1 wC j = �I i=1 wR i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' Note that in a) we have multiplied σij by IJ to unify theoretically the covariance analysis with the CA analysis, see the subsection on double cen- tering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' We factorize the interactions in (1), (2) and (3) by singular value decom- position (SVD) or taxicab SVD (TSVD) as τij = k � α=1 fα(i)gα(j)/δα, (4) where fα(i) is the ith row principal coordinate and gα(j) is the jth co- lumn principal coordinate along the αth principal dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' Also δα is the dispersion measure of the αth principal axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' Remark 2 : a) In the SVD case the parameters (fα(i), gα(j), δα) satisfy the conditions : for α, β = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=', k δ2 α = I � i=1 f 2 α (i)mr i = J � j=1 g2 α(j)mc j 0 = I � i=1 fα(i)mr i = J � j=1 gα(j)mc j 0 = I � i=1 fα(i)fβ(i)mr i = J � j=1 gα(j)gβ(j)mc j for α ̸= β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' b) In the TSVD case the parameters (fα(i), gα(j), δα) satisfy the conditions : for α, β = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=', k δα = I � i=1 |fα|(i)mr i = J � j=1 |gα(j)|mc j 0 = I � i=1 fα(i)mr i = J � j=1 gα(j)mc j 5 0 = I � i=1 fα(i) sign(fβ(i))mr i = J � j=1 gα(j) sign(gβ(j))mc j for α > β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' A description of TSVD can be found in Choulakian (2006, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' Remark 3 a) In the case (τij) = (IJσij), the bilinear decomposition (4) is also named interbattery analysis first proposed by Tucker (1958) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' later on, Tenenhaus and Augendre (1996) reintroduced it within correspondence analysis circles, where they showed that the Tucker decomposition by SVD produced on some correspondence tables more interesting (interpretable) structure than CA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' b) In the case (τij) = (∆ij), the CA decomposition has many interpreta- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' Essentially, for data analysis, purposes Benz´ecri (1973) interpreted it as weighted principal components analysis of row and column profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' Ano- ther useful interpretation of CA, comparable to Tucker interbattery analysis, is Hotelling(1936)’s canonical correlation analysis, see Lancaster (1958) and Goodman (1991, 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' c) In the case (τij) = (λ(pij, wR i , wC j )) and (MI, MJ) = (Diag(wR i ), Diag(wC j )), where (wR i , wC j ) are prespecified ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' we note this case by TLRA or LRA ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' for an example of general prespecified weights see Egozcue and Pawlowsky-Glahn (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' For the important particular case λ(pij, wR i , wC j ) = λ(pij, 1/I, 1/J), we get uniformly-weighted (or taxicab) logratio analysis uwLRA (or uwTLRA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' d) We are concerned with the property of scale dependence or indepen- dence of the three interaction indices (1), (2) and (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' The three indices depend on pij where pij = nij/ � i,j nij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' To emphasize this dependence, we ex- press any one of of the three interaction indices by τij(nij) = τ(pij, mR i , mC j ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' Following Yule (1912), we state the following Definition 1 : An interaction index τij(nij) is scale invariant if τij(nij) = τij(ainijbj) for arbitrary scales ai > 0 and bj > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' It is evident that neither of the three indices with data dependent mar- ginal weights (pi+, p+j) are scale invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' Concerning the association index (3) we have the following lemma, see Choulakian (2022) or Choulakian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' Lemma 1 : a) The association index (3) with prespecified weights (wR i , wC j ) 6 is scale invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' That is, λ(nij, wR i , wC j ) = λ(ainijbj/n∗, wR i , wC j ), for arbitrary scales ai > 0 and bj > 0 and n∗ = � i,j ainijbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' b) To a first-order approximation, λ(nij, wR i , wC j ) ≈ ( pij wC j wR i − pi+ wR i − p+j wC j +1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='3 Double centering The interaction matrix (τij) is double centered, that is : 0 = I � i=1 mr i mc jτij = J � j=1 mr i mc jτij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' (5) According to Tukey (1977, chapter 10), there are two kinds of double cen- tering row-PLUS-column and row-TIMES-column ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' we name them additive and multiplicative, that we describe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' We consider the triplet (Y, MI, MJ), where Yij = h(pij) where h is a function and (MI, MJ) = (Diag(mr i), Diag(mc j)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' We compute the three means, two marginals and the total : Yi+ = �J j=1 Yijmc j, Y+j = �I i=1 Yijmr i and Y++ = �J j=1 �I i=1 Yijmr imc j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' The multiplicative double centering is τij = Yij − Yi+Y+j Y++ , (6) where rank(τij) = rank(Yij) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' The additive double centering is τij = Yij − Yi+ − Y+j + Y++, (7) where rank(τij) = rank(Yij) − 1 or rank(Yij) − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' An evident difference between the two types of double centering is that (7) is invariant to a row or column additive constants ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' Definition 1 and Lemma1a become a corollary to this fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' We consider two functional forms of Yij = h(pij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' 7 a) Yij = h(pij) = pij mr i mc j is the density function of the joint probability measure pij with respect to the product measure mr i mc j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' First, the cova- riance interaction (τij) = (IJσij) is obtained from (6), when mr i = 1/I and mc j = 1/J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' So the rank(IJσij) = rank(IJpij) −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' Second, the CA interaction (τij) = (∆ij) is obtained either from (6) or from (7), when mr i = pi+ and mc j = p+j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' So the rank(∆ij) = rank( pij pi+p+j ) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' Third, the right-hand side of Lemma 1b is obtained from (7) when mr i = wR i and mc j = wC j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' b) Yij = h(pij) = log pij for pij > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' First, the log interaction (τij) = (λ(pij, wR i , wC j )) is obtained from (7), when mr i = wR i and mc j = wC j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' So the rank (λ(pij, wR i , wC j )) = rank(log pij) − 1, as in Goodman (1991, Table 11) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' or rank(log pij) − 2, as in Goodman (1991, Table 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' Second, to our knowledge the log interaction obtained from (6) has not been applied yet, most probably due to the fact that it is not row and column scale invariant ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' see also subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' Choulakian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' (2023) used the quality of the sign of the residuals index, QSR, to choose an optimal case among the different cases mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' 3 CA of power transformed strictly positive data Here, we state Greenacre’s Theorem and provide a mathematical proof in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' Let (p(α) ij = nα ij � i,j nα ij ) for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=', I and j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=', J be the correspon- dence table of the power transformed strictly positive data, and (∆(α) ij = p(α) ij −p(α) i+ p(α) +j p(α) i+ p(α) +j ) the CA interaction matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' Theorem (Greenacre (2010, Result 1) Under the assumption nij > 0, for α → 0 and α > 0 ∆(α) ij = p(α) ij − p(α) i+ p(α) +j p(α) i+ p(α) +j ≈ αλ(pij, wR i = 1/I, wC j = 1/J) = α(log pij + 1 IJ � i,j log pij − 1 I � i log pij − 1 J � j log pij) Furthermore, p(α) i+ ≈ 1/I for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=', I and p(α) +j ≈ 1/J for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' 8 A way to see the theorem is to look at the following sequence of approxi- mations using Lemma1b, and, the fact that p(α) i+ ≈ 1/I for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=', I and p(α) +j ≈ 1/J for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=', J as α → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' λ(p(α) ij , wR i = 1/I, wC j = 1/J) = αλ(pij, wR i = 1/I, wC j = 1/J) ≈ (IJp(α) ij + 1 − Ip(α) i+ − Jp(α) +j ) ≈ (IJp(α) ij − 1) ≈ ∆(α) ij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='1 Remarks a) The Theorem is interesting, but not useful in empirical contexts, be- cause one can directly compute the log interactions λ(pij, wR i = 1/I, wC j = 1/J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' b) In the above Theorem, the assumption nij > 0 is fundamental : if α = 0, then nα ij = n0 ij = 1 for all (i, j);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' that is, the rank of the matrix (nα ij = n0 ij = 1) is 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' thus both sides of the equation in the Theorem are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' c) In the proof of the above Theorem in the appendix, one sees that lim α→0 ∆(α) ij α = λ(pij, wR i = 1/I, wC j = 1/J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' d) Assume nij ≥ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' then another power transformation is the logarithmic transformation, also known as the Box-Cox transformation, L = (lognij = limα→0 1 α(nα ij − 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' We have not seen any application of CA to L, see the subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='3 on double centering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' Note that the assumption in Greenacre’s Theorem nij > 0 is weaker than the assumption nij ≥ 1 for the use of the log transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' Goodman’s RC model, based on the log transformation, has been developed for the analysis of contingency tables (tables with strictly positive counts), and not for compositional data where often data in % have strictly positive values less than 1, besides having positive values larger than 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' see for instance, the archeological compositional CUPS data in Greenacre and Lewi (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' e) There is some kind of kinship between Greenacre’s Theorem and Good- man’s marginal free CA (mfCA), see Goodman (1996, equation (46)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' In mfCA of a probability table P = (pij) with row and column marginals (pi+) 9 and (p+j) respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' CA is applied to a matrix Q = (qij = aipijbj) related to P in two steps : Step 1 : by the scale invariance property of the log interaction index,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' see Lemma1a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' under the assumption nij > 0 there exists a unique proba- bility matrix Q =(qij) which is related to P = (pij) via the strictly positive scales (ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' bj),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' that keeps Yule’s association between the i-th row and the j-th column unchanged ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' that is, λ(pij, wR i = 1/I, wC j = 1/J) = λ(qij = aipijbj, wR i = qi+ = 1/I, wC j = q+j = 1/J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' Note that Q has uniform row and column marginal weights similar to (p(α) ij ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' Step 2, mfCA of P is CA representation of Q IJqij − 1 = k � α=1 fα(i)gα(j)/δα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' For further details, see Choulakian (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' 4 CA of power transformed data set having zero valued cells It is quite common that large compositional data sets or contingency tables have zero valued cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' For this case, let m be the number of zero valued cells ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' so IJ −m is the number of strictly positive valued cells in the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' We see that for such tables the log transformation discussed in Remark b is not valid, while the simple power transformation nα ij for α ∈ [0, 1] is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' That is, the matrix (lim nα ij) as α → 0 converges to an indicator matrix Z = (zij), where zij = 1 if nij > 0 and zij = 0 if nij = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' A comparison of this case with the observation in Remark a) under the assumption nij > 0 is insightful, where the indicator matrix becomes Z = (zij = 1) of rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' Here, it is insightful to consider two distinct cases : Sparse tables containing multiple zeros in at least two nonproportional rows or nonproportional columns, and tables with exactly one column (or row) having at least one zero valued cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' In the latter case the zero entries exhibit a dominating influence very similar to the cases of a heavyweight cell and a heavyweight column (or row) discussed by Benz´ecri (1979) and Lebart (1979) in CA, and by Choulakian (2008) in TCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='1 Tables with one column having m zero-valued cells Here we discuss the case where there are m for m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=', I −1 zero valued cells in one column of the indicator matrix Z of size I × J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' By Benz´ecri’s principle of distributional equivalence property which states that in CA (or TCA) proportional rows or columns can be merged, CA (or TCA) of Z is identical to CA (or TCA) of the 2 × 2 contingency table R = � 0 m(J − 1) (I − m) (I − m)(J − 1) � , (8) which has only one CA (or TCA) principal dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' Lemma 2 a) In CA of R the dispersion, named inertia, ρ2 1 = m IJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='. b) In TCA of R the taxicab dispersion δ1 = 4m(J−1)(I−m) (IJ−m)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' Proof of a) : By equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='4 in Goodman (1996), for a 2×2 contingency table the ”ninindependence” based on the correlation coefficient is ρ2 1 = (p11p22 − p12p21)2 p1+p2+p+1p+2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' (9) By replacing the probability values of the elements of R in (9), we get ρ2 1 = m2(J − 1)2(I − m)2 (I − m)m(J − 1)(J − 1)I(I − m)J = m IJ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' Proof of b) : We have to calculate the cross-covariance values σij = pij − pi+p+j for i = 1, 2 and j = 1, 2 of R Σ = � σ11 σ12 σ21 σ22 � = � σ11 −σ11 −σ11 σ11 � , (10) because (σij = pij − pi+p+j) is row and column centered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' 11 There is one principal axis u′ 1 = (1 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' So, δ1 = ||Σu1||1 = 4|σ11| = 4m(J − 1)(I − m) (IJ − m)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='2 Sparse tables Suppose Z = (zij) is an incidence matrix, a presence-absence data set, where zij = 0 means level j is absent in the i-th individual, zij = 1 means level j is present in the i-th individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' CA (or TCA) is a popular method for the analysis of such tables, see for an example Choulakian and Abou-Samra (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' Putting pij = zij/ � i,j zij and supposing that the marginals p+j > 0 and pi+ > 0, CA (or TCA) data reconstruction formula is pij = p+jpi+(1 + k � α=1 fα(i)gα(j)/δα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' (11) Now suppose we apply uniformly weighted LRA (or TLRA) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' observing log2(zij+ 1) = zij, then the data reconstruction formula becomes pij = p+j/I + pi+/J − 1/(IJ) + k � α=1 fα(i)gα(j)/δα;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' (12) a familiar one known as a FANOVA ( factor analysis and analysis of variance), see Mandel (1971).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' To choose the ”best” between (11) and (12) we use the quality of the signs of the residuals index, (QSR), within Taxicab framework, see Choula- kian(2021) and Choulakian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' 5 Examples Here we analyze three publicly available contingency tables and provide summary results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' For the computations we use the two R packages ca and TaxicabCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' 12 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='1 Example 1 : Author data set Greenacre and Lewi (2009) discussed the author contingency table that has one zero valued count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' This data set is of size 12 × 26 and is included in the correspondence analysis ca in R package by Greenacre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' We consider the power transformed data set N(α) = (nα ij) for α = 10ˆ(−4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' this value of α is used by Greenacre (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' By the ca in R package the first two dispersion-inertia values of N(α) are : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='003204, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' while by Lemma2a, via the merged indicator matrix R of size 2 × 2, with m = 1, I = 12 and J = 26, we get the value of the first principal inertia : 1/(12 ∗ 26) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='003205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' By the TaxicabCA in R package the first two dispersion values of N(α) are : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='011369, 9e − 06;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' while by Lemma2b, via the R matrix of size 2 × 2, with m = 1, I = 12 and J = 26, we get the value of the first principal taxicab dispersion : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='011373.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' This example shows the dominant influence of one zero-valued cell in N(α) = (nα ij) for α = 10ˆ(−4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' The next example shows that similar result is also obtained for multiple zero-valued cells in one column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='2 Example 2 : RBGlass1 data set RBGlass1 is a compositional data set of size 105 × 11 that has m = 26 zeros in the variable Sb column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' It is found in the R package archdata by Carlson and Roth (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' We consider the power transformed data set N(α) = (nα ij) for α = 10ˆ(−4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' By the ca in R package the first two dispersion-inertia values of N(α) are : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='0225, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' while by Lemma2a, via the merged indicator matrix R of size 2 × 2, with m = 26, I = 105 and J = 11, we get the value of the first principal inertia : 26/(11 ∗ 105) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='022511.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' By the TaxicabCA in R package the first two dispersion values of N(α) are : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='0645, 7e − 06;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' 13 while by Lemma2b, via the R matrix of size 2×2, with m = 26, I = 105 and J = 11, we get the value of the first principal taxicab dispersion : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='0645.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='3 Example 3 : Rodent abundance data We consider the rodent data set of size 28 by 9 found in the R package TaxicabCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' This is an abundance data set of 9 species of rats in 28 cities in California.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' Choulakian (2017) analyzed it by comparing the CA and TCA maps ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' Choulakian (2021) showed that it has quasi-2-blocks diagonal struc- ture ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' furthermore Choulakian (2022) analyzed it by Goodman’s marginal-free CA and marginal-free TCA methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' Let N be the original data set of size 28 × 9 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' the apparent percentage of zero counts in N is 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='27%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' The function ”CombineCollinearRowsCols” in the package TaxicabCA in R merges the rows and the columns of N, which are proportional ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' we see that the size of the Nmerged is 21 × 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' So within the CA framework the real percentage of zero counts in N is the percentage of zero counts in Nmerged, which is 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='73%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' Similarly the real size of the indicator matrix Z is the size of the Zmerged, which is 14 × 9, whose % of zeros is 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='32%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' We consider the power transformed data set N(α) = (nα ij) for α = 10ˆ(−4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' The ca in R package produced exactly the same maps applied to N(α) and Zmerged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' Similarly, the TaxicabCA in R package produced exactly the same maps applied to N(α) and Zmerged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' The four maps can be seen by applying the R code in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' 6 Conclusion We attempted to have a simple clear picture on CA of power transfor- med or the Box-Cox transformed data initiated since 2009 by Grenacre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' We distinguished two types of data sets : strictly positive and with zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' In par- ticular, we showed the dominant influence of the zero entries in the CA of power transformed data when the power goes to zero ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' in this case the power transformed data set becomes almost a 0-1 indicator matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' An alternative approach is to add a positive constant to any zero-valued cell, then use the log transformation as in the development of (11) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' see, among others, Lubbe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' (2021) and Choulakian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' 14 Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' Choulakian’s research has been supported by NSERC of Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' Appendix 1: The R code The execution of the R code will produce the numerical results and the maps discussed in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' The R code uses the following three packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' a) The ca package, by Greenacre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' (2022), does CA and produces the CA map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' b) The TaxicabCA package, by Allard and Choulakian (2019), does TCA and produces the TCA map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' c) The archdata package, by Carlson and Roth (2022) for the RBGlasse1 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' # install packages install.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='packages(c(”ca”, ”TaxicabCA”, ”archdata”)) library(TaxicabCA) library(ca) library(archdata) #Choose a data set dataMatrix = as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='matrix(rodent) dataMatrix <- t(author) data(RBGlass1) dataMatrix <- RBGlass1[,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' -1] dim(dataMatrix) #Compute dataMerged and IndicatorM dataMerged <- CombineCollinearRowsCols(dataMatrix,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' rows = T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' cols = T) dim(dataMerged) IndicatorM = 1-(dataMatrix == 0) IndicMerged <- CombineCollinearRowsCols(IndicatorM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' rows = T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' cols = T) dim(IndicMerged) #Compute dataPowered and its marginals alpha <- 10ˆ(-4) dataPowered <- dataMatrixˆalpha sum(dataPowered) apply(dataPowered,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' function(x) sum(x)) 15 apply(dataPowered,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' function(x) sum(x)) #CA map of rodent dataset plot(ca(dataMatrix)) plot(ca(dataPowered)) plot(ca(IndicatorM)) # TCA maps tca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='Data <- tca(dataMatrix, nAxes=2,algorithm = ”exhaustive”) plot(tca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='Data, axes = c(1, 2),labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='rc = c(2, 2)) tca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='Data <- tca(dataPowered, nAxes=2,algorithm = ”exhaustive”) plot(tca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='Data, axes = c(1, 2),labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='rc = c(2, 2)) tca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='Data <- tca(IndicatorM, nAxes=2,algorithm = ”exhaustive”) plot(tca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='Data, axes = c(1, 2),labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='rc = c(2, 2)) Appendix 2: Theorem (Greenacre (2010, Result 1) Under the assumption nij > 0, ∆(α) ij = p(α) ij − p(α) i+ p(α) +j p(α) i+ p(α) +j ≈ αλ(pij, wR i = 1/I, wC j = 1/J) = α(log pij + 1 IJ � i,j log pij − 1 I � i log pij − 1 J � j log pij) for α → 0 and α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' Proof : By Maclaurin-Taylor series expansion, in the neighborhood of α = 0, nα ij = exp(α log nij) = 1 + α log x + O(α2), where r(α) = O(h(α)) 16 means limα→0 | r(α) h(α)| = constant > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' So for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=', I and j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=', J p(α) ij = nα ij � i,j nα ij = 1 + α log nij + O(α2) IJ + � i,j α log nij + O(α2) (A1) = [1 + α log nij + O(α2)] � IJ + α � i,j log nij + O(α2) � � IJ + α � i,j log nij + O(α2) �2 = IJ + αIJ log nij + α � i,j log nij + O(α2) [IJ + O(α)]2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' (A2) By (5a) we have p(α) i+ = � j p(α) ij = J + α � j log nij + O(α2) IJ + O(α) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' (A3) p(α) +j = I + α � i log nij + O(α2) IJ + O(α) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' (A4) p(α) i+ p(α) +j = IJ + αJ � i log nij + αI � j log nij + O(α2) [IJ + O(α)]2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' By (A1) and (A4), we have ∆(α) ij = p(α) ij − p(α) i+ p(α) +j p(α) i+ p(α) +j = � IJ + αIJ log nij + α � i,j log nij + O(α2) � − � IJ + αJ � i log nij + αI � j log nij + O(α2) � IJ + αJ � i log nij + αI � j log nij + O(α2) = (αIJ IJ ) log nij + 1 IJ � i,j log nij − 1 I � i log nij − 1 J � j log nij + O(α) 1 + α 1 I � i log nij + α 1 J � j log nij + O(α2) = αλ(pij, wR i = 1/I, wC j = 1/J) + O(α) 1 + O(α) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' 17 which is the required result by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=' Corollary By (A3) and (A4), we see that p(α) i+ = 1/I + O(α) for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=', I and p(α) +j = 1/J + O(α) for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfaPwm/content/2301.01364v1.pdf'} +page_content=', J.' metadata={'source': 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--git a/d9E1T4oBgHgl3EQfyAW2/content/tmp_files/2301.03429v1.pdf.txt b/d9E1T4oBgHgl3EQfyAW2/content/tmp_files/2301.03429v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..22c090cbabca21876905db1de169992e626b3499 --- /dev/null +++ b/d9E1T4oBgHgl3EQfyAW2/content/tmp_files/2301.03429v1.pdf.txt @@ -0,0 +1,2314 @@ +arXiv:2301.03429v1 [math.AP] 9 Jan 2023 +Local null controllability of a cubic Ginzburg-Landau equation with +dynamic boundary conditions +Nicol´as Carre˜no∗ +Alberto Mercado† +Roberto Morales‡ +January 10, 2023 +Abstract +This paper deals with controllability properties of a cubic Ginzburg-Landau equation with +dynamic boundary conditions. More precisely, we prove a local null controllability result by +using a single control supported in a small subset of the domain. In order to achieve this result, +we firstly linearize the system around the origin and we analyze it by the duality approach and +an appropriate Carleman estimate. Then, by using an inverse function theorem, the local null +controllability of the nonlinear system is proven. +Keyword: Controllability, Ginzburg-Landau equation, Dynamic boundary conditions. +MSC(2020) 93B05, 35Q56, 93B07. +1 +Introduction and main results +1.1 +Introduction +Let Ω ⊂ RN (N ⩾ 2) be a bounded domain with boundary Γ := ∂Ω of class C2. +Given the +parameters a, b, c > 0, α, γ ∈ R \ {0}, we consider the following cubic Ginzburg-Landau equation +with dynamic boundary conditions + + + + + + + + + + + +∂tu − a(1 + αi)∆u + c(1 + γi)|u|2u = +1ωh, +in Ω × (0, T), +∂tuΓ + a(1 + αi)∂νu − b(1 + αi)∆ΓuΓ + c(1 + γi)|uΓ|2uΓ = 0, +on Γ × (0, T), +u = uΓ, +on Γ × (0, T), +(u(0), uΓ(0)) = (u0, uΓ,0), +in Ω × Γ. +(1.1) +Here, (u, uΓ) is the state of the system, (u0, uΓ,0) the initial conditions and h ∈ L2(ω ×(0, T); C) +is a control acting on ω ⊂ Ω. We denote by ∆Γ the Laplace-Beltrami operator on Γ and by ∂νy the +normal derivative associated to the outward normal ν of Ω. +We notice that (1.1) can be seen as a coupled system in the variables (u, uΓ), which is controlled +by a single control h in a (small) subset ω of Ω. +This means that the first equation of (1.1) +∗Departamento de Matem´atica, Universidad T´ecnica Federico Santa Mar´ıa, Casilla 110-V, Valpara´ıso, Chile e-mail: +nicolas.carrenog@usm.cl. Partially supported by Fondecyt 1211292. +†Departamento de Matem´atica, Universidad T´ecnica Federico Santa Mar´ıa, Casilla 110-V, Valpara´ıso, Chile e-mail: +alberto.mercado@usm.cl. Partially supported by Fondecyt 1211292 and ANID Millennium Science Initiative Program, +Code NCN19-161. +‡Departamento de Matem´atica, Universidad T´ecnica Federico Santa Mar´ıa, Casilla 110-V, Valpara´ıso, Chile e-mail: +roberto.moralesp@usm.cl. Partially supported by Fondecyt 3200830. +1 + +is controlled directly by the action of the control, while the second equation is being controlled +through the side condition u = uΓ on Γ × (0, T). +The main objective of this work is to obtain the local null controllability of system (1.1), i.e., we +will prove the existence of a number δ > 0 such that, for every initial state (u0, uΓ,0) ∈ X (where +X is an appropriate Banach space) which fulfills +∥(u0, uΓ,0)∥X ⩽ δ, +we can find a control h ∈ L2(ω × (0, T)) such that the associated solution (u, uΓ) of (1.1) satisfies +y(·, T) = 0, in Ω, +yΓ(·, T) = 0, on Γ. +1.2 +Previous results +The cubic complex Ginzburg-Landau equation is one of the most studied nonlinear equations used +to model physical phenomena. This equation has been used to describe several phenomena ranging +from nonlinear waves, second-order phase transitions, superconductivity, superfluidity and Bose- +Einstein condensation to liquid crystals and strings in field theory. For a detailed description of +relevant applications in different fields, see [6] and [18]. +The existence and uniqueness of nonlinear Ginzburg-Landau equations with Dirichlet or periodic +boundary conditions have been intensely investigated in several papers. For instance, we refer to +[21], [16], [17], [12], [9], [8], [7] and the references therein. Concerning controllability properties of +the Ginzburg-Landau equation with Dirichlet boundary conditions, only a few papers have been +devoted to the study of the controllability of such problems. In [1], the stabilization of the linearized +Ginzburg-Landau model with Dirichlet boundary conditions around an unstable equilibrium state +is studied. Moreover, in [12], the author develop a Carleman inequality for an operator of the form +(a + ib)∂t + div(A · ∇), +with A being a smooth, uniformly elliptic matrix, and a null controllability result for the linear +PDE with a distributed control. In 2009, L. Rosier and B.-Y. Zhang in [25] proved a controllability +result for the nonlinear case. +In this case, the control acts on a part of the boundary and the +proof is based on a suitable Carleman estimate for the linear adjoint system. Then, combining a +fixed-point argument together with the theory of sectorial operators, the authors obtained a local +controllability result for a wide class of nonlinearities. In particular, controllability results for the +cubic and quintic Complex Ginzburg-Landau are provided. +Recently, some results on inverse problems and controllability issues have been obtained for +PDEs with dynamic boundary conditions, see for instance [22], [19], [4], [2], [3], and [20] for the +heat equation, [15] for the wave equation, and [24] for the Schr¨odinger operator. In these works, +the authors has been used the duality equivalence to prove the associated observability inequality +by using Carleman estimates. At this level, we point out that it is not evident at all that such +systems can be controlled by the action of a single control due to the tangential derivative terms. In +fact, in the case of the linear wave equation with mixed boundary conditions (oscilatory boundary +conditions and Dirichlet boundary conditions) [15], the authors obtain exact controllability results +where the control region is on the whole boundary (and therefore on the whole system). On the other +hand, in a similar setting for the Schr¨odinger equation, in [24] is obtained the exact controllability +with a control acting only in a part of the boundary, by using a particular Carleman weight adapted +to the geometric properties of the domain, which allows to estimate boundary terms. +Concerning the Ginzburg-Landau equation with dynamic boundary conditions, we mention [11], +where well-posedness of linear/nonlinear of such models is obtained and long time behavior of +2 + +solutions is characterized when Lipschitz nonlinearities are considered. However, to the best of the +authors’ knowledge, this is the first time that the null controllability for the cubic Ginzburg-Landau +is studied. +In the following subsection we shall introduce some functional spaces used in the context of +PDE’s with dynamic boundary conditions. +1.3 +General setting +In this section, we set up the notation and terminology used in this paper. The set Γ = ∂Ω can +be seen as an (N − 1)-dimensional compact Riemannian submanifold equipped by the Riemmanian +metric g induced by the natural embedding Γ ⊂ RN. +In addition, we shall denote by dS the +(N − 1)-Lebesgue measure for Γ. +Since we are considering dynamic boundary conditions, we need to define some differential +operators on Γ, which can be defined in terms of the metric g. However, for our purposes, it will be +enough to use the most important properties of the underlaying operators and spaces. The details +can be found, for instance, in [27]. For the sake of completeness, we recall some of those properties. +The tangential gradient ∇Γ of yΓ at each point x ∈ Γ can be seen as the projection of the +standard Euclidean gradient ∇y onto the tangent space of Γ at x ∈ Γ, where yΓ is the trace of y on +Γ, i.e., we have the following equation +∇ΓyΓ = ∇y − ν∂νy, +where y = yΓ on Γ and ∂νy is the normal derivative associated to the outward normal ν. In this +way, the tangential divergence divΓ in Γ is defined by +divΓ(FΓ) : H1(Γ; R) → R, +yΓ �→ − +� +Γ +FΓ · ∇ΓyΓdS. +The Laplace-Beltrami operator is given by ∆ΓyΓ := div(∇ΓyΓ), for all yΓ ∈ H2(Γ; R). +In +particular, the surface divergence theorem holds: +� +Γ +∆ΓyΓzΓdS = − +� +Γ +∇ΓyΓ · ∇ΓzΓdS, +∀yΓ ∈ H2(Γ; R), +zΓ ∈ H1(Γ; R). +In order to simplify the notation, here and subsequently, the function spaces refer to complex- +valued functions unless otherwise stated. +For 1 ⩽ p ⩽ +∞, we consider the Banach space Lp := Lp(Ω) × Lp(Γ), endowed by the norm +given by the relation +∥(u, uΓ)∥2 +Lp := ∥u∥2 +Lp(Ω) + ∥uΓ∥2 +Lp(Γ). +In particular, for p = 2, the space L2 := L2(Ω) × L2(Γ) is a (real) Hilbert space equipped with +the scalar product +⟨(u, uΓ), (v, vΓ)⟩L2 := ℜ +� +Ω +uvdx + ℜ +� +Γ +uΓvΓdσ. +For k ∈ N, we also introduce the space +Hk := {(y, yΓ) ∈ Hk(Ω) × Hk(Γ) ; y +�� +Γ = yΓ}, +where Hk(Ω) and Hk(Γ) are the usual Sobolev spaces. +3 + +1.4 +Main result +Our main result is given by the following theorem: +Theorem 1.1. Suppose that N = 2 or N = 3. Let a, b, c > 0, α, γ ∈ R \ {0}. Then, for every +T > 0 and ω ⋐ Ω, there exists δ > 0 such that, for every (y0, yΓ,0) ∈ H1 satisfying +∥(u0, uΓ,0)∥H1 ⩽ δ, +there exists a control h ∈ L2(ω ×(0, T)) such that the unique corresponding solution (u, uΓ) of (1.1) +satisfies +u(·, T) = 0, in Ω, +uΓ(·, T) = 0, on Γ, +i.e., the nonlinear system (1.1) is locally null controllable by a single control for an arbitrary control +domain. +To prove Theorem 1.1 we first deduce a null controllability result for a linear system associated +to (1.1): + + + + + + + + + + + +∂ty − a(1 + αi)∆y = f + +1ωh, +in Ω × (0, T), +∂tyΓ + a(1 + αi)∂νy − b(1 + αi)∆ΓyΓ = fΓ, +on Γ × (0, T), +y = yΓ, +on Γ × (0, T), +(y, yΓ)(0) = (y0, yΓ,0), +in Ω × Γ, +(1.2) +where (f, fΓ) will be taken to decrease exponentially to zero in t = T. Then, we prove a new +Carleman estimate for the adjoint system of (1.2) (see (4.2) below). This will provide existence +(and uniqueness) to a variational problem, from which we define a solution (y, yΓ, h) to (1.2) such +that y(T) = 0 in Ω. Moreover, the solution is such that eC/(T−t)(y, yΓ, h) ∈ L2 × L2(ω × (0, T)), for +some constant C > 0. Finally, by an inverse mapping theorem, we deduce the null controllability +for the nonlinear system. +The rest of the paper is organized as follows. +In Section 2 we establish the existence and +uniqueness of solutions of systems like (1.1) and (1.2). In Section 3, we prove a suitable Carleman +estimate for the Ginzburg-Landau operator with dynamic boundary conditions. In Section 4 we +prove the observability estimate for the adjoint system and prove the null controllability of (1.2). +Finally, in Section 5 we prove the Theorem 1.1. +2 +Existence and uniqueness of solutions +In this section, we present new results concerning existence and uniqueness for the Ginzburg-Landau +equations with dynamic boundary conditions. +2.1 +Linear problem +We devote to study the Cauchy problem + + + + + + + + + + + +Lu = f, +in Ω × (0, T), +N(u, uΓ) = fΓ, +on Γ × (0, T), +u = uΓ, +on Γ × (0, T), +(u(0), uΓ(0)) = (u0, uΓ,0), +in Ω × Γ, +(2.1) +4 + +where +Lu := ∂tu − a(1 + αi)∆u, +N(u, uΓ) := ∂tuΓ + a(1 + αi)∂νu − b(1 + αi)∆ΓuΓ. +(2.2) +The problem (2.1) can be seen in the abstract form +� +U ′(t) = AGLU(t) + F(t), +t ∈ (0, T), +U(0) = U0, +(2.3) +where AGL : D(AGL) ⊂ L2 → L2 is the operator defined by +AGL(U) := +� +a(1 + αi)∆u +−a(1 + αi)∂νu + b(1 + αi)∆ΓuΓ +� +, +∀ U := +� u +uΓ +� +∈ D(AGL), +(2.4) +with domain +D(AGL) := +� +U = +� u +uΓ +� +∈ H1 : (∆u, ∆ΓuΓ) ∈ L2 +� += H2, +where the last equivalence is justified in [10] (see also [14]). It is easy to see that AGL can be written +as +AGL = (1 + αi)AW , +D(AW) = D(AGL) = H2, +where AW is the Wentzell-Laplacian operator defined in [23]. Then, arguing as in [26, Section 2] +we have the following result: +Proposition 2.1. The operator AGL defined in (2.4) is densely defined and generates an analytic +semigroup (etAGL)t⩾0 in L2. +According to Proposition 2.1, the existence and uniqueness of strong solutions of (2.3) in the +usual sense are guaranteed. In the next subsection, we provide existence and uniqueness of solutions +in appropriate spaces by energy estimates and density arguments. +2.2 +A priori estimates +Proposition 2.2. Suppose that (u0, uΓ,0) ∈ L2 and (f, fΓ) ∈ L2(0, T; L2). Then, the mild solution +(u, uΓ) of (2.1) belongs to C0([0, T]; L2) ∩ L2(0, T; H1). Moreover, there exists a constant C1 > 0 +such that the associated solution (u, uΓ) of (2.1) satisfies +∥(u, uΓ)∥C0([0,T];L2) + ∥(u, uΓ)∥L2(0,T;H1) ⩽ C1∥(f, fΓ)∥L2(0,T;L2) + C1∥(u0, uΓ,0)∥L2. +(2.5) +Proof. Firstly, we multiply by u the first equation of (2.1) and we integrate in Ω × (0, T). Secondly, +we multiply the second equation of (2.1) by uΓ and integrate on Γ × (0, T). Next, we add these +identities and take the real part on the obtained equation. After integration by parts, this yields +1 +2 +d +dt +�� +Ω +|u(t)|2dx + +� +Γ +|uΓ(t)|2dS +� ++ a +� +Ω +|∇u(t)|2dx + b +� +Γ +|∇ΓuΓ(t)|2dS +=ℜ +� +Ω +f(t)u(t)dx + ℜ +� +Γ +fΓ(t)uΓ(t)dS. +By Young’s inequality, it is easy to check that +∥(u, uΓ)∥2 +C0([0,T];L2) + ∥(u, uΓ)∥2 +L2(0,T;H1) ⩽ C∥(f, fΓ)∥2 +L2(0,T;L2) + C∥(u0, uΓ,0)∥2 +L2, +which clearly implies (2.5). +5 + +Proposition 2.3. Let (u, uΓ,0) ∈ H1 and (f, fΓ) ∈ L2(0, T; L2). Then, the associated weak solution +(u, uΓ) of (2.1) belongs to H1(0, T; L2) ∩ C0([0, T]; H1) ∩ L2(0, T; H2). Moreover, there exists a +constant C2 > 0 such that (u, uΓ) satisfies +∥(u, uΓ)∥H1(0,T;L2) + ∥(u, uΓ)∥C0([0,T];H1) + ∥(u, uΓ)∥L2(0,T;H2) +⩽C2∥(f, fΓ)∥L2(0,T;L2) + C2∥(u0, uΓ,0)∥H1. +(2.6) +Proof. The proof is divided into three steps. +• Step 1: Our first task is to obtain an L2 estimates for ∂tu and ∂tuΓ, respectively. In order to do +that, we multiply the first equation of (2.1) by (1 − αi)∂tu and integrate in Ω × (0, T). In addition, +we multiply the second equation of (2.1) by (1−αi)∂tuΓ and integrate on Γ×(0, T). Then, we sum +up these identities and take the real part. This yields +� +Ω +|∂tu(t)|2dx + +� +Γ +|∂tuΓ(t)|2dS + 1 +2a(1 + α2) d +dt +� +Ω +|∇u|2dx + 1 +2b(1 + α2) d +dt +� +Γ +|∇ΓuΓ|2dS +=ℜ +� +Ω +(1 − αi)f∂tudx + ℜ +� +Γ +(1 − αi)fΓ∂tuΓdS. +This implies that +∥(u, uΓ)∥2 +H1(0,T;L2) + ∥(u, uΓ)∥2 +C0([0,T];H1) ⩽ C∥(f, fΓ)∥2 +L2(0,T;L2) + C∥(u0, uΓ,0)∥2 +H1. +• Step 2: In this step, we derive L2(H2) estimates for the solution (u, uΓ). To do this, we firstly +point out that the estimate of ∂tu implies that +∥∆u∥L2(0,T;L2(Ω)) ⩽ C∥(f, fΓ)∥L2(0,T;L2) + C∥(u0, uΓ,0)∥H1. +By elliptic regularity applied to the first equation of (2.1), we have +∥u(t)∥H2(Ω) ⩽ C∥f(t)∥L2(Ω) + C∥∂tu(t)∥L2(Ω) + C∥uΓ(t)∥H3/2(Γ). +Integrating on t ∈ [0, T], we deduce that +∥u∥L2(0,T;H2(Ω)) ⩽ C∗∥(f, fΓ)∥L2(0,T;L2) + C∗∥(u0, uΓ,0)∥H1 + C∗∥uΓ∥L2(0,T;H3/2(Γ)), +(2.7) +for some constant C∗ > 0. Now, from the second equation of (2.1), we deduce that +∥uΓ∥L2(0,T;H2(Γ)) ⩽C∥fΓ∥L2(0,T;L2(Γ)) + C∥∂νu∥L2(0,T;L2(Γ)) + C∥∂tuΓ∥L2(0,T;L2(Γ)) +⩽C∥(f, fΓ)∥L2(0,T;L2) + C∥(u0, uΓ,0)∥H1 + C∥∂νu∥L2(0,T;L2(Γ)). +(2.8) +Moreover, by interpolation inequalities, we have that for every 0 < s < 1/2 and ε > 0, there are +positive constants Cs and Cε such that +∥∂νu∥L2(0,T;L2(Γ)) ⩽ Cs∥u∥L2(0,T;H3/2+s(Ω)) ⩽ Cε∥u∥L2(0,T;H1(Ω)) + ε∥u∥L2(0,T;H2(Ω)). +(2.9) +Combining (2.7), (2.9) and (2.9) together with estimate (2.5), we get +∥u∥L2(0,T;H2(Ω)) ⩽ C∥(f, fΓ)∥L2(0,T;L2(Ω)) + C∥(u0, uΓ,0)∥H1, +(2.10) +where we have chosen ε > 0 small enough. With the estimate (2.10) at hand, by (2.9) we can assert +that +∥∂νu∥L2(0,T;L2(Γ)) ⩽ C∥(f, fΓ)∥L2(0,T;L2) + C∥(u0, uΓ,0)∥H1. +(2.11) +Thus, substituting (2.11) into (2.8), we conclude that +∥uΓ∥L2(0,T;H2(Γ)) ⩽ C∥(f, fΓ)∥L2(0,T;L2) + C∥(u0, uΓ,0)∥H1. +6 + +Proposition 2.4. Let (f, fΓ) ∈ L2(0, T; L2) and (u0, uΓ,0) ∈ L2. Then, the associated weak solution +(u, uΓ) satisfies +∥( +√ +t∂tu, +√ +t∂tuΓ)∥L2(0,T;L2) + ∥( +√ +tu, +√ +tuΓ)∥L2(0,T;H2) + ∥( +√ +tu, +√ +tuΓ)∥C0([0,T];H1) +⩽C∥(f, fΓ)∥L2(0,T;L2(Ω)) + C∥(u0, uΓ,0)∥L2. +Proof. The proof is a slight modification of the arguments used in of Proposition 2.3. For this +reason, we omit the details. +Remark 2.5. We point out that Proposition 2.4 implies that, for each T > 0, (u0, uΓ,0) ∈ L2, +(f, fΓ) ∈ L2(0, T; L2) and ε > 0, the weak solution (u, uΓ) of (2.1) satisfies +(u, uΓ) ∈ H1(ε, T; L2) ∩ L2(ε, T; H2) ∩ C0([ε, T]; H1). +Moreover, there exists a constant C > 0 (independent of ε) such that +∥(u, uΓ)∥H1(ε,T;L2) + ∥(u, uΓ)∥L2(ε,T;H2) + ∥(u, uΓ)∥C0([ε,T];H1) +⩽C∥(u0, uΓ)∥L2 + C∥(f, fΓ)∥L2(0,T;L2). +2.3 +Nonlinear problem +In this subsection, we prove a local existence theorem for the nonlinear problem + + + + + + + + + + + +Lu + c(1 + γi)|u|2u = f, +in Ω × (0, T), +N(u, uΓ) + c(1 + γi)|uΓ|2uΓ = fΓ, +in Γ × (0, T), +u = uΓ, +on Γ × (0, T), +(u(0), uΓ(0)) = (u0, uΓ,0), +in Ω × Γ, +(2.12) +where L and N are given by (2.2), with a, b, c > 0 and α, γ ̸= 0. +Proposition 2.6. Let N = 2 or N = 3. +There exist ε > 0 and C > 0 such that, for every +(f, fΓ) ∈ L2(0, T; L2), (u0, uΓ,0) ∈ H1 such that +∥(f, fΓ)∥L2(0,T;L2) + ∥(u0, uΓ,0)∥H1 ⩽ ε, +(2.13) +there exists a unique solution (u, uΓ) of (2.12) which satisfies +∥(u, uΓ)∥C0([0,T];H1) + ∥(u, uΓ)∥L2(0,T;H2) ⩽ C +� +∥(f, fΓ)∥L2(0,T;L2) + ∥(u0, uΓ,0)∥H1 +� +. +Proof. We denote B = C0([0, T]; H1)∩L2(0, T; H2). Given (f, fΓ) ∈ L2(0, T; L2) and (u0, uΓ,0) ∈ H1, +for each (z, zΓ) ∈ B, we consider the system + + + + + + + + + + + +Lu = f − c(1 + γi)|z|2z, +in Ω × (0, T), +N(u, uΓ) = fΓ − c(1 + γi)|zΓ|2zΓ, +on Γ × (0, T), +u = uΓ, +on Γ × (0, T), +(u(0), uΓ(0)) = (u0, uΓ,0), +in Ω × Γ. +(2.14) +From Proposition 2.3 and the fact that B ֒→ L6(0, T; L6), we have that (2.14) has a unique +solution (u, uΓ) ∈ B. +Hence, we can define the map F : B → B given by F(z, zΓ) = (u, uΓ). +Moreover, we also have that there exists D > 0 such that +∥F(z, zΓ)∥B ⩽ D +� +∥(f, fΓ)∥L2(0,T;L2) + ∥(u0, uΓ,0)∥H1 + ∥(z, zΓ)∥3 +B +� +. +(2.15) +7 + +Clearly, (u, uΓ) ∈ B is a solution of (2.12) if and only if it is a fixed point of the map F. We +will show that there exists R > 0 such that the restriction of F to the closed ball BR := {(z, zΓ) ∈ +B : ∥(z, zΓ)∥B ⩽ R} is a contraction from BR into BR. Then, the proof will follow from a classic +fixed point result. +Indeed, from (2.15) and assumption (2.13), for each (z, zΓ) ∈ BR we have +∥F(z, zΓ)∥B ⩽ D +� +ε + R3� +. +(2.16) +Moreover, for each (z, zΓ), (w, wΓ) ∈ BR, taking into account the equations satisfied by F(z, zΓ)− +F(w, wΓ), from Proposition 2.3 we have that +∥F(z, zΓ) − F(w, wΓ)∥2 +B ⩽D1∥|z|2z − |w|2w, |zΓ|2zΓ − |wΓ|2wΓ∥2 +L2(0,T;L2) +⩽D1 +� +∥z − w∥2 +L∞(0,T;L4(Ω)) +� +∥z2 + zw∥2 +L2(0,T;L4(Ω)) + ∥w∥4 +L4(0,T;L8(Ω)) +� ++∥zΓ − wΓ∥2 +L∞(0,T;L4(Γ)) +� +∥z2 +Γ + zΓwΓ∥2 +L2(0,T;L4(Γ)) + ∥wΓ∥4 +L4(0,T;L8(Γ)) +�� +, +and then, taking into account that B ֒→ L∞(0, T; L4) ∩ L4(0, T; L8), we get that +∥F(z, zΓ) − F(w, wΓ)∥B ⩽ D2R2∥(z, zΓ) − (w, wΓ)∥B. +(2.17) +Therefore, in order to conclude, we choose R > 0 such that R2 < min +� 1 +2D, 1 +D2 +� +and ε > 0 such +that ε ⩽ +R +2D. From (2.16), (2.17) and the Banach Fixed point Theorem, we get the existence of a +unique fixed point of F. +3 +A new Carleman estimate for the linear Ginzburg-Landau equa- +tion with dynamic boundary conditions +In this section, we deduce a Carleman estimate for the linear Ginzburg-Landau operator with +dynamic boundary conditions. We start introducing the weight functions that we shall use. For +this propose, we recall the following +Lemma 3.1. Given a nonempty open set ω′ ⋐ Ω, there exists a function η0 ∈ C2(Ω) such that +η0 > 0, in Ω, +η0 = 0, on Γ, +|∇η0| > 0, in Ω \ ω′. +Given ω′ ⋐ Ω, we take η0 with respect to ω′ as in Lemma 3.1. For λ, m > 1, we define +ϕ(x, t) :=(t(T − t))−1 � +e2λm∥η0∥∞ − eλ(m∥η0∥∞+η0(x))� +, +∀(x, t) ∈ Ω × (0, T), +ξ(x, t) :=(t(T − t))−1eλ(m∥η0∥∞+η0(x)), +∀(x, t) ∈ Ω × (0, T). +Theorem 3.2. There exist constants C, λ0, s0 > 0 such that for all λ ⩾ λ0 and s ⩾ s0, we have +� T +0 +� +Ω +e−2sϕ � +s3λ4ξ3|v|2 + sλ2ξ|∇v|2 + s−1|∂tv|2 + s−1|∆v|2� +dxdt ++ +� T +0 +� +Γ +e−2sϕ(s3λ3ξ3|vΓ|2 + sλξ|∇ΓvΓ|2 + sλ|∂νv|2 + |∂tvΓ|2 + |∆ΓvΓ|2)dSdt +⩽Cs3λ4 +� T +0 +� +ω +e−2sϕξ3|v|2dxdt + C +� T +0 +� +Ω +e−2sϕ|L∗(v)|2dxdt ++ C +� T +0 +� +Γ +e−2sϕ|N ∗(v, vΓ)|2dSdt, +(3.1) +8 + +for all (v, vΓ) ∈ H1(0, T; L2) ∩ L2(0, T; H2), where +L∗(v) = ∂tv + a(1 − αi)∆v, +N ∗(v, vΓ) := ∂tvΓ − a(1 − αi)∂νv + b(1 − αi)∆ΓvΓ. +(3.2) +Proof. For convenience, the proof has been divided into several steps: +• Step 1. Conjugation. For simplicity, we consider v ∈ C∞(Ω × [0, T]), vΓ = v, λ ⩾ λ1 ⩾ 1, and +s ⩾ s0 ⩾ 1. define +w :=e−sϕv, +in Ω × (0, T), +f :=e−sϕ(∂tv + a(1 − αi)∆v), +in Ω × (0, T), +fΓ :=e−sϕ(∂tv − a(1 − αi)∂νv + b(1 − αi)∆Γv), +on Γ × (0, T). +Straightforward computations show that +∇ϕ = −∇ξ = −λξ∇η0, +∆ϕ = −λ2ξ|∇η0|2 − λξ∆η0, +∂νϕ = −λξ∂νη0, +with ∂νη0 ⩽ c < 0 on Γ, for some constant c > 0 and +∇Γϕ = ∇Γξ = 0, +∆Γϕ = ∆Γξ = 0. +Then, in Ω × (0, T) we have the following identity: +f =∂tw + a(1 − αi)∆w + as2λ2(1 − αi)|∇η0|2ξ2w − asλ2(1 − αi)|∇η0|2ξw +− asλ(1 − αi)∆η0ξw − 2asλ(1 − αi)ξ∇η0 · ∇w + s∂tϕw. +(3.3) +On the other hand, on Γ × (0, T) we have +fΓ =∂tw − a(1 − αi)∂νw + asλ(1 − αi)∂νη0ξw + b(1 − αi)∆Γw ++ s∂tϕw +(3.4) +Now, we define +P1w :=a(s2λ2|∇η0|2ξ2w + ∆w) + aαi(2sλξ∇η0 · ∇w + (sλ2|∇η0|2 + sλ∆η0)ξw) ++ s∂tϕw +P2w := − a(2sλξ∇η0 · ∇w + (sλ2|∇η0|2 + sλ∆η0)ξw) − aαi(s2λ2|∇η0|2ξ2w + ∆w) ++ ∂tw +Rw :=f. +and +PΓ,1w :=b∆Γw − 2a2 +b αisλ∂νη0ξw + ∂tϕw, +PΓ,2w := −αb∆Γw + 2a2 +b sλ∂νη0w + ∂tw, +RΓw :=fΓ − a(1 − αi)sλ∂νη0ξw + a(1 − αi)∂νw + 2a2 +b (1 − αi)sλ∂νη0ξw. +Then, (3.3) and (3.4) can be written as +P1w + P2w = Rw, +in Ω × (0, T), +(3.5) +PΓ,1w + PΓ,2w = RΓw, +on Γ × (0, T). +(3.6) +9 + +Then, taking the L2(Ω × (0, T))-norm in (3.5) and the L2(Γ × (0, T))-norm in (3.6), we obtain +∥P1w∥2 +L2(Ω×(0,T)) + ∥P2w∥2 +L2(Ω×(0,T)) + ∥PΓ,1w∥2 +L2(Γ×(0,T)) + ∥PΓ,2w∥2 +L2(Γ×(0,T)) ++ 2⟨P1w, P2w⟩L2(Ω×(0,T)) + 2⟨PΓ,1w, PΓ,2w⟩L2(Γ×(0,T)) +=∥Rw∥2 +L2(Ω×(0,T)) + ∥RΓw∥2 +L2(Γ×(0,T)). +• Step 2. Computations of the L2 products in Ω × (0, T). In this step, we devote to compute the +terms +⟨P1w, P2w⟩L2(Ω×(0,T)) = +3 +� +j,k=1 +Ijk, +where we have used the notation Ijk to denote the inner product between the jth-term of P1w and +the kth-term of P2w. +Firstly, the term I11 can be written as +I11 = − a2ℜ +� T +0 +� +Ω +(s2λ2|∇η0|2ξ2w + ∆w)(2sλξ∇η0 · ∇w + (sλ2|∇η0|2 + sλ∆η0)ξw)dxdt +=I(1) +11 + I(2) +11 + I(3) +11 + I(4) +11 . +Integration by parts yields +I(1) +11 = − 2a2s3λ3ℜ +� T +0 +� +Ω +|∇η0|2ξ3w∇η0 · ∇wdxdt +=2a2s3λ3ℜ +� T +0 +� +Ω +∇ +� +|∇η0|2� +· ∇η0ξ3|w|2dxdt + 6a2s3λ4 +� T +0 +� +Ω +|∇η0|4ξ3|w|2dxdt +− I(1) +11 − 2a2s3λ3 +� T +0 +� +Γ +|∇η0|2∂νη0ξ3|w|2dxdt + 2a2s3λ3 +� T +0 +� +Ω +|∇η0|2∆η0ξ3|w|2dxdt. +Then, I(1) +11 is given by +I(1) +11 =a2s3λ3 +� T +0 +� +Ω +� +∇(|∇η0|2) · ∇η0 + |∇η0|2∆η0� +ξ3|w|2dxdt ++ 3a2s3λ4 +� T +0 +� +Ω +|∇η0|4ξ3|w|2dxdt − a2s3λ3 +� T +0 +� +Γ +(∂νη0)3xi3|w|2dSdt. +On the other hand, we notice that +I(2) +11 = −a2 +� T +0 +� +Ω +(s3λ4|∇η0|4 + s3λ3|∇η0|2∆η0)ξ3|w|2dxdt +Integrating by parts in space, we have +I(3) +11 = − a2ℜ +� T +0 +� +Ω +∆w(2sλξ∇η0 · ∇w)dxdt +=2a2sλ2 +� T +0 +� +Ω +ξ|∇η0 · ∇w|2dxdt + 2a2sλℜ +� T +0 +� +Ω +ξ∇w · ∇(∇η0 · ∇w)dxdt +− 2a2sλ +� T +0 +� +Γ +ξ∂νη0|∂νw|2dSdt. +10 + +Using the identity +∇ψ · ∇(∇η0 · ∇w) = ∇2η0(∇w, ∇w) + 1 +2∇η0 · ∇(|∇w|2), +in Ω × (0, T), +we notice that +2a2sλℜ +� T +0 +� +Ω +ξ∇w · ∇(∇η0 · ∇w)dxdt +=2a2sλ +� T +0 +� +Ω +ξ∇2η0(∇w, ∇w)dxdt − a2sλ +� T +0 +� +Ω +∆η0ξ|∇w|2dxdt +− a2sλ2 +� T +0 +� +Ω +ξ|∇η0|2|∇w|2dxdt + a2sλ +� T +0 +� +Γ +∂νη0ξ(|∇Γw|2 + |∂νw|2)dSdt. +Then, I(3) +11 is given by +I(3) +11 =2a2sλ2 +� T +0 +� +Ω +ξ|∇η0 · ∇w|2dxdt + 2a2sλ +� T +0 +� +Ω +ξ∇2η0(∇w, ∇w)dxdt +− a2sλ +� T +0 +� +Ω +∆η0ξ|∇w|2dxdt − a2sλ2 +� T +0 +� +Ω +ξ|∇η0|2|∇w|2dxdt +− a2sλ +� T +0 +� +Γ +∂νη0ξ +� +|∂νw|2 − |∇Γw|2� +dSdt. +After integration by parts, I(4) +11 can be written as follows: +I(4) +11 =a2ℜ +� T +0 +� +Ω +ξw∇w · ∇(sλ2|∇η0|2 + sλ∆η0)dxdt ++ a2ℜ +�� T +0 +� +Ω +λξ +� +sλ3|∇η0|2 + sλ2∆η0� +w∇η0 · ∇wdxdt ++ a2 +�� T +0 +� +Ω +(sλ2|∇η0|2 + sλ∆η0)ξ|∇w|2dxdt +− a2ℜ +�� T +0 +� +Γ +(sλ2|∂νη0|2 + sλ∆η0)ξ(∂νw)wdSdt. +Then, I11 can be estimated as +I11 ⩾C +� T +0 +� +Ω +(s3λ4ξ3|w|2 + sλ2ξ|∇η0 · ∇w|2)dxdt ++ Csλ +� T +0 +� +Γ +(ξ|∂νw|2 + a2sλ∂νη0ξ|∇Γw|2)dSdt +− Cs3λ4 +� T +0 +� +ω +ξ3|w|2dxdt − X11, +where X11 satisfies the following upper bound: +X11 ⩽C +� T +0 +� +Ω +(s3λ3ξ3|w|2 + sλξ|∇w|2)dxdt ++ C +� T +0 +� +Γ +� +s2λ3ξ3|w|2 + λξ|∂νw|2� +dSdt. +11 + +It is clear that +I12 =0. +Moreover, we write I13 as +I13 =as2λ2ℜ +� T +0 +� +Ω +|∇η0|2ξ2w∂twdxdt + aℜ +� T +0 +� +Ω +∆w∂twdxdt +=I(1) +13 + I(2) +13 . +Integrating by parts on time and using the fact that w = 0 in t = 0 and t = T, we have +I(1) +13 = − as2λ2 +� T +0 +� +Ω +|∇η0|2ξ∂tξ|w|2dxdt. +In the same manner, I(2) +13 gives +I(2) +13 =aℜ +� T +0 +� +Γ +∂νw∂twdSdt. +Then, I13 is given by +I13 ⩾ − Cs2λ3 +� T +0 +� +Ω +ξ3|w|2dxdt − C +� T +0 +� +Γ +(s1/2ξ|∂νw|2 − s−1/2ξ−1|∂tw|2)dSdt. +Moreover, from the definition of I21, we see that +I21 =0 +Similar to the computations of I11, I22 can be estimated as follows: +I22 ⩾C +� T +0 +� +Ω +� +s3λ4ξ3|w|2 + sλ2ξ|∇η0 · ∇w|2� +dxdt ++ C +� T +0 +� +Γ +(s3λ3ξ3|w|2 + Csλξ|∂νw|2 + a2α2sλ∂νη0ξ|∇Γw|2)dSdt +− Cs3λ4 +� T +0 +� +ω +ξ3|w|2dSdt − X22, +where X22 satisfies +X22 ⩽C +� T +0 +� +Ω +(s3λ3ξ3|w|2 + Csλξ|∇w|2)dxdt ++ C +� T +0 +� +Γ +� +s2λ3ξ2|w|2 + λξ|∂νw|2� +dSdt. +To compute the term I23, we follow [25]. Then, we write +I23 =1 +2aαi +� T +0 +� +Ω +(2sλξ∇η0 · ∇w + (sλ2|∇η0|2 + sλ∆η0)ξw)∂twdxdt +− 1 +2aαi +� T +0 +� +Ω +(2sλξ∇η0 · ∇w + (sλ2|∇η0|2 + sλ∆η0)ξw)∂twdxdt +=I(1) +23 + I(2) +23 . +(3.7) +12 + +On one hand, integrating by parts in time we have +I(1) +23 := − 1 +2aαi +�� +Q +(2sλ∂tξ∇η0 · ∇w + 2sλξ∇η0 · ∇∂tw)wdxdt +− 1 +2aαi +�� +Q +� +(sλ2|∇η0|2 + sλ∆η0)∂tξ|w|2 + (sλ2|∇η0|2 + sλ∆η0)ξw∂tw +� +dxdt. +(3.8) +On the other hand, integration by parts in space yields +I(2) +23 =1 +2aαi +� T +0 +� +Ω +� +sλ2|∇η0|2ξw∂tw + sλξw∇η0 · ∇∂tw + sλ∆η0ξw∂tw +� +dxdt +− 1 +2aαi +� T +0 +� +Ω +(sλ2|∇η0|2 + sλ∆η0)ξw∂twdxdt − aαsλi +� T +0 +� +Γ +∂νη0ξw∂twdSdt. +(3.9) +Then, substituting (3.8) and (3.9) into (3.7) we can estimate I23 be below in the following way +I23 ⩾ − C +� T +0 +� +Ω +� +s2λ2ξ3|w|2 + λξ|∇w|2� +dxdt +− C +� T +0 +� +Γ +(s2λ3ξ3|w|2 + λ−1ξ−1|∂tw|2)dSdt. +The term I31 is +I31 = − 2as2λℜ +� T +0 +� +Ω +ξ∂tϕw∇η0 · ∇wdxdt − a +� T +0 +� +Ω +(s2λ2|∇η0|2 + s2λ∆η0)∂tϕξ|w|2dxdt +where the first term can be computed as +− 2as2λℜ +� T +0 +� +Ω +ξ∂tϕw∇η0 · ∇wdxdt +=as2λ2 +� T +0 +� +Ω +(|∇η0|2ξ∂tϕ + ξ∇η0 · ∇(∂tϕ) − ∆η0ξ∂tϕ)|w|2dxdt +− 2as2λ +� T +0 +� +Γ +∂νη0ξ∂tϕ|w|2dSdt +Then, I31 is estimated as +I31 ⩾ − Cs2λ2 +� T +0 +� +Ω +ξ3|w|2dxdt − Cs2λ2 +� T +0 +� +Γ +ξ3|w|2dSdt. +Now, for I32, it is clear that +I32 = − aαsℑ +� T +0 +� +Ω +w∇(∂tϕ) · ∇wdxdt + aαsℑ +� T +0 +� +Γ +∂tϕw∂νwdSdt +⩾ − C +� T +0 +� +Ω +� +ξ3|w|2 + ξ|∇w|2� +dxdt +− C +� T +0 +� +Γ +� +sξ3|w|2 + sξ|∂νw|2� +dSdt. +13 + +In addition, since w = 0 in t = 0 and t = T, we have +I33 ⩾ −Cs +� T +0 +� +Ω +ξ3|w|2dxdt. +According to the above estimates, take s and λ > 0 large enough, to deduce that +⟨P1w, P2w⟩L2(Ω×(0,T)) +⩾Cs3λ4 +� T +0 +� +Ω +ξ3|w|2dxdt + C +� T +0 +� +Γ +(s3λ3ξ3|w|2 + sλξ|∂νw|2)dSdt ++ a2(1 + α2)sλ +� T +0 +� +Γ +∂νη0ξ|∇Γw|2dSdt − Cs3λ4 +� T +0 +� +ω +ξ3|w|2dxdt − ˜X, +(3.10) +for s ⩾ s1 > 0 and λ ⩾ λ1 > 0, where ˜X satisfies +˜X ⩽ Csλ +� T +0 +� +Ω +ξ|∇w|2dxdt + Cs−1(1 + λ−1) +� T +0 +� +Γ +ξ−1|∂tw|2dSdt. +Before going any further, let us point out that the integral term |∇Γw|2 is negative. However, +in the next step we obtain additional terms to solve this problem. +• Step 3. Computations of the L2 products on Γ × (0, T). In this step, we compute the terms +ℜ +� T +0 +� +Ω +PΓ,1wPΓ,2wdxdt = +3 +� +j=1 +3 +� +k=1 +Jjk, +where Jjk denotes the L2 real inner product of the jth of P1w with the kth-term of P2w. Firstly, we +have +J11 =b2αℑ +� T +0 +� +Γ +|∆Γw|2dSdt = 0. +Secondly, by the surface divergence theorem, we get +J12 =2a2sλℜ +� T +0 +� +Γ +(ξw∇Γ(∂νη0) · ∇Γw)dSdt − 2a2sλ +� T +0 +� +Γ +∂νη0ξ|∇Γw|2dSdt +⩾ − 2a2sλ +� T +0 +� +Γ +∂νη0ξ|∇Γw|2dSdt − C +� T +0 +� +Γ +(s2λ2ξ|w|2 + ξ|∇Γw|2)dSdt. +Moreover, integrating by parts in time and space, we can assert that +J13 =0. +The term J21 is +J21 = − 2a2α2sλℜ +� T +0 +� +Γ +ξw∇Γ(∂νη0) · ∇ΓwdSdt − 2a2αsλ +� T +0 +� +Γ +∂νη0ξ|∇Γw|2dSdt +⩾ − 2a2α2sλ +� T +0 +� +Γ +∂νη0ξ|∇Γw|2dSdt − C +� T +0 +� +Γ +(s2λ2ξ|w|2 + ξ|∇Γw|2)dSdt. +Furthermore, by definition, J22 is +J22 = 0 +14 + +For J23 we have +J23 ⩾ − C +� T +0 +� +Γ +� +s5/2λ5/2ξ3|w|2 + s−1/2λ−1/2ξ−1|∂tw|2� +dSdt. +by the surface divergence theorem and the fact that ∇Γϕ = 0 on Γ × (0, T). we deduce that +J31 =0. +Moreover, by definition J32 is given by +J32 ⩾ −Csλ +� T +0 +� +Γ +ξ3|w|2dSdt. +On the other hand, integration by parts yields +J33 ⩾ − C +� T +0 +� +Γ +ξ3|w|2dSdt. +According to these estimates, we can assert that +⟨PΓ,1w, PΓ,2w⟩L2(Γ×(0,T)) ⩾ −2a2(1 + α2)sλ +� T +0 +� +Γ +∂νη0ξ|∇Γw|2dSdt − Y, +(3.11) +where Y satisfies +Y ⩽C +� T +0 +� +Γ +(s5/2λ5/2ξ3|w|2 + s−1/2λ−1/2ξ−1|∂tw|2)dSdt ++ C +� T +0 +� +Γ +(s−1/2λ−1/2ξ−1|∆Γw|2 + ξ|∇Γw|2)dSdt. +From (3.10) and (3.11), and using that ∂νη0 ⩽ −c < 0 on Γ × (0, T) we deduce that +∥P1w∥2 +L2(Ω×(0,T)) + ∥P2w∥2 +L2(Ω×(0,T)) + ∥PΓ,1w∥2 +L2(Γ×(0,T)) + ∥PΓ,2w∥2 +L2(Γ×(0,T)) ++ C +� T +0 +� +Ω +(s3λ4ξ3|w|2 + sλ2ξ|∇η0 · ∇w|2)dxdt + C +� T +0 +� +Γ +(s3λ3ξ3|w|2 + sλξ|∇Γw|2)dSdt +⩽∥f∥2 +L2(Ω×(0,T)) + ∥fΓ∥2 +L2(Γ×(0,T)) + Z, +where Z can be bounded as +Z ⩽Csλ +� T +0 +� +Ω +ξ|∇w|2dxdt + Cs−1(1 + λ−1) +� T +0 +� +Γ +ξ−1|∂tw|2dSdt ++ C +� T +0 +� +Γ +(s5/2λ5/2ξ3|w|2 + ξ|∇Γw|2 + |∂νw|2)dSdt ++ C +� T +0 +� +Γ +(s−1/2λ−1/2|∆Γw|2 + s−1/2λ−1/2ξ−1|∂tw|2)dSdt. +Now, taking s, λ > 0 large enough we obtain +∥P1w∥2 +L2(Ω×(0,T)) + ∥P2w∥2 +L2(Ω×(0,T)) + ∥PΓ,1w∥2 +L2(Γ×(0,T)) + ∥PΓ,2w∥2 +L2(Γ×(0,T)) ++ C +� T +0 +� +Ω +(s3λ4ξ3|w|2 + sλ2ξ|∇η0 · ∇w|2)dxdt ++ C +� T +0 +� +Γ +(s3λ3ξ3|w|2 + sλ∂νη0ξ|∇Γw|2)dSdt +⩽∥f∥2 +L2(Ω×(0,T)) + ∥fΓ∥2 +L2(Γ×(0,T)) + s3λ4 +� T +0 +� +ω +ξ3|w|2dxdt + ˜Z, +(3.12) +15 + +where ˜Z is given by +˜Z ⩽C +� T +0 +� +Γ +(s−1/2λ−1/2ξ−1|∆Γw|2 + (s−1/2λ−1/2 + s−1)ξ−1|∂tw|2)dSdt ++ Csλ +� T +0 +� +Ω +ξ|∇w|2dxdt. +From the bounds of ˜Z, it is clear that we need to absorb the terms of |∇v| in Ω, ∆w and ∂tw +on Γ × (0, T). To do this, we shall use an indirect estimate using the definitions given in (3.5) and +(3.6). +• Step 4. Estimates of additional terms in the bulk. In this step, we compute the terms ∆w, ∇w +and ∂tw in Ω × (0, T). More precisely, the purpose of this step is to prove the following inequality: +� T +0 +� +Ω +� +s−1ξ−1|∆w|2 + s−1ξ−1|∂tw|2 + sλ2|∇w|2� +dxdt +⩽C +� T +0 +� +Ω +� +s3λ4ξ|w|2 + sλ2ξ|∇η0 · ∇w|2 + s−1ξ−1|P1w|2 + s−1ξ−1|P2w|2� +dxdt ++ C +� T +0 +� +Γ +� +s3λ3|w|2 + sλ|∂νw|2� +dSdt. +(3.13) +In order to do that, we firstly estimate the term of ∆w. From the definition of P1, it is clear +that +s−1 +� T +0 +� +Ω +ξ−1|∆w|2dxdt +⩽C +� T +0 +� +Ω +� +s−1ξ−1|P1w|2 + s3λ4|w|2 + sλ2ξ|∇η0 · ∇w|2� +dxdt. +(3.14) +Secondly, we estimate ∇w as follows: +sλ2 +� T +0 +� +Ω +ξ|∇w|2dxdt = − sλ2ℜ +� T +0 +� +Ω +w∇ξ · ∇wdxdt + sλ2ℜ +� T +0 +� +Γ +ξw∂νwdSdt +− sλ2ℜ +� T +0 +� +Ω +ξw∆wdxdt +Then, by Young’s inequality we obtain +sλ2 +� T +0 +� +Ω +ξ|∇w|2dxdt ⩽C +� T +0 +� +Ω +� +s3λ4ξ3|w|2 + sλ2ξ|∇η0 · ∇w|2 + s−1ξ−1|∆w|2� +dxdt ++ C +� T +0 +� +Γ +(s3λ3|w|2 + sλ|∂νw|2)dSdt. +(3.15) +Next, we estimate the term of ∂tw directly from the definition of P2: +s−1 +� T +0 +� +Ω +ξ−1|∂tw|2dxdt ⩽ C +� T +0 +� +Ω +� +s3λ4ξ|w|2 + sλ2ξ|∇η0 · ∇w|2 + s−1ξ−1|∆w|2� +dxdt. +(3.16) +Thus, combining (3.14), (3.15) and (3.16), we easily get (3.13). +16 + +• Step 5. Estimates of ∆Γw and ∂tw on Γ × (0, T). In this step, we want to prove that +� T +0 +� +Γ +� +ξ−1|∂tw|2 + ξ−1|∆Γw|2� +dSdt +⩽C +� T +0 +� +Γ +� +s2λ2|w|2 + ξ−1|∂νw|2 + ξ−1|PΓ,1w|2 + ξ−1|PΓ,2w|2� +dSdt. +(3.17) +From one hand, by definition of PΓ,1, we have +� T +0 +� +Γ +ξ−1|∆Γw|2dSdt ⩽ C +� T +0 +� +Γ +� +ξ−1|PΓ,1w|2 + s2λ2ξ3|w|2 + ξ−1|∂νw|2� +dSdt. +(3.18) +On the other hand, using the definition of PΓ,2 we deduce that +� T +0 +� +Γ +ξ−1|∂tw|2dSdt ⩽ C +� T +0 +� +Γ +� +ξ−1|∆Γw|2 + sλξ|w|2 + ξ−1|∂νw|2 + ξ−1|PΓ,2w|2� +dSdt. +(3.19) +Combining (3.18) and (3.19), we obtain (3.17). +Finally, combining (3.13), (3.17) and (3.12), and taking s, λ > 0, we obtain +� T +0 +� +Ω +(s3λ4ξ3|w|2 + sλ2ξ|∇w|2 + s−1ξ−1|∂tw|2 + s−1ξ−1|∆w|2)dxdt ++ +� T +0 +� +Γ +� +s3λ3ξ3|w|2 + sλ|∂νw|2 + sλ|∇Γw|2 + ξ−1|∆Γw|2 + ξ−1|∂tw|2� +dSdt +⩽C∥Rw∥2 +L2(Ω×(0,T)) + C∥RΓw∥2 +L2(Γ×(0,T)) + Cs3λ4 +� T +0 +� +ω +ξ3|w|2dxdt. +Finally, we come back to the original variable and conclude the desired inequality. +4 +Observability and null controllability of the linear system +The goal of this section is to prove a null controllability result for the linear Ginzburg-Landau + + + + + + + + + + + +L(y) = f + +1ωh, +in Ω × (0, T), +N(y, yΓ) = fΓ, +in Γ × (0, T), +y = yΓ, +on Γ × (0, T), +(y(0), yΓ(0)) = (y0, yΓ,0), +in Ω × Γ. +(4.1) +where the operators L and N were defined in (2.2). Naturally, the null controllability for (4.1) can +be expressed in the following terms: +Definition 4.1. We say that (4.1) is null controllable in L2 if for all T > 0, (y0, yΓ,0) ∈ L2, there +exists a control h ∈ L2(ω × (0, T)) such that the associated solution (y, yΓ) of (4.1) satisfies +y(·, T) = 0, in Ω, +yΓ(·, T) = 0, on Γ. +Following the classical duality between controllability and observability in the context of parabolic +equations (see e.g. [13]), the null controllability of (4.1) is equivalent to prove a suitable observabil- +ity inequality for its adjoint system. Then, thanks to this inequality and the Lax-Milgram lemma, +we allow us to deduce the null controllability of (4.1), which is crucial for the proof of the Theorem +(1.1) and an interesting result by itself. +17 + +4.1 +Observability inequality +As we explained above, we introduce the adjoint system + + + + + + + + + + + +L∗z = g, +in Ω × (0, T), +N ∗(z, zΓ)z = gΓ, +on Γ × (0, T), +z = zΓ, +on Γ × (0, T), +(z(T), zΓ(T)) = (zT , zΓ,T ), +in Ω × Γ. +(4.2) +where L∗ and N ∗ are given by (3.2). We point out that, if (zT , zΓ,T ) ∈ L2, then the associated weak +solution (z, zΓ) belongs to C0([0, T]; L2) ∩ L2(0, T; H1). This is done by results of section 2. +To formulate the result of this subsection, we shall define some additional functions. For t ∈ +(0, T), we define +µ(t) := + + + + + +4 +T 2 , +if t ∈ (0, T/2], +1 +t(T − t), +if t ∈ (T/2, T), +ˇϕ(t) :=µ(t) min +x∈Ω +� +e2sλm∥η0∥∞ − eλ(m∥η0∥∞+η0(x))� +, +t ∈ (0, T), +ˆϕ(t) :=µ(t) max +x∈Ω +� +e2sλm∥η0∥∞ − eλ(m∥η0∥∞+η0(x))� +, +t ∈ (0, T), +ˇξ(t) :=µ(t) min +x∈Ω +eλ(m∥η0∥∞+η0(x)), +t ∈ (0, T), +ˆξ(t) :=µ(t) max +x∈Ω +eλ(m∥η0∥∞+η0(x)), +t ∈ (0, T). +Proposition 4.2 (Observability inequality). Let N ⩾ 2. Suppose that (zT , zΓ,T ) ∈ L2 and (g, gΓ) ∈ +L2(0, T; L2). Then, there exists a constant C > 0 such that the associated weak solution (z, zΓ) ∈ +C0([0, T]; L2) ∩ L2(0, T; H1) of (4.2) satisfies +� +Ω +|z(0)|2dx + +� +Γ +|zΓ(0)|2dS + +� T +0 +� +Ω +e−2s ˆϕ(ˇξ3|z|2 + ˇξ|∇z|2)dxdt ++ +� T +0 +� +Γ +e−2s ˆϕ(ˇξ3|zΓ|2 + ˇξ|∇ΓzΓ|2)dSdt +⩽C +� T +0 +� +Ω +e−2s ˇϕ|g|2dxdt + C +� T +0 +� +Γ +e−2s ˇϕ|gΓ|2dSdt + C +� T +0 +� +ω +e−2s ˇϕ ˆξ3|z|2dxdt. +(4.3) +Proof. Consider a cut-off function θ ∈ C1([0, T]; R) such that +0 ⩽ θ(t) ⩽ 1, +∀t ∈ [0, T], +θ(t) = 1, +∀t ∈ [0, T/2], +θ(t) = 0, +∀t ∈ [3T/4, T]. +Then, the new variables (˜z, ˜zΓ) = (θz, θzΓ) satisfies + + + + + + + + + + + +L∗˜z = θg + θ′z, +in Ω × (0, T), +N ∗(˜z, ˜zΓ) = θgΓ + θ′zΓ, +on Γ × (0, T), +˜z = ˜zΓ, +on Γ × (0, T), +(˜z(T), ˜zΓ(T)) = (0, 0), +in Ω × Γ. +18 + +Then, by estimate (2.2), we can assert that +∥(˜z, ˜zΓ)∥2 +L∞(0,T;L2) + ∥(˜z, ˜zΓ)∥2 +L2(0,T;H1) +⩽C∥(θg, θgΓ)∥2 +L2(0,T;L2) + C∥(θ′z, θ′zΓ)∥2 +L2(0,T;L2). +In particular, we can deduce that +∥(z(0), zΓ(0))∥2 +L2 + ∥(z, zΓ)∥2 +L2(0,T/2;H1) +⩽C∥(g, gΓ)∥2 +L2(0,3T/4;L2) + C∥(z, zΓ)∥2 +L2(T/2,3T/4;L2). +Since e−2s ˆϕ ⩾ C > 0, for each t ∈ [0, T/2], we have +� +Ω +|z(0)|2dx + +� +Γ +|zΓ(0)|2dS + +� T/2 +0 +� +Ω +e−2s ˆϕ(|z|2 + |∇z|2)dxdt ++ +� T/2 +0 +� +Γ +e−2s ˆϕ(|zΓ|2 + |∇Γz|2)dSdt +⩽C +� 3T/4 +0 +� +Ω +|g|2dxdt + C +� 3T/4 +0 +� +Γ +|gΓ|2dSdt + C +� 3T/4 +T/2 +� +Ω +|z|2dxdt ++ C +� 3T/4 +T/2 +� +Γ +|zΓ|2dSdt. +(4.4) +In order to estimate the last two terms of (4.4), we use the fact that +0 < C ⩽ e−2sϕ(x,t)ξ3, +∀t ∈ (T/2, 3T/4), +to obtain +� 3T/4 +T/2 +� +Ω +|z|2dxdt + +� 3T/4 +T/2 +� +Γ +|zΓ|2dSdt +⩽ +� 3T/4 +T/2 +� +Ω +e−2sϕξ3|z|2dxdt + C +� 3T/4 +T/2 +� +Γ +e−2sϕξ3|zΓ|2dSdt +⩽C +� T +0 +� +Ω +e−2sϕ|g|2dxdt + C +� T +0 +� +Γ +e−2sϕ|gΓ|2dSdt + C +� T +0 +� +ω +e−2sϕξ3|z|2dxdt, +(4.5) +where we have used the Carleman estimate in Theorem 3.2 with s > 0 and λ > 0 fixed. Now, +combining (4.4) and (4.5) and using that − ˇϕ(t) ⩾ −ϕ(x, t) for all (x, t) ∈ Ω × (0, T), +� +Ω +|z(0)|2dx + +� +Γ +|zΓ(0)|2dS + +� T/2 +0 +� +Ω +e−2s ˆϕ(ˇξ3|z|2 + ˇξ|∇z|2)dxdt ++ +� T/2 +0 +� +Γ +e−2s ˆϕ(ˇξ3|zΓ|2 + ˇξ|∇ΓzΓ|2)dSdt +⩽C +� T +0 +� +Ω +e−2s ˇϕ|g|2dxdt + C +� T +0 +� +Γ +e−2s ˇϕ|gΓ|2dSdt + C +� T +0 +� +ω +e−2s ˇϕ ˆξ3|z|2dxdt. +(4.6) +On the other hand, by Carleman estimate (3.1) again, the following inequality holds +� T +T/2 +� +Ω +e−2s ˆϕ(ˇξ3|z|2 + ˇξ|∇z|2)dxdt + +� T +T/2 +� +Γ +e−2s ˆϕ(ˇξ3|zΓ|2 + ˇξ|∇ΓzΓ|2)dSdt +⩽C +� T +0 +� +Ω +e−2s ˇϕ|g|2dxdt + C +� T +0 +� +Γ +e−2s ˇϕ|gΓ|2dSdt + C +� T +0 +� +ω +e−2s ˇϕ ˆξ3|z|2dxdt. +(4.7) +Finally, adding inequalities (4.6) and (4.7), we deduce the observability inequality (4.3). This +ends the proof of the Proposition 4.2. +19 + +4.2 +Proof of the null controllability for the linear system +From the observability inequality (4.3), we can deduce the null controllability of the linear system +(4.1). In the following, for r ∈ [1, +∞], a linear space H and a measurable function ρ : (0, T) −→ R, +we shall consider the notation +Lr(ρ(0, T); H) = {y ∈ Lr(0, T; H); ρy ∈ Lr(0, T; H)}. +Consider the Banach space V defined by +V := {(y, yΓ, h) :(y, yΓ) ∈ L2(es ˇϕ(0, T); L2), h1ω ∈ L2(es ˇϕ ˆξ−3/2(0, T); L2(Ω)), +(L(y) − +1ωh, N(y, yΓ)) ∈ L2(es ˆϕ ˇξ−3/2(0, T); L2), +(y, yΓ) ∈ L2(e +1 +3s ˆϕ(0, T); H2) ∩ L∞(e +1 +3 s ˆϕ(0, T); H1)}, +(4.8) +endowed by its natural norm: +∥(y, yΓ, h)∥2 +V :=∥es ˇϕ(y, yΓ)∥2 +L2(0,T;L2) + ∥es ˇϕ ˆξ3h1ω∥2 +L2(Ω×(0,T)) ++ ∥es ˆϕ ˇξ−3/2(L(y) − +1ωh, N(y, yΓ))∥2 +L2(0,T;L2) ++ ∥e +1 +3s ˆϕ(y, yΓ)∥2 +L2(0,T;H2) + ∥e +1 +3s ˆϕ(y, yΓ)∥2 +L∞(0,T;H1). +Proposition 4.3. Let (y0, yΓ,0) ∈ L2 and assume that +(f, fΓ) ∈ L2(es ˆϕˇξ−3/2(0, T); L2). +(4.9) +Then, we can find a control h such that the associated solution (y, yΓ) of (4.1) satisfies (y, yΓ, h) ∈ V. +In particular, we have +y(·, T) = 0, in Ω, +yΓ(·, T) = 0, on Γ. +Proof. For our purposes, we define +P0 := +� +(z, zΓ) ∈ C∞(Ω × [0, T]) × C∞(Γ × [0, T]) : z +�� +Γ = zΓ on Γ × [0, T] +� +. +Let +a : P0 × P0 → R be the bilinear form +a((z, zΓ), (w, wΓ)) :=ℜ +� T +0 +� +Ω +e−2s ˇϕL∗zL∗(w)dxdt + ℜ +� T +0 +� +Γ +e−2s ˇϕN ∗(z, zΓ)N ∗(w, wΓ)dSdt ++ ℜ +� T +0 +� +ω +e−2s ˇϕˆξ3zwdxdt, +∀(z, zΓ), (w, wΓ) ∈ P0 × P0. +We also define the linear form ℓ : P0 → R as +ℓ(w, wΓ) :=ℜ +� +Ω +y0w(0)dx + ℜ +� +Γ +yΓ,0wΓ(0)dS + ℜ +� T +0 +� +Ω +fwdxdt ++ ℜ +� T +0 +� +Γ +fΓwΓdSdt, +∀(w, wΓ) ∈ P0. +In view of the observability inequality (4.3), it is clear that +∥(z, zΓ)∥ +a := +� +a((z, zΓ), (z, zΓ)), +∀(z, zΓ) ∈ P0, +20 + +defines a norm in P0. Let P be the completion of (P0, ∥·∥ +a). Then, +a(·, ·) is well-defined, continuous +and again definite positive on P. On the other hand, by (4.9) and the observability inequality (4.3), +it is easy to see that +|ℓ(w, wΓ)| ⩽ C∥(w, wΓ)∥ +a, +∀(w, wΓ) ∈ P0, +i.e. ℓ is continuous in P0 and, by density, in P. Thus, by Lax-Milgram Theorem, the variational +problem +� +Find (z, zΓ) ∈ P such that +a((z, zΓ), (w, wΓ)) = ℓ(w, wΓ), +∀(w, wΓ) ∈ P, +possesses exactly one solution (z∗, z∗ +Γ). Let (y∗, y∗ +Γ, h∗) defined by + + + + + +y∗ := e−2s ˇϕL∗(z∗), +in Ω × (0, T), +y∗ +Γ := e−2s ˇϕN ∗(z∗, z∗ +Γ), +on Γ × (0, T), +h∗ := e−2s ˇϕˆξ3z∗, +in ω × (0, T). +We point out that (y∗, y∗ +Γ, h∗) satisfies +� T +0 +� +Ω +e2s ˇϕ|y∗|2dxdt + +� T +0 +� +Γ +e2s ˇϕ|y∗ +Γ|2dSdt + +� T +0 +� +ω +e2s ˇϕˆξ−3|h∗|2dxdt += a((z∗, z∗ +Γ), (z∗, z∗ +Γ)) < ∞, +and also that +ℜ +� T +0 +� +Ω +y∗L∗(w)dxdt + ℜ +� T +0 +� +Γ +y∗ +ΓN ∗(w, wΓ)dSdt + ℜ +� T +0 +� +ω +h∗wdxdt +=ℜ +� +Ω +y0w(0)dx + ℜ +� +Γ +yΓ,0wΓ(0)dS + ℜ +� T +0 +� +Ω +fwdxdt ++ ℜ +� T +0 +� +Γ +fΓwΓdSdt, +(4.10) +i.e., from (4.10) we deduce that (y∗, y∗ +Γ) ∈ C0([0, T]; L2) ∩ L2(0, T; H1) is a distributional solution +with control h∗ and initial datum (y0, yΓ,0) such that y∗(·, T) = 0 in Ω and yΓ(·, T) = 0 on Γ. +It remains to check that +(y∗, y∗ +Γ) ∈ L2(e +1 +3s ˆϕ(0, T); H2) ∩ L∞(e +1 +3s ˆϕ(0, T); H1). +(4.11) +In order to do that, we notice that the new variables +(y⋆, y⋆ +Γ) := (e +1 +3s ˆϕy∗, e +1 +3s ˆϕy∗ +Γ) +solves the problem + + + + + + + + + + + +L(y⋆) = e +1 +3s ˆϕ(f + h1ω) + (e +1 +3 s ˆϕ)ty∗, +in Ω × (0, T), +N(y⋆, y⋆ +Γ) = e +1 +3s ˆϕfΓ + (e +1 +3s ˆϕ)ty∗ +Γ, +in Γ × (0, T), +y⋆ = y⋆ +Γ, +on Γ × (0, T), +(y(0), yΓ(0)) = e +1 +3s ˆϕ(0)(y0, yΓ,0), +in Ω × Γ. +(4.12) +21 + +Since +��� +� +e +1 +3s ˆϕ� +t y∗��� ⩽ C ˆξ2e +1 +3 s ˆϕ|y∗| ⩽ Ces ˇϕ|y⋆| ∈ L2(0, T; L2), +and +� +e +1 +3 s ˆϕ(f + h1ω), e +1 +3s ˆϕfΓ +� +∈ L2(0, T; L2), +and since e +1 +3 s ˆϕ(0)(y0, yΓ,0) ∈ H1, it follows from Proposition 2.3 that the solution (y⋆, y⋆ +Γ) of (4.12) +satisfies +(y⋆, y⋆ +Γ) ∈ L2(0, T; H2 ∩ C0([0, T]; H1)), +which is equivalently to (4.11). We conclude that (y∗, y∗ +Γ, h∗) ∈ V. This ends the proof of the +Proposition 4.3. +5 +Proof of the Theorem 1.1 +To prove the main theorem of this article, we shall use a local inversion argument. This will be +done by using the following local inversion mapping theorem in Banach spaces (see for instance [5], +page 107). +Theorem 5.1. Let B1 and B2 be two Banach spaces and let A : B1 → B2 be a C1(B1; B2) function. +Suppose that b1 ∈ B1, b2 = A(b1) and that A′(b1) : B1 → B2 is surjective. Then, there exists δ > 0 +such that, for every b′ ∈ B2 satisfying ∥b′ − b2∥B2 < δ, there exists a solution of the equation +A(b) = b′, +b ∈ B1. +Moreover, there exists a constant C > 0 such that +∥b1 − b∥B1 ⩽ C∥b2 − b′∥B2. +Now, we have all the ingredients to prove the Theorem 1.1. +Proof of the Theorem 1.1. Consider the spaces +B1 := V, +B2 := L2(es ˆϕ ˇξ−3/2(0, T); L2) × L2, +where V is the linear space defined in (4.8). +Consider A : B1 → B2 be the operator defined by +A(u, uΓ, h) = +� +Lu + c(1 + γi)|u|2u − +1ωh, N(u, uΓ) + c(1 + γi)|uΓ|2uΓ, u(0), uΓ(0) +� +. +We choose, b1 = (0, 0, 0) and b2 = A(0, 0, 0) = (0, 0, u0, uΓ,0). In order to use the Theorem 5.1, we +shall check that the following two assertions: +(a) A′(0, 0, 0) : B1 → B2 is surjective. +(b) A is an operator of class C1 from B1 to B2. +22 + +A simple computation shows that for all (u∗, u∗ +Γ, h∗) ∈ B1, +A′(u, uΓ, h)(u∗, u∗ +Γ, h∗) =(L(u∗) + 3c(1 + γi)|u|2u∗ − +1ωh∗, N(u∗, u∗ +Γ) ++ 3c(1 + γi)|uΓ|2u∗ +Γ, u∗(0), u∗ +Γ(0)). +(5.1) +In particular, taking (u, uΓ, h) = (0, 0, 0) in (5.1), we have +A′(0, 0, 0)(u∗, u∗ +Γ, h∗) = (L(u∗) − +1ωh∗, N(u∗, u∗ +Γ), u∗(0), u∗ +Γ(0)) . +Now, it is clear that, in view of the Proposition 4.3, the operator A′(0, 0, 0) is surjective. +On the other hand, to prove that A ∈ C1(B1, B2), it is sufficient to check that the map +(u1, uΓ,1, h1), (u2, uΓ,2, h2), (u3, uΓ,3, h3) �→ (u1u2u3, uΓ,1uΓ,2uΓ,3), +is continuous from V × V × V to L2(es ˇϕ(0, T); L2). Then, according to H¨older inequality and using +that L2(0, T; H2) ∩ L∞(0, T; H1) ֒→ L6(0, T; L9) ֒→ L6(0, T; L6), we have +∥es ˆϕ ˇξ−3/2(u1u2u3, uΓ,1uΓ,2uΓ,3)∥L2(0,T;L2) +⩽C +3 +� +j=1 +∥e +1 +3 s ˆϕ(uj, uΓ,j)∥L6(0,T;L6) +⩽C +3 +� +j=1 +∥e +1 +3 s ˆϕ(uj, uΓ,j)∥L2(0,T;H2)∩L∞(0,T;H1) +⩽C +3 +� +j=1 +∥(uj, uΓ,j, hj)∥V. +(5.2) +Therefore, we have that A ∈ C1(B1, B2), and we can apply Theorem 5.1 to guarantee the +existence of δ > 0 such that ∥(u0, uΓ,0)∥H1 ⩽ δ, one can find (y, yΓ, h) ∈ B1 := V such that the +associated solution (u, uΓ) (with initial datum (u0, uΓ,0)) satisfies +y(·, T) = 0, in Ω, +yΓ(·, T) = 0, on Γ. +This, ends the proof of the Theorem 1.1. +Acknowledgments +References +[1] Ole Morten Aamo, Andrey Smyshlyaev, and Miroslav Krsti´c. Boundary control of the linearized +Ginzburg-Landau model of vortex shedding. SIAM J. Control Optim., 43(6):1953–1971, 2005. +[2] El Mustapha Ait Ben Hassi, Salah-Eddine Chorfi, and Lahcen Maniar. 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Discrete carleman +estimates and application to controllability for a fully-discrete parabolic operator with dynamic +boundary conditions. arXiv preprint arXiv:2209.14351, 2022. +[21] C. David Levermore and Marcel Oliver. The complex Ginzburg-Landau equation as a model +problem. +In Dynamical systems and probabilistic methods in partial differential equations +(Berkeley, CA, 1994), volume 31 of Lectures in Appl. Math., pages 141–190. Amer. Math. +Soc., Providence, RI, 1996. +[22] Lahcen Maniar, Martin Meyries, and Roland Schnaubelt. Null controllability for parabolic +equations with dynamic boundary conditions. Evol. Equ. Control Theory, 6(3):381–407, 2017. +[23] Lahcen Maniar, Martin Meyries, and Roland Schnaubelt. Null controllability for parabolic +equations with dynamic boundary conditions. Evolution Equations & Control Theory, 6(3):381– +407, 2017. +[24] Alberto Mercado and Roberto Morales. Exact controllability for a schr¨odinger equation with +dynamic boundary conditions. Submitted, 2022. +[25] Lionel Rosier and Bing-Yu Zhang. Null controllability of the complex Ginzburg-Landau equa- +tion. Ann. Inst. H. Poincar´e C Anal. Non Lin´eaire, 26(2):649–673, 2009. +[26] Lionel Rosier and Bing-Yu Zhang. Null controllability of the complex ginzburg-landau equation. +In Annales de l’IHP Analyse non lin´eaire, volume 26, pages 649–673, 2009. +[27] Michael E. Taylor. Partial differential equations I. Basic theory, volume 115 of Applied Math- +ematical Sciences. Springer, New York, second edition, 2011. +25 + diff --git a/d9E1T4oBgHgl3EQfyAW2/content/tmp_files/load_file.txt b/d9E1T4oBgHgl3EQfyAW2/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..06338dae8e830fcc1237ce056ae3c6e8ded9aba4 --- /dev/null +++ b/d9E1T4oBgHgl3EQfyAW2/content/tmp_files/load_file.txt @@ -0,0 +1,797 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf,len=796 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='03429v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='AP] 9 Jan 2023 Local null controllability of a cubic Ginzburg-Landau equation with dynamic boundary conditions Nicol´as Carre˜no∗ Alberto Mercado† Roberto Morales‡ January 10, 2023 Abstract This paper deals with controllability properties of a cubic Ginzburg-Landau equation with dynamic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' More precisely, we prove a local null controllability result by using a single control supported in a small subset of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' In order to achieve this result, we firstly linearize the system around the origin and we analyze it by the duality approach and an appropriate Carleman estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Then, by using an inverse function theorem, the local null controllability of the nonlinear system is proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Keyword: Controllability, Ginzburg-Landau equation, Dynamic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' MSC(2020) 93B05, 35Q56, 93B07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' 1 Introduction and main results 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1 Introduction Let Ω ⊂ RN (N ⩾ 2) be a bounded domain with boundary Γ := ∂Ω of class C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Given the parameters a, b, c > 0, α, γ ∈ R \\ {0}, we consider the following cubic Ginzburg-Landau equation with dynamic boundary conditions \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ∂tu − a(1 + αi)∆u + c(1 + γi)|u|2u = 1ωh, in Ω × (0, T), ∂tuΓ + a(1 + αi)∂νu − b(1 + αi)∆ΓuΓ + c(1 + γi)|uΓ|2uΓ = 0, on Γ × (0, T), u = uΓ, on Γ × (0, T), (u(0), uΓ(0)) = (u0, uΓ,0), in Ω × Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1) Here, (u, uΓ) is the state of the system, (u0, uΓ,0) the initial conditions and h ∈ L2(ω ×(0, T);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' C) is a control acting on ω ⊂ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' We denote by ∆Γ the Laplace-Beltrami operator on Γ and by ∂νy the normal derivative associated to the outward normal ν of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' We notice that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1) can be seen as a coupled system in the variables (u, uΓ), which is controlled by a single control h in a (small) subset ω of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' This means that the first equation of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1) ∗Departamento de Matem´atica, Universidad T´ecnica Federico Santa Mar´ıa, Casilla 110-V, Valpara´ıso, Chile e-mail: nicolas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='carrenog@usm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Partially supported by Fondecyt 1211292.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' †Departamento de Matem´atica, Universidad T´ecnica Federico Santa Mar´ıa, Casilla 110-V, Valpara´ıso, Chile e-mail: alberto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='mercado@usm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Partially supported by Fondecyt 1211292 and ANID Millennium Science Initiative Program, Code NCN19-161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' ‡Departamento de Matem´atica, Universidad T´ecnica Federico Santa Mar´ıa, Casilla 110-V, Valpara´ıso, Chile e-mail: roberto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='moralesp@usm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Partially supported by Fondecyt 3200830.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' 1 is controlled directly by the action of the control, while the second equation is being controlled through the side condition u = uΓ on Γ × (0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' The main objective of this work is to obtain the local null controllability of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=', we will prove the existence of a number δ > 0 such that, for every initial state (u0, uΓ,0) ∈ X (where X is an appropriate Banach space) which fulfills ∥(u0, uΓ,0)∥X ⩽ δ, we can find a control h ∈ L2(ω × (0, T)) such that the associated solution (u, uΓ) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1) satisfies y(·, T) = 0, in Ω, yΓ(·, T) = 0, on Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='2 Previous results The cubic complex Ginzburg-Landau equation is one of the most studied nonlinear equations used to model physical phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' This equation has been used to describe several phenomena ranging from nonlinear waves, second-order phase transitions, superconductivity, superfluidity and Bose- Einstein condensation to liquid crystals and strings in field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' For a detailed description of relevant applications in different fields, see [6] and [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' The existence and uniqueness of nonlinear Ginzburg-Landau equations with Dirichlet or periodic boundary conditions have been intensely investigated in several papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' For instance, we refer to [21], [16], [17], [12], [9], [8], [7] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Concerning controllability properties of the Ginzburg-Landau equation with Dirichlet boundary conditions, only a few papers have been devoted to the study of the controllability of such problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' In [1], the stabilization of the linearized Ginzburg-Landau model with Dirichlet boundary conditions around an unstable equilibrium state is studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Moreover, in [12], the author develop a Carleman inequality for an operator of the form (a + ib)∂t + div(A · ∇), with A being a smooth, uniformly elliptic matrix, and a null controllability result for the linear PDE with a distributed control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' In 2009, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Rosier and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Zhang in [25] proved a controllability result for the nonlinear case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' In this case, the control acts on a part of the boundary and the proof is based on a suitable Carleman estimate for the linear adjoint system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Then, combining a fixed-point argument together with the theory of sectorial operators, the authors obtained a local controllability result for a wide class of nonlinearities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' In particular, controllability results for the cubic and quintic Complex Ginzburg-Landau are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Recently, some results on inverse problems and controllability issues have been obtained for PDEs with dynamic boundary conditions, see for instance [22], [19], [4], [2], [3], and [20] for the heat equation, [15] for the wave equation, and [24] for the Schr¨odinger operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' In these works, the authors has been used the duality equivalence to prove the associated observability inequality by using Carleman estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' At this level, we point out that it is not evident at all that such systems can be controlled by the action of a single control due to the tangential derivative terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' In fact, in the case of the linear wave equation with mixed boundary conditions (oscilatory boundary conditions and Dirichlet boundary conditions) [15], the authors obtain exact controllability results where the control region is on the whole boundary (and therefore on the whole system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' On the other hand, in a similar setting for the Schr¨odinger equation, in [24] is obtained the exact controllability with a control acting only in a part of the boundary, by using a particular Carleman weight adapted to the geometric properties of the domain, which allows to estimate boundary terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Concerning the Ginzburg-Landau equation with dynamic boundary conditions, we mention [11], where well-posedness of linear/nonlinear of such models is obtained and long time behavior of 2 solutions is characterized when Lipschitz nonlinearities are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' However, to the best of the authors’ knowledge, this is the first time that the null controllability for the cubic Ginzburg-Landau is studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' In the following subsection we shall introduce some functional spaces used in the context of PDE’s with dynamic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='3 General setting In this section, we set up the notation and terminology used in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' The set Γ = ∂Ω can be seen as an (N − 1)-dimensional compact Riemannian submanifold equipped by the Riemmanian metric g induced by the natural embedding Γ ⊂ RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' In addition, we shall denote by dS the (N − 1)-Lebesgue measure for Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Since we are considering dynamic boundary conditions, we need to define some differential operators on Γ, which can be defined in terms of the metric g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' However, for our purposes, it will be enough to use the most important properties of the underlaying operators and spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' The details can be found, for instance, in [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' For the sake of completeness, we recall some of those properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' The tangential gradient ∇Γ of yΓ at each point x ∈ Γ can be seen as the projection of the standard Euclidean gradient ∇y onto the tangent space of Γ at x ∈ Γ, where yΓ is the trace of y on Γ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=', we have the following equation ∇ΓyΓ = ∇y − ν∂νy, where y = yΓ on Γ and ∂νy is the normal derivative associated to the outward normal ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' In this way, the tangential divergence divΓ in Γ is defined by divΓ(FΓ) : H1(Γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' R) → R, yΓ �→ − � Γ FΓ · ∇ΓyΓdS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' The Laplace-Beltrami operator is given by ∆ΓyΓ := div(∇ΓyΓ), for all yΓ ∈ H2(Γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' In particular, the surface divergence theorem holds: � Γ ∆ΓyΓzΓdS = − � Γ ∇ΓyΓ · ∇ΓzΓdS, ∀yΓ ∈ H2(Γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' R), zΓ ∈ H1(Γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' In order to simplify the notation, here and subsequently, the function spaces refer to complex- valued functions unless otherwise stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' For 1 ⩽ p ⩽ +∞, we consider the Banach space Lp := Lp(Ω) × Lp(Γ), endowed by the norm given by the relation ∥(u, uΓ)∥2 Lp := ∥u∥2 Lp(Ω) + ∥uΓ∥2 Lp(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' In particular, for p = 2, the space L2 := L2(Ω) × L2(Γ) is a (real) Hilbert space equipped with the scalar product ⟨(u, uΓ), (v, vΓ)⟩L2 := ℜ � Ω uvdx + ℜ � Γ uΓvΓdσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' For k ∈ N, we also introduce the space Hk := {(y, yΓ) ∈ Hk(Ω) × Hk(Γ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' y �� Γ = yΓ}, where Hk(Ω) and Hk(Γ) are the usual Sobolev spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='4 Main result Our main result is given by the following theorem: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Suppose that N = 2 or N = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Let a, b, c > 0, α, γ ∈ R \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Then, for every T > 0 and ω ⋐ Ω, there exists δ > 0 such that, for every (y0, yΓ,0) ∈ H1 satisfying ∥(u0, uΓ,0)∥H1 ⩽ δ, there exists a control h ∈ L2(ω ×(0, T)) such that the unique corresponding solution (u, uΓ) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1) satisfies u(·, T) = 0, in Ω, uΓ(·, T) = 0, on Γ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=', the nonlinear system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1) is locally null controllable by a single control for an arbitrary control domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' To prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1 we first deduce a null controllability result for a linear system associated to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1): \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ∂ty − a(1 + αi)∆y = f + 1ωh, in Ω × (0, T), ∂tyΓ + a(1 + αi)∂νy − b(1 + αi)∆ΓyΓ = fΓ, on Γ × (0, T), y = yΓ, on Γ × (0, T), (y, yΓ)(0) = (y0, yΓ,0), in Ω × Γ, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='2) where (f, fΓ) will be taken to decrease exponentially to zero in t = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Then, we prove a new Carleman estimate for the adjoint system of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='2) (see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='2) below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' This will provide existence (and uniqueness) to a variational problem, from which we define a solution (y, yΓ, h) to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='2) such that y(T) = 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Moreover, the solution is such that eC/(T−t)(y, yΓ, h) ∈ L2 × L2(ω × (0, T)), for some constant C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Finally, by an inverse mapping theorem, we deduce the null controllability for the nonlinear system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' In Section 2 we establish the existence and uniqueness of solutions of systems like (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' In Section 3, we prove a suitable Carleman estimate for the Ginzburg-Landau operator with dynamic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' In Section 4 we prove the observability estimate for the adjoint system and prove the null controllability of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Finally, in Section 5 we prove the Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' 2 Existence and uniqueness of solutions In this section, we present new results concerning existence and uniqueness for the Ginzburg-Landau equations with dynamic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1 Linear problem We devote to study the Cauchy problem \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 Lu = f, in Ω × (0, T), N(u, uΓ) = fΓ, on Γ × (0, T), u = uΓ, on Γ × (0, T), (u(0), uΓ(0)) = (u0, uΓ,0), in Ω × Γ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1) 4 where Lu := ∂tu − a(1 + αi)∆u, N(u, uΓ) := ∂tuΓ + a(1 + αi)∂νu − b(1 + αi)∆ΓuΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='2) The problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1) can be seen in the abstract form � U ′(t) = AGLU(t) + F(t), t ∈ (0, T), U(0) = U0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='3) where AGL : D(AGL) ⊂ L2 → L2 is the operator defined by AGL(U) := � a(1 + αi)∆u −a(1 + αi)∂νu + b(1 + αi)∆ΓuΓ � , ∀ U := � u uΓ � ∈ D(AGL), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='4) with domain D(AGL) := � U = � u uΓ � ∈ H1 : (∆u, ∆ΓuΓ) ∈ L2 � = H2, where the last equivalence is justified in [10] (see also [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' It is easy to see that AGL can be written as AGL = (1 + αi)AW , D(AW) = D(AGL) = H2, where AW is the Wentzell-Laplacian operator defined in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Then, arguing as in [26, Section 2] we have the following result: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' The operator AGL defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='4) is densely defined and generates an analytic semigroup (etAGL)t⩾0 in L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' According to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1, the existence and uniqueness of strong solutions of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='3) in the usual sense are guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' In the next subsection, we provide existence and uniqueness of solutions in appropriate spaces by energy estimates and density arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='2 A priori estimates Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Suppose that (u0, uΓ,0) ∈ L2 and (f, fΓ) ∈ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Then, the mild solution (u, uΓ) of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1) belongs to C0([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' L2) ∩ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' H1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Moreover, there exists a constant C1 > 0 such that the associated solution (u, uΓ) of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1) satisfies ∥(u, uΓ)∥C0([0,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='L2) + ∥(u, uΓ)∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='H1) ⩽ C1∥(f, fΓ)∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='L2) + C1∥(u0, uΓ,0)∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='5) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Firstly, we multiply by u the first equation of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1) and we integrate in Ω × (0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Secondly, we multiply the second equation of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1) by uΓ and integrate on Γ × (0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Next, we add these identities and take the real part on the obtained equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' After integration by parts, this yields 1 2 d dt �� Ω |u(t)|2dx + � Γ |uΓ(t)|2dS � + a � Ω |∇u(t)|2dx + b � Γ |∇ΓuΓ(t)|2dS =ℜ � Ω f(t)u(t)dx + ℜ � Γ fΓ(t)uΓ(t)dS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' By Young’s inequality, it is easy to check that ∥(u, uΓ)∥2 C0([0,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='L2) + ∥(u, uΓ)∥2 L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='H1) ⩽ C∥(f, fΓ)∥2 L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='L2) + C∥(u0, uΓ,0)∥2 L2, which clearly implies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' 5 Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Let (u, uΓ,0) ∈ H1 and (f, fΓ) ∈ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Then, the associated weak solution (u, uΓ) of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1) belongs to H1(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' L2) ∩ C0([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' H1) ∩ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' H2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Moreover, there exists a constant C2 > 0 such that (u, uΓ) satisfies ∥(u, uΓ)∥H1(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='L2) + ∥(u, uΓ)∥C0([0,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='H1) + ∥(u, uΓ)∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='H2) ⩽C2∥(f, fΓ)∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='L2) + C2∥(u0, uΓ,0)∥H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='6) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' The proof is divided into three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Step 1: Our first task is to obtain an L2 estimates for ∂tu and ∂tuΓ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' In order to do that, we multiply the first equation of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1) by (1 − αi)∂tu and integrate in Ω × (0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' In addition, we multiply the second equation of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1) by (1−αi)∂tuΓ and integrate on Γ×(0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Then, we sum up these identities and take the real part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' This yields � Ω |∂tu(t)|2dx + � Γ |∂tuΓ(t)|2dS + 1 2a(1 + α2) d dt � Ω |∇u|2dx + 1 2b(1 + α2) d dt � Γ |∇ΓuΓ|2dS =ℜ � Ω (1 − αi)f∂tudx + ℜ � Γ (1 − αi)fΓ∂tuΓdS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' This implies that ∥(u, uΓ)∥2 H1(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='L2) + ∥(u, uΓ)∥2 C0([0,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='H1) ⩽ C∥(f, fΓ)∥2 L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='L2) + C∥(u0, uΓ,0)∥2 H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Step 2: In this step, we derive L2(H2) estimates for the solution (u, uΓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' To do this, we firstly point out that the estimate of ∂tu implies that ∥∆u∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='L2(Ω)) ⩽ C∥(f, fΓ)∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='L2) + C∥(u0, uΓ,0)∥H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' By elliptic regularity applied to the first equation of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1), we have ∥u(t)∥H2(Ω) ⩽ C∥f(t)∥L2(Ω) + C∥∂tu(t)∥L2(Ω) + C∥uΓ(t)∥H3/2(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Integrating on t ∈ [0, T], we deduce that ∥u∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='H2(Ω)) ⩽ C∗∥(f, fΓ)∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='L2) + C∗∥(u0, uΓ,0)∥H1 + C∗∥uΓ∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='H3/2(Γ)), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='7) for some constant C∗ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Now, from the second equation of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1), we deduce that ∥uΓ∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='H2(Γ)) ⩽C∥fΓ∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='L2(Γ)) + C∥∂νu∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='L2(Γ)) + C∥∂tuΓ∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='L2(Γ)) ⩽C∥(f, fΓ)∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='L2) + C∥(u0, uΓ,0)∥H1 + C∥∂νu∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='L2(Γ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='8) Moreover, by interpolation inequalities, we have that for every 0 < s < 1/2 and ε > 0, there are positive constants Cs and Cε such that ∥∂νu∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='L2(Γ)) ⩽ Cs∥u∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='H3/2+s(Ω)) ⩽ Cε∥u∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='H1(Ω)) + ε∥u∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='H2(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='9) Combining (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='7), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='9) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='9) together with estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='5), we get ∥u∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='H2(Ω)) ⩽ C∥(f, fΓ)∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='L2(Ω)) + C∥(u0, uΓ,0)∥H1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='10) where we have chosen ε > 0 small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' With the estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='10) at hand, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='9) we can assert that ∥∂νu∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='L2(Γ)) ⩽ C∥(f, fΓ)∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='L2) + C∥(u0, uΓ,0)∥H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='11) Thus, substituting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='11) into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='8), we conclude that ∥uΓ∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='H2(Γ)) ⩽ C∥(f, fΓ)∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='L2) + C∥(u0, uΓ,0)∥H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' 6 Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Let (f, fΓ) ∈ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' L2) and (u0, uΓ,0) ∈ L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Then, the associated weak solution (u, uΓ) satisfies ∥( √ t∂tu, √ t∂tuΓ)∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='L2) + ∥( √ tu, √ tuΓ)∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='H2) + ∥( √ tu, √ tuΓ)∥C0([0,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='H1) ⩽C∥(f, fΓ)∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='L2(Ω)) + C∥(u0, uΓ,0)∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' The proof is a slight modification of the arguments used in of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' For this reason, we omit the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' We point out that Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='4 implies that, for each T > 0, (u0, uΓ,0) ∈ L2, (f, fΓ) ∈ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' L2) and ε > 0, the weak solution (u, uΓ) of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1) satisfies (u, uΓ) ∈ H1(ε, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' L2) ∩ L2(ε, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' H2) ∩ C0([ε, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' H1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Moreover, there exists a constant C > 0 (independent of ε) such that ∥(u, uΓ)∥H1(ε,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='L2) + ∥(u, uΓ)∥L2(ε,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='H2) + ∥(u, uΓ)∥C0([ε,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='H1) ⩽C∥(u0, uΓ)∥L2 + C∥(f, fΓ)∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='3 Nonlinear problem In this subsection, we prove a local existence theorem for the nonlinear problem \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 Lu + c(1 + γi)|u|2u = f, in Ω × (0, T), N(u, uΓ) + c(1 + γi)|uΓ|2uΓ = fΓ, in Γ × (0, T), u = uΓ, on Γ × (0, T), (u(0), uΓ(0)) = (u0, uΓ,0), in Ω × Γ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='12) where L and N are given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='2), with a, b, c > 0 and α, γ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Let N = 2 or N = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' There exist ε > 0 and C > 0 such that, for every (f, fΓ) ∈ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' L2), (u0, uΓ,0) ∈ H1 such that ∥(f, fΓ)∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='L2) + ∥(u0, uΓ,0)∥H1 ⩽ ε, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='13) there exists a unique solution (u, uΓ) of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='12) which satisfies ∥(u, uΓ)∥C0([0,T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='H1) + ∥(u, uΓ)∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='H2) ⩽ C � ∥(f, fΓ)∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='L2) + ∥(u0, uΓ,0)∥H1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' We denote B = C0([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' H1)∩L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' H2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Given (f, fΓ) ∈ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' L2) and (u0, uΓ,0) ∈ H1, for each (z, zΓ) ∈ B, we consider the system \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 Lu = f − c(1 + γi)|z|2z, in Ω × (0, T), N(u, uΓ) = fΓ − c(1 + γi)|zΓ|2zΓ, on Γ × (0, T), u = uΓ, on Γ × (0, T), (u(0), uΓ(0)) = (u0, uΓ,0), in Ω × Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='14) From Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='3 and the fact that B ֒→ L6(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' L6), we have that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='14) has a unique solution (u, uΓ) ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Hence, we can define the map F : B → B given by F(z, zΓ) = (u, uΓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Moreover, we also have that there exists D > 0 such that ∥F(z, zΓ)∥B ⩽ D � ∥(f, fΓ)∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='L2) + ∥(u0, uΓ,0)∥H1 + ∥(z, zΓ)∥3 B � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='15) 7 Clearly, (u, uΓ) ∈ B is a solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='12) if and only if it is a fixed point of the map F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' We will show that there exists R > 0 such that the restriction of F to the closed ball BR := {(z, zΓ) ∈ B : ∥(z, zΓ)∥B ⩽ R} is a contraction from BR into BR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Then, the proof will follow from a classic fixed point result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Indeed, from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='15) and assumption (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='13), for each (z, zΓ) ∈ BR we have ∥F(z, zΓ)∥B ⩽ D � ε + R3� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='16) Moreover, for each (z, zΓ), (w, wΓ) ∈ BR, taking into account the equations satisfied by F(z, zΓ)− F(w, wΓ), from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='3 we have that ∥F(z, zΓ) − F(w, wΓ)∥2 B ⩽D1∥|z|2z − |w|2w, |zΓ|2zΓ − |wΓ|2wΓ∥2 L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='L2) ⩽D1 � ∥z − w∥2 L∞(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='L4(Ω)) � ∥z2 + zw∥2 L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='L4(Ω)) + ∥w∥4 L4(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='L8(Ω)) � +∥zΓ − wΓ∥2 L∞(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='L4(Γ)) � ∥z2 Γ + zΓwΓ∥2 L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='L4(Γ)) + ∥wΓ∥4 L4(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='L8(Γ)) �� , and then, taking into account that B ֒→ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' L4) ∩ L4(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' L8), we get that ∥F(z, zΓ) − F(w, wΓ)∥B ⩽ D2R2∥(z, zΓ) − (w, wΓ)∥B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='17) Therefore, in order to conclude, we choose R > 0 such that R2 < min � 1 2D, 1 D2 � and ε > 0 such that ε ⩽ R 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='16), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='17) and the Banach Fixed point Theorem, we get the existence of a unique fixed point of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' 3 A new Carleman estimate for the linear Ginzburg-Landau equa- tion with dynamic boundary conditions In this section, we deduce a Carleman estimate for the linear Ginzburg-Landau operator with dynamic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' We start introducing the weight functions that we shall use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' For this propose, we recall the following Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Given a nonempty open set ω′ ⋐ Ω, there exists a function η0 ∈ C2(Ω) such that η0 > 0, in Ω, η0 = 0, on Γ, |∇η0| > 0, in Ω \\ ω′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Given ω′ ⋐ Ω, we take η0 with respect to ω′ as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' For λ, m > 1, we define ϕ(x, t) :=(t(T − t))−1 � e2λm∥η0∥∞ − eλ(m∥η0∥∞+η0(x))� , ∀(x, t) ∈ Ω × (0, T), ξ(x, t) :=(t(T − t))−1eλ(m∥η0∥∞+η0(x)), ∀(x, t) ∈ Ω × (0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' There exist constants C, λ0, s0 > 0 such that for all λ ⩾ λ0 and s ⩾ s0, we have � T 0 � Ω e−2sϕ � s3λ4ξ3|v|2 + sλ2ξ|∇v|2 + s−1|∂tv|2 + s−1|∆v|2� dxdt + � T 0 � Γ e−2sϕ(s3λ3ξ3|vΓ|2 + sλξ|∇ΓvΓ|2 + sλ|∂νv|2 + |∂tvΓ|2 + |∆ΓvΓ|2)dSdt ⩽Cs3λ4 � T 0 � ω e−2sϕξ3|v|2dxdt + C � T 0 � Ω e−2sϕ|L∗(v)|2dxdt + C � T 0 � Γ e−2sϕ|N ∗(v, vΓ)|2dSdt, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1) 8 for all (v, vΓ) ∈ H1(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' L2) ∩ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' H2), where L∗(v) = ∂tv + a(1 − αi)∆v, N ∗(v, vΓ) := ∂tvΓ − a(1 − αi)∂νv + b(1 − αi)∆ΓvΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='2) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' For convenience, the proof has been divided into several steps: Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' For simplicity, we consider v ∈ C∞(Ω × [0, T]), vΓ = v, λ ⩾ λ1 ⩾ 1, and s ⩾ s0 ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' define w :=e−sϕv, in Ω × (0, T), f :=e−sϕ(∂tv + a(1 − αi)∆v), in Ω × (0, T), fΓ :=e−sϕ(∂tv − a(1 − αi)∂νv + b(1 − αi)∆Γv), on Γ × (0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Straightforward computations show that ∇ϕ = −∇ξ = −λξ∇η0, ∆ϕ = −λ2ξ|∇η0|2 − λξ∆η0, ∂νϕ = −λξ∂νη0, with ∂νη0 ⩽ c < 0 on Γ, for some constant c > 0 and ∇Γϕ = ∇Γξ = 0, ∆Γϕ = ∆Γξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Then, in Ω × (0, T) we have the following identity: f =∂tw + a(1 − αi)∆w + as2λ2(1 − αi)|∇η0|2ξ2w − asλ2(1 − αi)|∇η0|2ξw − asλ(1 − αi)∆η0ξw − 2asλ(1 − αi)ξ∇η0 · ∇w + s∂tϕw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='3) On the other hand, on Γ × (0, T) we have fΓ =∂tw − a(1 − αi)∂νw + asλ(1 − αi)∂νη0ξw + b(1 − αi)∆Γw + s∂tϕw (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='4) Now, we define P1w :=a(s2λ2|∇η0|2ξ2w + ∆w) + aαi(2sλξ∇η0 · ∇w + (sλ2|∇η0|2 + sλ∆η0)ξw) + s∂tϕw P2w := − a(2sλξ∇η0 · ∇w + (sλ2|∇η0|2 + sλ∆η0)ξw) − aαi(s2λ2|∇η0|2ξ2w + ∆w) + ∂tw Rw :=f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' and PΓ,1w :=b∆Γw − 2a2 b αisλ∂νη0ξw + ∂tϕw, PΓ,2w := −αb∆Γw + 2a2 b sλ∂νη0w + ∂tw, RΓw :=fΓ − a(1 − αi)sλ∂νη0ξw + a(1 − αi)∂νw + 2a2 b (1 − αi)sλ∂νη0ξw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Then, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='4) can be written as P1w + P2w = Rw, in Ω × (0, T), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='5) PΓ,1w + PΓ,2w = RΓw, on Γ × (0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='6) 9 Then, taking the L2(Ω × (0, T))-norm in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='5) and the L2(Γ × (0, T))-norm in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='6), we obtain ∥P1w∥2 L2(Ω×(0,T)) + ∥P2w∥2 L2(Ω×(0,T)) + ∥PΓ,1w∥2 L2(Γ×(0,T)) + ∥PΓ,2w∥2 L2(Γ×(0,T)) + 2⟨P1w, P2w⟩L2(Ω×(0,T)) + 2⟨PΓ,1w, PΓ,2w⟩L2(Γ×(0,T)) =∥Rw∥2 L2(Ω×(0,T)) + ∥RΓw∥2 L2(Γ×(0,T)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Computations of the L2 products in Ω × (0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' In this step, we devote to compute the terms ⟨P1w, P2w⟩L2(Ω×(0,T)) = 3 � j,k=1 Ijk, where we have used the notation Ijk to denote the inner product between the jth-term of P1w and the kth-term of P2w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Firstly, the term I11 can be written as I11 = − a2ℜ � T 0 � Ω (s2λ2|∇η0|2ξ2w + ∆w)(2sλξ∇η0 · ∇w + (sλ2|∇η0|2 + sλ∆η0)ξw)dxdt =I(1) 11 + I(2) 11 + I(3) 11 + I(4) 11 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Integration by parts yields I(1) 11 = − 2a2s3λ3ℜ � T 0 � Ω |∇η0|2ξ3w∇η0 · ∇wdxdt =2a2s3λ3ℜ � T 0 � Ω ∇ � |∇η0|2� ∇η0ξ3|w|2dxdt + 6a2s3λ4 � T 0 � Ω |∇η0|4ξ3|w|2dxdt − I(1) 11 − 2a2s3λ3 � T 0 � Γ |∇η0|2∂νη0ξ3|w|2dxdt + 2a2s3λ3 � T 0 � Ω |∇η0|2∆η0ξ3|w|2dxdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Then, I(1) 11 is given by I(1) 11 =a2s3λ3 � T 0 � Ω � ∇(|∇η0|2) · ∇η0 + |∇η0|2∆η0� ξ3|w|2dxdt + 3a2s3λ4 � T 0 � Ω |∇η0|4ξ3|w|2dxdt − a2s3λ3 � T 0 � Γ (∂νη0)3xi3|w|2dSdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' On the other hand, we notice that I(2) 11 = −a2 � T 0 � Ω (s3λ4|∇η0|4 + s3λ3|∇η0|2∆η0)ξ3|w|2dxdt Integrating by parts in space, we have I(3) 11 = − a2ℜ � T 0 � Ω ∆w(2sλξ∇η0 · ∇w)dxdt =2a2sλ2 � T 0 � Ω ξ|∇η0 · ∇w|2dxdt + 2a2sλℜ � T 0 � Ω ξ∇w · ∇(∇η0 · ∇w)dxdt − 2a2sλ � T 0 � Γ ξ∂νη0|∂νw|2dSdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' 10 Using the identity ∇ψ · ∇(∇η0 · ∇w) = ∇2η0(∇w, ∇w) + 1 2∇η0 · ∇(|∇w|2), in Ω × (0, T), we notice that 2a2sλℜ � T 0 � Ω ξ∇w · ∇(∇η0 · ∇w)dxdt =2a2sλ � T 0 � Ω ξ∇2η0(∇w, ∇w)dxdt − a2sλ � T 0 � Ω ∆η0ξ|∇w|2dxdt − a2sλ2 � T 0 � Ω ξ|∇η0|2|∇w|2dxdt + a2sλ � T 0 � Γ ∂νη0ξ(|∇Γw|2 + |∂νw|2)dSdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Then, I(3) 11 is given by I(3) 11 =2a2sλ2 � T 0 � Ω ξ|∇η0 · ∇w|2dxdt + 2a2sλ � T 0 � Ω ξ∇2η0(∇w, ∇w)dxdt − a2sλ � T 0 � Ω ∆η0ξ|∇w|2dxdt − a2sλ2 � T 0 � Ω ξ|∇η0|2|∇w|2dxdt − a2sλ � T 0 � Γ ∂νη0ξ � |∂νw|2 − |∇Γw|2� dSdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' After integration by parts, I(4) 11 can be written as follows: I(4) 11 =a2ℜ � T 0 � Ω ξw∇w · ∇(sλ2|∇η0|2 + sλ∆η0)dxdt + a2ℜ �� T 0 � Ω λξ � sλ3|∇η0|2 + sλ2∆η0� w∇η0 · ∇wdxdt + a2 �� T 0 � Ω (sλ2|∇η0|2 + sλ∆η0)ξ|∇w|2dxdt − a2ℜ �� T 0 � Γ (sλ2|∂νη0|2 + sλ∆η0)ξ(∂νw)wdSdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Then, I11 can be estimated as I11 ⩾C � T 0 � Ω (s3λ4ξ3|w|2 + sλ2ξ|∇η0 · ∇w|2)dxdt + Csλ � T 0 � Γ (ξ|∂νw|2 + a2sλ∂νη0ξ|∇Γw|2)dSdt − Cs3λ4 � T 0 � ω ξ3|w|2dxdt − X11, where X11 satisfies the following upper bound: X11 ⩽C � T 0 � Ω (s3λ3ξ3|w|2 + sλξ|∇w|2)dxdt + C � T 0 � Γ � s2λ3ξ3|w|2 + λξ|∂νw|2� dSdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' 11 It is clear that I12 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Moreover, we write I13 as I13 =as2λ2ℜ � T 0 � Ω |∇η0|2ξ2w∂twdxdt + aℜ � T 0 � Ω ∆w∂twdxdt =I(1) 13 + I(2) 13 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Integrating by parts on time and using the fact that w = 0 in t = 0 and t = T, we have I(1) 13 = − as2λ2 � T 0 � Ω |∇η0|2ξ∂tξ|w|2dxdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' In the same manner, I(2) 13 gives I(2) 13 =aℜ � T 0 � Γ ∂νw∂twdSdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Then, I13 is given by I13 ⩾ − Cs2λ3 � T 0 � Ω ξ3|w|2dxdt − C � T 0 � Γ (s1/2ξ|∂νw|2 − s−1/2ξ−1|∂tw|2)dSdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Moreover, from the definition of I21, we see that I21 =0 Similar to the computations of I11, I22 can be estimated as follows: I22 ⩾C � T 0 � Ω � s3λ4ξ3|w|2 + sλ2ξ|∇η0 · ∇w|2� dxdt + C � T 0 � Γ (s3λ3ξ3|w|2 + Csλξ|∂νw|2 + a2α2sλ∂νη0ξ|∇Γw|2)dSdt − Cs3λ4 � T 0 � ω ξ3|w|2dSdt − X22, where X22 satisfies X22 ⩽C � T 0 � Ω (s3λ3ξ3|w|2 + Csλξ|∇w|2)dxdt + C � T 0 � Γ � s2λ3ξ2|w|2 + λξ|∂νw|2� dSdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' To compute the term I23, we follow [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Then, we write I23 =1 2aαi � T 0 � Ω (2sλξ∇η0 · ∇w + (sλ2|∇η0|2 + sλ∆η0)ξw)∂twdxdt − 1 2aαi � T 0 � Ω (2sλξ∇η0 · ∇w + (sλ2|∇η0|2 + sλ∆η0)ξw)∂twdxdt =I(1) 23 + I(2) 23 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='7) 12 On one hand, integrating by parts in time we have I(1) 23 := − 1 2aαi �� Q (2sλ∂tξ∇η0 · ∇w + 2sλξ∇η0 · ∇∂tw)wdxdt − 1 2aαi �� Q � (sλ2|∇η0|2 + sλ∆η0)∂tξ|w|2 + (sλ2|∇η0|2 + sλ∆η0)ξw∂tw � dxdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='8) On the other hand, integration by parts in space yields I(2) 23 =1 2aαi � T 0 � Ω � sλ2|∇η0|2ξw∂tw + sλξw∇η0 · ∇∂tw + sλ∆η0ξw∂tw � dxdt − 1 2aαi � T 0 � Ω (sλ2|∇η0|2 + sλ∆η0)ξw∂twdxdt − aαsλi � T 0 � Γ ∂νη0ξw∂twdSdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='9) Then, substituting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='8) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='9) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='7) we can estimate I23 be below in the following way I23 ⩾ − C � T 0 � Ω � s2λ2ξ3|w|2 + λξ|∇w|2� dxdt − C � T 0 � Γ (s2λ3ξ3|w|2 + λ−1ξ−1|∂tw|2)dSdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' The term I31 is I31 = − 2as2λℜ � T 0 � Ω ξ∂tϕw∇η0 · ∇wdxdt − a � T 0 � Ω (s2λ2|∇η0|2 + s2λ∆η0)∂tϕξ|w|2dxdt where the first term can be computed as − 2as2λℜ � T 0 � Ω ξ∂tϕw∇η0 · ∇wdxdt =as2λ2 � T 0 � Ω (|∇η0|2ξ∂tϕ + ξ∇η0 · ∇(∂tϕ) − ∆η0ξ∂tϕ)|w|2dxdt − 2as2λ � T 0 � Γ ∂νη0ξ∂tϕ|w|2dSdt Then, I31 is estimated as I31 ⩾ − Cs2λ2 � T 0 � Ω ξ3|w|2dxdt − Cs2λ2 � T 0 � Γ ξ3|w|2dSdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Now, for I32, it is clear that I32 = − aαsℑ � T 0 � Ω w∇(∂tϕ) · ∇wdxdt + aαsℑ � T 0 � Γ ∂tϕw∂νwdSdt ⩾ − C � T 0 � Ω � ξ3|w|2 + ξ|∇w|2� dxdt − C � T 0 � Γ � sξ3|w|2 + sξ|∂νw|2� dSdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' 13 In addition, since w = 0 in t = 0 and t = T, we have I33 ⩾ −Cs � T 0 � Ω ξ3|w|2dxdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' According to the above estimates, take s and λ > 0 large enough, to deduce that ⟨P1w, P2w⟩L2(Ω×(0,T)) ⩾Cs3λ4 � T 0 � Ω ξ3|w|2dxdt + C � T 0 � Γ (s3λ3ξ3|w|2 + sλξ|∂νw|2)dSdt + a2(1 + α2)sλ � T 0 � Γ ∂νη0ξ|∇Γw|2dSdt − Cs3λ4 � T 0 � ω ξ3|w|2dxdt − ˜X, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='10) for s ⩾ s1 > 0 and λ ⩾ λ1 > 0, where ˜X satisfies ˜X ⩽ Csλ � T 0 � Ω ξ|∇w|2dxdt + Cs−1(1 + λ−1) � T 0 � Γ ξ−1|∂tw|2dSdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Before going any further, let us point out that the integral term |∇Γw|2 is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' However, in the next step we obtain additional terms to solve this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Computations of the L2 products on Γ × (0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' In this step, we compute the terms ℜ � T 0 � Ω PΓ,1wPΓ,2wdxdt = 3 � j=1 3 � k=1 Jjk, where Jjk denotes the L2 real inner product of the jth of P1w with the kth-term of P2w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Firstly, we have J11 =b2αℑ � T 0 � Γ |∆Γw|2dSdt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Secondly, by the surface divergence theorem, we get J12 =2a2sλℜ � T 0 � Γ (ξw∇Γ(∂νη0) · ∇Γw)dSdt − 2a2sλ � T 0 � Γ ∂νη0ξ|∇Γw|2dSdt ⩾ − 2a2sλ � T 0 � Γ ∂νη0ξ|∇Γw|2dSdt − C � T 0 � Γ (s2λ2ξ|w|2 + ξ|∇Γw|2)dSdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Moreover, integrating by parts in time and space, we can assert that J13 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' The term J21 is J21 = − 2a2α2sλℜ � T 0 � Γ ξw∇Γ(∂νη0) · ∇ΓwdSdt − 2a2αsλ � T 0 � Γ ∂νη0ξ|∇Γw|2dSdt ⩾ − 2a2α2sλ � T 0 � Γ ∂νη0ξ|∇Γw|2dSdt − C � T 0 � Γ (s2λ2ξ|w|2 + ξ|∇Γw|2)dSdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Furthermore, by definition, J22 is J22 = 0 14 For J23 we have J23 ⩾ − C � T 0 � Γ � s5/2λ5/2ξ3|w|2 + s−1/2λ−1/2ξ−1|∂tw|2� dSdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' by the surface divergence theorem and the fact that ∇Γϕ = 0 on Γ × (0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' we deduce that J31 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Moreover, by definition J32 is given by J32 ⩾ −Csλ � T 0 � Γ ξ3|w|2dSdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' On the other hand, integration by parts yields J33 ⩾ − C � T 0 � Γ ξ3|w|2dSdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' According to these estimates, we can assert that ⟨PΓ,1w, PΓ,2w⟩L2(Γ×(0,T)) ⩾ −2a2(1 + α2)sλ � T 0 � Γ ∂νη0ξ|∇Γw|2dSdt − Y, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='11) where Y satisfies Y ⩽C � T 0 � Γ (s5/2λ5/2ξ3|w|2 + s−1/2λ−1/2ξ−1|∂tw|2)dSdt + C � T 0 � Γ (s−1/2λ−1/2ξ−1|∆Γw|2 + ξ|∇Γw|2)dSdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='10) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='11),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' and using that ∂νη0 ⩽ −c < 0 on Γ × (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' T) we deduce that ∥P1w∥2 L2(Ω×(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='T)) + ∥P2w∥2 L2(Ω×(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='T)) + ∥PΓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1w∥2 L2(Γ×(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='T)) + ∥PΓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='2w∥2 L2(Γ×(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='T)) + C � T 0 � Ω (s3λ4ξ3|w|2 + sλ2ξ|∇η0 · ∇w|2)dxdt + C � T 0 � Γ (s3λ3ξ3|w|2 + sλξ|∇Γw|2)dSdt ⩽∥f∥2 L2(Ω×(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='T)) + ∥fΓ∥2 L2(Γ×(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='T)) + Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' where Z can be bounded as Z ⩽Csλ � T 0 � Ω ξ|∇w|2dxdt + Cs−1(1 + λ−1) � T 0 � Γ ξ−1|∂tw|2dSdt + C � T 0 � Γ (s5/2λ5/2ξ3|w|2 + ξ|∇Γw|2 + |∂νw|2)dSdt + C � T 0 � Γ (s−1/2λ−1/2|∆Γw|2 + s−1/2λ−1/2ξ−1|∂tw|2)dSdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Now, taking s, λ > 0 large enough we obtain ∥P1w∥2 L2(Ω×(0,T)) + ∥P2w∥2 L2(Ω×(0,T)) + ∥PΓ,1w∥2 L2(Γ×(0,T)) + ∥PΓ,2w∥2 L2(Γ×(0,T)) + C � T 0 � Ω (s3λ4ξ3|w|2 + sλ2ξ|∇η0 · ∇w|2)dxdt + C � T 0 � Γ (s3λ3ξ3|w|2 + sλ∂νη0ξ|∇Γw|2)dSdt ⩽∥f∥2 L2(Ω×(0,T)) + ∥fΓ∥2 L2(Γ×(0,T)) + s3λ4 � T 0 � ω ξ3|w|2dxdt + ˜Z, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='12) 15 where ˜Z is given by ˜Z ⩽C � T 0 � Γ (s−1/2λ−1/2ξ−1|∆Γw|2 + (s−1/2λ−1/2 + s−1)ξ−1|∂tw|2)dSdt + Csλ � T 0 � Ω ξ|∇w|2dxdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' From the bounds of ˜Z, it is clear that we need to absorb the terms of |∇v| in Ω, ∆w and ∂tw on Γ × (0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' To do this, we shall use an indirect estimate using the definitions given in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='5) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Estimates of additional terms in the bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' In this step, we compute the terms ∆w, ∇w and ∂tw in Ω × (0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' More precisely, the purpose of this step is to prove the following inequality: � T 0 � Ω � s−1ξ−1|∆w|2 + s−1ξ−1|∂tw|2 + sλ2|∇w|2� dxdt ⩽C � T 0 � Ω � s3λ4ξ|w|2 + sλ2ξ|∇η0 · ∇w|2 + s−1ξ−1|P1w|2 + s−1ξ−1|P2w|2� dxdt + C � T 0 � Γ � s3λ3|w|2 + sλ|∂νw|2� dSdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='13) In order to do that, we firstly estimate the term of ∆w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' From the definition of P1, it is clear that s−1 � T 0 � Ω ξ−1|∆w|2dxdt ⩽C � T 0 � Ω � s−1ξ−1|P1w|2 + s3λ4|w|2 + sλ2ξ|∇η0 · ∇w|2� dxdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='14) Secondly, we estimate ∇w as follows: sλ2 � T 0 � Ω ξ|∇w|2dxdt = − sλ2ℜ � T 0 � Ω w∇ξ · ∇wdxdt + sλ2ℜ � T 0 � Γ ξw∂νwdSdt − sλ2ℜ � T 0 � Ω ξw∆wdxdt Then, by Young’s inequality we obtain sλ2 � T 0 � Ω ξ|∇w|2dxdt ⩽C � T 0 � Ω � s3λ4ξ3|w|2 + sλ2ξ|∇η0 · ∇w|2 + s−1ξ−1|∆w|2� dxdt + C � T 0 � Γ (s3λ3|w|2 + sλ|∂νw|2)dSdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='15) Next, we estimate the term of ∂tw directly from the definition of P2: s−1 � T 0 � Ω ξ−1|∂tw|2dxdt ⩽ C � T 0 � Ω � s3λ4ξ|w|2 + sλ2ξ|∇η0 · ∇w|2 + s−1ξ−1|∆w|2� dxdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='16) Thus, combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='14), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='15) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='16), we easily get (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' 16 Step 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Estimates of ∆Γw and ∂tw on Γ × (0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' In this step, we want to prove that � T 0 � Γ � ξ−1|∂tw|2 + ξ−1|∆Γw|2� dSdt ⩽C � T 0 � Γ � s2λ2|w|2 + ξ−1|∂νw|2 + ξ−1|PΓ,1w|2 + ξ−1|PΓ,2w|2� dSdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='17) From one hand, by definition of PΓ,1, we have � T 0 � Γ ξ−1|∆Γw|2dSdt ⩽ C � T 0 � Γ � ξ−1|PΓ,1w|2 + s2λ2ξ3|w|2 + ξ−1|∂νw|2� dSdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='18) On the other hand, using the definition of PΓ,2 we deduce that � T 0 � Γ ξ−1|∂tw|2dSdt ⩽ C � T 0 � Γ � ξ−1|∆Γw|2 + sλξ|w|2 + ξ−1|∂νw|2 + ξ−1|PΓ,2w|2� dSdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='19) Combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='18) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='19), we obtain (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Finally, combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='13), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='17) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='12), and taking s, λ > 0, we obtain � T 0 � Ω (s3λ4ξ3|w|2 + sλ2ξ|∇w|2 + s−1ξ−1|∂tw|2 + s−1ξ−1|∆w|2)dxdt + � T 0 � Γ � s3λ3ξ3|w|2 + sλ|∂νw|2 + sλ|∇Γw|2 + ξ−1|∆Γw|2 + ξ−1|∂tw|2� dSdt ⩽C∥Rw∥2 L2(Ω×(0,T)) + C∥RΓw∥2 L2(Γ×(0,T)) + Cs3λ4 � T 0 � ω ξ3|w|2dxdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Finally, we come back to the original variable and conclude the desired inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' 4 Observability and null controllability of the linear system The goal of this section is to prove a null controllability result for the linear Ginzburg-Landau \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 L(y) = f + 1ωh, in Ω × (0, T), N(y, yΓ) = fΓ, in Γ × (0, T), y = yΓ, on Γ × (0, T), (y(0), yΓ(0)) = (y0, yΓ,0), in Ω × Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1) where the operators L and N were defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Naturally, the null controllability for (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1) can be expressed in the following terms: Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' We say that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1) is null controllable in L2 if for all T > 0, (y0, yΓ,0) ∈ L2, there exists a control h ∈ L2(ω × (0, T)) such that the associated solution (y, yΓ) of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1) satisfies y(·, T) = 0, in Ω, yΓ(·, T) = 0, on Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Following the classical duality between controllability and observability in the context of parabolic equations (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' [13]), the null controllability of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1) is equivalent to prove a suitable observabil- ity inequality for its adjoint system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Then, thanks to this inequality and the Lax-Milgram lemma, we allow us to deduce the null controllability of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1), which is crucial for the proof of the Theorem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1) and an interesting result by itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' 17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1 Observability inequality As we explained above, we introduce the adjoint system \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 L∗z = g, in Ω × (0, T), N ∗(z, zΓ)z = gΓ, on Γ × (0, T), z = zΓ, on Γ × (0, T), (z(T), zΓ(T)) = (zT , zΓ,T ), in Ω × Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='2) where L∗ and N ∗ are given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' We point out that, if (zT , zΓ,T ) ∈ L2, then the associated weak solution (z, zΓ) belongs to C0([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' L2) ∩ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' H1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' This is done by results of section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' To formulate the result of this subsection, we shall define some additional functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' For t ∈ (0, T), we define µ(t) := \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 4 T 2 , if t ∈ (0, T/2], 1 t(T − t), if t ∈ (T/2, T), ˇϕ(t) :=µ(t) min x∈Ω � e2sλm∥η0∥∞ − eλ(m∥η0∥∞+η0(x))� , t ∈ (0, T), ˆϕ(t) :=µ(t) max x∈Ω � e2sλm∥η0∥∞ − eλ(m∥η0∥∞+η0(x))� , t ∈ (0, T), ˇξ(t) :=µ(t) min x∈Ω eλ(m∥η0∥∞+η0(x)), t ∈ (0, T), ˆξ(t) :=µ(t) max x∈Ω eλ(m∥η0∥∞+η0(x)), t ∈ (0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='2 (Observability inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Let N ⩾ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Suppose that (zT , zΓ,T ) ∈ L2 and (g, gΓ) ∈ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Then, there exists a constant C > 0 such that the associated weak solution (z, zΓ) ∈ C0([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' L2) ∩ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' H1) of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='2) satisfies � Ω |z(0)|2dx + � Γ |zΓ(0)|2dS + � T 0 � Ω e−2s ˆϕ(ˇξ3|z|2 + ˇξ|∇z|2)dxdt + � T 0 � Γ e−2s ˆϕ(ˇξ3|zΓ|2 + ˇξ|∇ΓzΓ|2)dSdt ⩽C � T 0 � Ω e−2s ˇϕ|g|2dxdt + C � T 0 � Γ e−2s ˇϕ|gΓ|2dSdt + C � T 0 � ω e−2s ˇϕ ˆξ3|z|2dxdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='3) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Consider a cut-off function θ ∈ C1([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' R) such that 0 ⩽ θ(t) ⩽ 1, ∀t ∈ [0, T], θ(t) = 1, ∀t ∈ [0, T/2], θ(t) = 0, ∀t ∈ [3T/4, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Then, the new variables (˜z, ˜zΓ) = (θz, θzΓ) satisfies \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 L∗˜z = θg + θ′z, in Ω × (0, T), N ∗(˜z, ˜zΓ) = θgΓ + θ′zΓ, on Γ × (0, T), ˜z = ˜zΓ, on Γ × (0, T), (˜z(T), ˜zΓ(T)) = (0, 0), in Ω × Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' 18 Then, by estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='2), we can assert that ∥(˜z, ˜zΓ)∥2 L∞(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='L2) + ∥(˜z, ˜zΓ)∥2 L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='H1) ⩽C∥(θg, θgΓ)∥2 L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='L2) + C∥(θ′z, θ′zΓ)∥2 L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' In particular, we can deduce that ∥(z(0), zΓ(0))∥2 L2 + ∥(z, zΓ)∥2 L2(0,T/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='H1) ⩽C∥(g, gΓ)∥2 L2(0,3T/4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='L2) + C∥(z, zΓ)∥2 L2(T/2,3T/4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Since e−2s ˆϕ ⩾ C > 0, for each t ∈ [0, T/2], we have � Ω |z(0)|2dx + � Γ |zΓ(0)|2dS + � T/2 0 � Ω e−2s ˆϕ(|z|2 + |∇z|2)dxdt + � T/2 0 � Γ e−2s ˆϕ(|zΓ|2 + |∇Γz|2)dSdt ⩽C � 3T/4 0 � Ω |g|2dxdt + C � 3T/4 0 � Γ |gΓ|2dSdt + C � 3T/4 T/2 � Ω |z|2dxdt + C � 3T/4 T/2 � Γ |zΓ|2dSdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='4) In order to estimate the last two terms of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='4), we use the fact that 0 < C ⩽ e−2sϕ(x,t)ξ3, ∀t ∈ (T/2, 3T/4), to obtain � 3T/4 T/2 � Ω |z|2dxdt + � 3T/4 T/2 � Γ |zΓ|2dSdt ⩽ � 3T/4 T/2 � Ω e−2sϕξ3|z|2dxdt + C � 3T/4 T/2 � Γ e−2sϕξ3|zΓ|2dSdt ⩽C � T 0 � Ω e−2sϕ|g|2dxdt + C � T 0 � Γ e−2sϕ|gΓ|2dSdt + C � T 0 � ω e−2sϕξ3|z|2dxdt, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='5) where we have used the Carleman estimate in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='2 with s > 0 and λ > 0 fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Now, combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='4) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='5) and using that − ˇϕ(t) ⩾ −ϕ(x, t) for all (x, t) ∈ Ω × (0, T), � Ω |z(0)|2dx + � Γ |zΓ(0)|2dS + � T/2 0 � Ω e−2s ˆϕ(ˇξ3|z|2 + ˇξ|∇z|2)dxdt + � T/2 0 � Γ e−2s ˆϕ(ˇξ3|zΓ|2 + ˇξ|∇ΓzΓ|2)dSdt ⩽C � T 0 � Ω e−2s ˇϕ|g|2dxdt + C � T 0 � Γ e−2s ˇϕ|gΓ|2dSdt + C � T 0 � ω e−2s ˇϕ ˆξ3|z|2dxdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='6) On the other hand, by Carleman estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1) again, the following inequality holds � T T/2 � Ω e−2s ˆϕ(ˇξ3|z|2 + ˇξ|∇z|2)dxdt + � T T/2 � Γ e−2s ˆϕ(ˇξ3|zΓ|2 + ˇξ|∇ΓzΓ|2)dSdt ⩽C � T 0 � Ω e−2s ˇϕ|g|2dxdt + C � T 0 � Γ e−2s ˇϕ|gΓ|2dSdt + C � T 0 � ω e−2s ˇϕ ˆξ3|z|2dxdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='7) Finally, adding inequalities (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='6) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='7), we deduce the observability inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' This ends the proof of the Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' 19 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='2 Proof of the null controllability for the linear system From the observability inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='3), we can deduce the null controllability of the linear system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' In the following, for r ∈ [1, +∞], a linear space H and a measurable function ρ : (0, T) −→ R, we shall consider the notation Lr(ρ(0, T);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' H) = {y ∈ Lr(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' H);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' ρy ∈ Lr(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' H)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Consider the Banach space V defined by V := {(y, yΓ, h) :(y, yΓ) ∈ L2(es ˇϕ(0, T);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' L2), h1ω ∈ L2(es ˇϕ ˆξ−3/2(0, T);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' L2(Ω)), (L(y) − 1ωh, N(y, yΓ)) ∈ L2(es ˆϕ ˇξ−3/2(0, T);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' L2), (y, yΓ) ∈ L2(e 1 3s ˆϕ(0, T);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' H2) ∩ L∞(e 1 3 s ˆϕ(0, T);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' H1)}, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='8) endowed by its natural norm: ∥(y, yΓ, h)∥2 V :=∥es ˇϕ(y, yΓ)∥2 L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='L2) + ∥es ˇϕ ˆξ3h1ω∥2 L2(Ω×(0,T)) + ∥es ˆϕ ˇξ−3/2(L(y) − 1ωh, N(y, yΓ))∥2 L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='L2) + ∥e 1 3s ˆϕ(y, yΓ)∥2 L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='H2) + ∥e 1 3s ˆϕ(y, yΓ)∥2 L∞(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='H1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Let (y0, yΓ,0) ∈ L2 and assume that (f, fΓ) ∈ L2(es ˆϕˇξ−3/2(0, T);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='9) Then, we can find a control h such that the associated solution (y, yΓ) of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1) satisfies (y, yΓ, h) ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' In particular, we have y(·, T) = 0, in Ω, yΓ(·, T) = 0, on Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' For our purposes, we define P0 := � (z, zΓ) ∈ C∞(Ω × [0, T]) × C∞(Γ × [0, T]) : z �� Γ = zΓ on Γ × [0, T] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Let a : P0 × P0 → R be the bilinear form a((z, zΓ), (w, wΓ)) :=ℜ � T 0 � Ω e−2s ˇϕL∗zL∗(w)dxdt + ℜ � T 0 � Γ e−2s ˇϕN ∗(z, zΓ)N ∗(w, wΓ)dSdt + ℜ � T 0 � ω e−2s ˇϕˆξ3zwdxdt, ∀(z, zΓ), (w, wΓ) ∈ P0 × P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' We also define the linear form ℓ : P0 → R as ℓ(w, wΓ) :=ℜ � Ω y0w(0)dx + ℜ � Γ yΓ,0wΓ(0)dS + ℜ � T 0 � Ω fwdxdt + ℜ � T 0 � Γ fΓwΓdSdt, ∀(w, wΓ) ∈ P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' In view of the observability inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='3), it is clear that ∥(z, zΓ)∥ a := � a((z, zΓ), (z, zΓ)), ∀(z, zΓ) ∈ P0, 20 defines a norm in P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Let P be the completion of (P0, ∥·∥ a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Then, a(·, ·) is well-defined, continuous and again definite positive on P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' On the other hand, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='9) and the observability inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='3), it is easy to see that |ℓ(w, wΓ)| ⩽ C∥(w, wΓ)∥ a, ∀(w, wΓ) ∈ P0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' ℓ is continuous in P0 and, by density, in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Thus, by Lax-Milgram Theorem, the variational problem � Find (z, zΓ) ∈ P such that a((z, zΓ), (w, wΓ)) = ℓ(w, wΓ), ∀(w, wΓ) ∈ P, possesses exactly one solution (z∗, z∗ Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Let (y∗, y∗ Γ, h∗) defined by \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 y∗ := e−2s ˇϕL∗(z∗), in Ω × (0, T), y∗ Γ := e−2s ˇϕN ∗(z∗, z∗ Γ), on Γ × (0, T), h∗ := e−2s ˇϕˆξ3z∗, in ω × (0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' We point out that (y∗, y∗ Γ, h∗) satisfies � T 0 � Ω e2s ˇϕ|y∗|2dxdt + � T 0 � Γ e2s ˇϕ|y∗ Γ|2dSdt + � T 0 � ω e2s ˇϕˆξ−3|h∗|2dxdt = a((z∗, z∗ Γ), (z∗, z∗ Γ)) < ∞, and also that ℜ � T 0 � Ω y∗L∗(w)dxdt + ℜ � T 0 � Γ y∗ ΓN ∗(w, wΓ)dSdt + ℜ � T 0 � ω h∗wdxdt =ℜ � Ω y0w(0)dx + ℜ � Γ yΓ,0wΓ(0)dS + ℜ � T 0 � Ω fwdxdt + ℜ � T 0 � Γ fΓwΓdSdt, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='10) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=', from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='10) we deduce that (y∗, y∗ Γ) ∈ C0([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' L2) ∩ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' H1) is a distributional solution with control h∗ and initial datum (y0, yΓ,0) such that y∗(·, T) = 0 in Ω and yΓ(·, T) = 0 on Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' It remains to check that (y∗, y∗ Γ) ∈ L2(e 1 3s ˆϕ(0, T);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' H2) ∩ L∞(e 1 3s ˆϕ(0, T);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' H1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='11) In order to do that, we notice that the new variables (y⋆, y⋆ Γ) := (e 1 3s ˆϕy∗, e 1 3s ˆϕy∗ Γ) solves the problem \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 L(y⋆) = e 1 3s ˆϕ(f + h1ω) + (e 1 3 s ˆϕ)ty∗, in Ω × (0, T), N(y⋆, y⋆ Γ) = e 1 3s ˆϕfΓ + (e 1 3s ˆϕ)ty∗ Γ, in Γ × (0, T), y⋆ = y⋆ Γ, on Γ × (0, T), (y(0), yΓ(0)) = e 1 3s ˆϕ(0)(y0, yΓ,0), in Ω × Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='12) 21 Since ��� � e 1 3s ˆϕ� t y∗��� ⩽ C ˆξ2e 1 3 s ˆϕ|y∗| ⩽ Ces ˇϕ|y⋆| ∈ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' L2), and � e 1 3 s ˆϕ(f + h1ω), e 1 3s ˆϕfΓ � ∈ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' L2), and since e 1 3 s ˆϕ(0)(y0, yΓ,0) ∈ H1, it follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='3 that the solution (y⋆, y⋆ Γ) of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='12) satisfies (y⋆, y⋆ Γ) ∈ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' H2 ∩ C0([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' H1)), which is equivalently to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' We conclude that (y∗, y∗ Γ, h∗) ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' This ends the proof of the Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' 5 Proof of the Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1 To prove the main theorem of this article, we shall use a local inversion argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' This will be done by using the following local inversion mapping theorem in Banach spaces (see for instance [5], page 107).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Let B1 and B2 be two Banach spaces and let A : B1 → B2 be a C1(B1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' B2) function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Suppose that b1 ∈ B1, b2 = A(b1) and that A′(b1) : B1 → B2 is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Then, there exists δ > 0 such that, for every b′ ∈ B2 satisfying ∥b′ − b2∥B2 < δ, there exists a solution of the equation A(b) = b′, b ∈ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Moreover, there exists a constant C > 0 such that ∥b1 − b∥B1 ⩽ C∥b2 − b′∥B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Now, we have all the ingredients to prove the Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Proof of the Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Consider the spaces B1 := V, B2 := L2(es ˆϕ ˇξ−3/2(0, T);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' L2) × L2, where V is the linear space defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Consider A : B1 → B2 be the operator defined by A(u, uΓ, h) = � Lu + c(1 + γi)|u|2u − 1ωh, N(u, uΓ) + c(1 + γi)|uΓ|2uΓ, u(0), uΓ(0) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' We choose, b1 = (0, 0, 0) and b2 = A(0, 0, 0) = (0, 0, u0, uΓ,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' In order to use the Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1, we shall check that the following two assertions: (a) A′(0, 0, 0) : B1 → B2 is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' (b) A is an operator of class C1 from B1 to B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' 22 A simple computation shows that for all (u∗, u∗ Γ, h∗) ∈ B1, A′(u, uΓ, h)(u∗, u∗ Γ, h∗) =(L(u∗) + 3c(1 + γi)|u|2u∗ − 1ωh∗, N(u∗, u∗ Γ) + 3c(1 + γi)|uΓ|2u∗ Γ, u∗(0), u∗ Γ(0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1) In particular, taking (u, uΓ, h) = (0, 0, 0) in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1), we have A′(0, 0, 0)(u∗, u∗ Γ, h∗) = (L(u∗) − 1ωh∗, N(u∗, u∗ Γ), u∗(0), u∗ Γ(0)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Now, it is clear that, in view of the Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='3, the operator A′(0, 0, 0) is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' On the other hand, to prove that A ∈ C1(B1, B2), it is sufficient to check that the map (u1, uΓ,1, h1), (u2, uΓ,2, h2), (u3, uΓ,3, h3) �→ (u1u2u3, uΓ,1uΓ,2uΓ,3), is continuous from V × V × V to L2(es ˇϕ(0, T);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Then, according to H¨older inequality and using that L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' H2) ∩ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' H1) ֒→ L6(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' L9) ֒→ L6(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' L6), we have ∥es ˆϕ ˇξ−3/2(u1u2u3, uΓ,1uΓ,2uΓ,3)∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='L2) ⩽C 3 � j=1 ∥e 1 3 s ˆϕ(uj, uΓ,j)∥L6(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='L6) ⩽C 3 � j=1 ∥e 1 3 s ˆϕ(uj, uΓ,j)∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='H2)∩L∞(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='H1) ⩽C 3 � j=1 ∥(uj, uΓ,j, hj)∥V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='2) Therefore, we have that A ∈ C1(B1, B2), and we can apply Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1 to guarantee the existence of δ > 0 such that ∥(u0, uΓ,0)∥H1 ⩽ δ, one can find (y, yΓ, h) ∈ B1 := V such that the associated solution (u, uΓ) (with initial datum (u0, uΓ,0)) satisfies y(·, T) = 0, in Ω, yΓ(·, T) = 0, on Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' This, ends the proof of the Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Acknowledgments 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Theory, 6(3):381– 407, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' [24] Alberto Mercado and Roberto Morales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Exact controllability for a schr¨odinger equation with dynamic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Submitted, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' [25] Lionel Rosier and Bing-Yu Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Null controllability of the complex Ginzburg-Landau equa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Poincar´e C Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Non Lin´eaire, 26(2):649–673, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' [26] Lionel Rosier and Bing-Yu Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Null controllability of the complex ginzburg-landau equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' In Annales de l’IHP Analyse non lin´eaire, volume 26, pages 649–673, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' [27] Michael E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Taylor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Partial differential equations I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Basic theory, volume 115 of Applied Math- ematical Sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' Springer, New York, second edition, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} +page_content=' 25' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E1T4oBgHgl3EQfyAW2/content/2301.03429v1.pdf'} diff --git a/dtE0T4oBgHgl3EQfWwBP/content/tmp_files/2301.02282v1.pdf.txt b/dtE0T4oBgHgl3EQfWwBP/content/tmp_files/2301.02282v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9f710ed74622499d7d737bbf3635e65744ddfc50 --- /dev/null +++ b/dtE0T4oBgHgl3EQfWwBP/content/tmp_files/2301.02282v1.pdf.txt @@ -0,0 +1,3551 @@ +Efficient full frequency GW for metals using a multipole approach for the dielectric +screening +Dario A. Leon1,2,3,∗ Andrea Ferretti2, Daniele Varsano2, Elisa Molinari1,2, and Claudia Cardoso2 +1 FIM Department, University of Modena & Reggio Emilia, 41125, Modena (Italy) +2S3 Centre, Istituto Nanoscienze, CNR, 41125, Modena (Italy) and +3Department of Mechanical Engineering and Technology Management, +Norwegian University of Life Sciences, 1430, ˚ +As (Norway) +The properties of metallic systems with important and structured excitations at low energies, such +as Cu, are challenging to describe with simple models like the plasmon pole approximation (PPA), +and more accurate and sometimes prohibitive full frequency approaches are usually required. In +this paper we propose a numerical approach to GW calculations on metals that takes into account +the frequency dependence of the screening via the multipole approximation (MPA), an accurate and +efficient alternative to current full-frequency methods that was recently developed and validated +for semiconductors and overcomes several limitations of PPA. We now demonstrate that MPA +can be successfully extended to metallic systems by optimizing the frequency sampling for this +class of materials and introducing a simple method to include the q → 0 limit of the intra-band +contributions. The good agreement between MPA and full frequency results for the calculations of +quasi-particle energies, polarizability, self-energy and spectral functions in different metallic systems +confirms the accuracy and computational efficiency of the method. Finally, we discuss the physical +interpretation of the MPA poles through a comparison with experimental electron energy loss spectra +for Cu. +I. +INTRODUCTION +Many-body perturbation theory provides accurate +methods to study the spectroscopic properties of con- +densed matter systems from first principles [1–3]. Cal- +culations often adopt the so-called GW +approxima- +tion [2, 4–8], for which the frequency integration in the +evaluation of the self-energy is crucial to the deploy- +ment of the method. The frequency dependence of the +screened potential, W, is often described within the plas- +mon pole approximation (PPA) [9–14], successfully ap- +plied to the calculation of quasi-particle energies of semi- +conductors [9], the homogeneous electron gas [15] and +simple metals as Al and Na [16–19], especially for quasi- +particles with energies close to the Fermi level. However, +the description of the self-energy and the spectral func- +tions for the whole range of frequencies is still challenging +and requires expensive full frequency (FF) approaches. +Despite its success, the use of PPA is problematic when +complex metals are concerned, even for the calculations +of quasi-particle energies [6]. Its applicability for tran- +sition and noble metals has often been disputed [6, 20], +since the approximation is based on the homogeneous +electron gas, for which PPA becomes exact in the long +wave-length limit +[4, 21, 22], while it is in principle +not strictly valid in the presence of strongly localized +d-bands. In fact, these metals present complex screen- +ing effects due to collective excitations [23, 24], which +result in highly structured energy-loss spectra whose de- +scription is unattainable with a single plasmon peak [24]. +Moreover, metals with relevant excitations at low ener- +gies, such as Cu, require a specially accurate description +∗ dario.alejandro.leon.valido@nmbu.no +of the low frequency regime, which makes it difficult to +determine the PPA parameters since it requires sampling +the polarizability at zero frequency [20]. +In this context, we have recently developed a multipole +approach (MPA) that naturally bridges from PPA to FF +treatments of the GW self-energy [25]. The method has +been implemented in the yambo code [26, 27] and was +validated for bulk semiconductors. We have shown that, +for semiconductors, MPA attains an accuracy compara- +ble to that of FF methods at a much lower computational +cost, while also circumventing several of the PPA short- +comings. Here we extend the assessment of MPA valid- +ity and performance to the case of metals. We do so by +computing quasi-particle energies, together with the full +frequency dependence of the self-energy and the spectral +function. +The approach is similar to the one used for +semiconductors [25], with only slight changes in the fre- +quency sampling strategy used in the multipole interpo- +lation. In the following, we show that MPA is accurate +for metallic systems, even in cases in which the use of +PPA is challenging. In addition to MPA, we also propose +a simple ab-initio method to include intra-band contri- +butions [28–32] to the dielectric function in the q → 0 +limit, absent in semiconductors. +Despite its virtually +zero computational cost, it significantly accelerates the +convergence of quasi-particle energies with respect to the +k-points grid, in systems where the intra-band contribu- +tions are dominant. +The paper is organized as follows: In Sec. II, we briefly +summarize the GW approximation and the MPA ap- +proach. +In the same Section, we further extend the +strategy used in the frequency sampling for the multipole +interpolation, with respect to the MPA implementation +presented in Ref. [25] for semiconductors. We also dis- +cuss the relevance of the inclusion of the intra-band con- +arXiv:2301.02282v1 [cond-mat.mtrl-sci] 5 Jan 2023 + +2 +tribution to the dielectric function in the limit q → 0. In +Sec. III we first present MPA calculations for simple met- +als and propose a simple way of including the aforemen- +tioned intra-band limit. We then describe in detail the +results obtained for Cu, a prototype challenging system +for PPA. Finally, in Sec. IV we summarize and discuss +the main conclusions of this work. +II. +METHODS +A. +Quasi-particle energies within GW +We adopt the GW approximation [2, 4–8] for the eval- +uation of the electron-electron self-energy, which is com- +puted via a frequency convolution of the one-particle +Green’s function G(ω) and the dynamical screened in- +teraction potential W(ω): +ΣGW (ω) = i +2π +� +∞ +−∞ +dω′e−iω′ηG(ω − ω′)W(ω′). +(1) +In the present work we limit ourselves to the G0W0 +approximation, although MPA, the method we want +to discuss here, can be exploited also within more +advanced approaches such as different self-consistent +GW +schemes [33–39], or methods including vertex- +corrections [35, 40–43] and cumulant expansions [44]. A +more comprehensive discussion of these aspects can be +found e.g. in Refs. [7, 8]. The present implementation +uses as a starting point single-particle energies and wave- +functions computed within Kohn-Sham (KS) density fun- +tional theory (DFT) to then build the non-interacting +single-particle Green’s function G0(ω) and the irreducible +polarizability, X0(ω). +The dressed polarizability, X(ω), and the screened in- +teraction, W(ω), are then numerically evaluated by solv- +ing the Dyson equation for each given frequency: +X(ω) = X0(ω) + X0(ω)vX(ω) +(2) +W(ω) = ε−1(ω)v = v + vX(ω)v, +where v is the bare Coulomb potential, ε the dielectric +function and, for simplicity, we have omitted the spatial, +non-local, degrees of freedom. +All the quantities have +to be thought as frequency dependent operators or ma- +trices of the form X(ω) = X(r, r′, ω), or, when using +a plane-wave basis set, XGG′(q, ω). The quasi-particle +(QP) energies ϵQP +m +are then computed either by numeri- +cally solving the exact QP equation, +ϵQP +m = ϵKS +m + ⟨ψKS +m |Σ(ϵQP +m ) − vKS +xc |ψKS +m ⟩, +(3) +or its linearized form: +ϵQP +m ≈ ϵKS +m + Zm⟨ψKS|Σ(ϵKS +m ) − vKS +xc |ψKS +m ⟩, +(4) +with the renormalization factors Zm given by +Zm = +� +1 − ⟨ψKS +m |∂Σ(ω) +∂ω +���� +ω=ϵKS +m +|ψKS +m ⟩ +�−1 +. +(5) +In the above equations we have made reference to the +Kohn-Sham eigelvaues and eigenvectors, ϵKS +m and |ψKS +m ⟩, +respectively. +A key quantity in the above formulation is the dy- +namical part of the inverse dielectric function, Y +≡ +ε−1 − I = vX, which determines the correlation part +of W, Wc ≡ W − v = Y v, and, through Eq. (1), the cor- +relation part of the self-energy, Σc. With the purpose of +avoiding the expensive numerical evaluation of the fre- +quency convolution in Σc, Eq. (1), as required e.g. by +full frequency real axis (FF-RA) approaches [20, 45] or +contour deformation (FF-CD) techniques [34, 46, 47], Y +or X have been the target of several analytical simplifica- +tions like the plasmon pole approximation (PPA) [9–13] +or the multipole approach (MPA) [25], briefly sketched +below. +B. +The multipole approach +The +multipole +approximation +is +inspired +by +the +Lehmann representation of the polarizability X. At the +independent particle level, X (equal to X0) is written in +a compact way as a sum of poles with vanishing imag- +inary part corresponding to all possible single particle +transitions (here considered at the Kohn-Sham level for +simplicity) of energy ΩKS and probability amplitude RKS: +X0(ω) = +NT +� +n +2RKS +n ΩKS +n +ω2 − (ΩKS +n )2 , +(6) +where Re[ΩKS +n +] is positive defined and Im[ΩKS +n +] → 0− to +ensure the correct time ordering. The sum is truncated +at a finite number of transitions (NT ) determined by the +number of bands included in the calculation. +The MPA approach provides an analytic continuation +for the dressed polarizability X to the complex frequency +plane, z ≡ ω + iϖ, by representing it as a sum of a few +complex poles np (usually of the order of 10 to 15), as +XMP(z) = +np +� +n +2RnΩn +z2 − Ω2n +. +(7) +Note that this representation is applied to each matrix +element in reciprocal space, XMP +GG′(q, z). +By considering Eq. (7) and the Lehmann representa- +tion for G0, the correlation part of the GW self-energy is +then integrated analytically and reads: +ΣMP +c +(ω) = +NB +� +m +np +� +n +PmvRn +� +fm +ω − Em + Ωn − iη + ++ +(1 − fm) +ω − Em − Ωn + iη +� +v. +(8) +where Pm are projectors over KS states, Em their +eigenenergies, and fm their occupations. The sum-over- +states is truncated at the maximum number of bands, + +3 +NB. This expression generalises the PPA solution to the +case of a multipole expansion for X(z), and bridges be- +tween PPA and an exact full-frequency approach by in- +creasing the number of poles in X. More details about +this procedure can be found in Ref. [25]. +C. +MPA sampling for metals +The poles and residues in Eq. (7) are obtained by nu- +merically evaluating X for a number of frequencies equal +to twice the number of poles and solving the resulting sys- +tem of equations (see details in Ref. [25]). Since the num- +ber of poles used in the MPA model, np, is much smaller +than the total number of electron-hole transitions of the +target polarizability, NT , the representation, and there- +fore the efficiency of the method, depends critically on the +frequency sampling used in the interpolation. For semi- +conductors, the so-called double parallel sampling proved +to be the most robust and accurate with respect to FF +calculations, with the fastest convergence with respect +to the number of poles. It runs along two parallel lines +above the real axis: +sDP = +� +z1: z1 +n = ωn + iϖ1 +z2: z2 +n = ωn + iϖ2, +n = 1, .., np +(9) +The first of the two branches is closer to the real axis (e.g. +with ϖ1 = 0.1 Ha), except for the first point, set exactly +at the origin of coordinates, z1 +1 = 0. The second branch +is located further away, typically at ϖ2 = 1 Ha. In a +simplified view, X sampled along the first line preserves +some of the structure of X in a region close to its poles, +while X sampled along the second line is simple enough +to be described with a few poles, and accounts for the +overall structure of X. A more detailed description can +be found in Ref. [25]. +In order to obtain a numerically stable and effective +sampling for metals we found that, at variance with the +semiconductor case [25], a small shift of the z1 +1 point (in +the origin) along the imaginary axis is needed, resulting +in z1 +1 = iϖ1, where ϖ1 = 10−5 Ha. The shift is done in +order to avoid numerical instabilities due to intra-band +transitions with energies close to zero. This is similar to +the PPA implementation for metals [26, 27], which adopts +a 10−8 Ha shift, but in this case along the positive real +axis instead of the imaginary axis. +A second difference with respect the strategy used +for semiconductors concerns the distribution of the fre- +quency sampling of X along the real axis. +For semi- +conductors [25], the frequency sampling is done in non- +uniform grids, in particular, a semi-homogeneous parti- +tion in powers of 2 that ranges from 0 to ωm, called linear +partition. Here, we generalize it to any possible exponent +α: +{ωn}α : +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +(0) , np = 1 +(0, 1) × ωm, np = 2 +� +0, 1 +2, 1 +�α +× ωm, np = 3 +� +0, 1 +4, 1 +2, 1 +�α +× ωm, np = 4 +� +0, 1 +8, 1 +4, 1 +2, 1 +�α +× ωm, np = 5 +� +0, 1 +8, 1 +4, 1 +2, 3 +4, 1 +�α +× ωm, np = 6 +� +0, 1 +8, 1 +4, 3 +8, 1 +2, 3 +4, 1 +�α +× ωm, np = 7 +... +(10) +The distribution described on Ref. [25] corresponds to +α = 1. As discussed below, there are cases (see for exam- +ple the case of copper in Fig. 4), in which X presents a +more complex structure at low frequencies and therefore +a denser sampling grid in that region is convenient. The +distribution corresponding to α = 2 concentrates more +points at low frequencies than the linear case, α = 1, +and permits to increase the accuracy of the X descrip- +tion without changing the frequency range, ωm, or in- +crease the number of poles used in MPA. In this work, +we adopt a quadratic partition, corresponding to α = 2, +for Al and Cu, and a linear one, α = 1, for Na. +D. +Intra-band contributions +Despite the success of the GW approximation, systems +with metallic screening present specific methodological +challenges, one being the inclusion of intra-band transi- +tions [31, 48]. Specifically, for partially filled bands, there +is a non-vanishing probability that an electron is excited +within the same band, i.e. within states with quantum +numbers k, n and k − q, m, with n = m. Notably, these +transitions play an important role, for example, in noble +metals [20, 49]. Both inter- and intra-band transitions +contribute to the irreducible polarizability as defined in +Eq. (6). However, the energy of the pole corresponding +to intra-band transitions decreases with q until it van- +ishes in the q → 0 limit. +Despite this behaviour, the +contribution to the inverse dielectric function in the case +of bulk metals is still finite, due to the divergence of the +Coulomb potential, which makes Y = vX not vanish- +ing for q → 0. +For this reason, in the case of metals +it is important to properly take this term into account, +since it cannot be simply evaluated as in the case of the +inter-band contributions. +In principle, it is possible to decrease the weight of +the q = 0 element, that contains only inter-band terms, +by systematically increasing the number of k-points in + +4 +the Brillouin zone (BZ) sampling. +However, the con- +tributions from the Fermi surface can dramatically slow +down the convergence with respect to the k-space sam- +pling [29], resulting in spurious gaps at the Fermi level +that vanish very slowly with increasing number of k- +points [32]. +Several approaches to include the intra- +band limit have been proposed. The ones based on ex- +plicit Fermi-surface integration [28, 30, 31] are, as ex- +plained above, computationally expensive since they re- +quire dense k-grids. +Alternatively, analytical models +based on a Taylor expansion of the dielectric function in +the small-q region, avoiding explicit Fermi-surface calcu- +lations, are able to remove the spurious gap at the Fermi +level with a limited number of k-points [32, 50, 51]. Nev- +ertheless, some of them may depend on ad hoc external +parameters. +A common approach to include the missing intra-band +contribution relies on the use of a phenomenological +Drude-like term added to the head of the irreducible +dielectric matrix in the q → 0 limit, YG=G′=0(q = +0, ω) [30]. In the long-wavelength limit, q → 0, the Drude +term for the independent particle dielectric function can +be written in the form [24, 28, 30, 52, 53] +YD(ω) = +ω2 +D +ω(ω + iγ) + O[q2], +(11) +where the Drude frequency, ωD (see Table II), is an input +parameter of the model and the relaxation frequency γ +is usually a free parameter set typically to γ = 0.1 eV. In +principle ωD can be determined fully ab-initio, resorting +to very dense k-point grids [20, 30] or to an interpolation +of the BZ, for instance with Wannier functions [54–57] or +the tetrahedron method [29, 30, 39, 58]. Alternatively, +experimental values can also be used when available. +In the next Sections we will discuss the possibility +to extrapolate a complex plasmon frequency (see Ta- +ble II) in the q → 0 limit from the frequency structure +of Y (q, ω) at finite q, which in general is a superposi- +tion of intra- and inter-band contributions. In a second +step, we will use a f-sum rule [24] in the same spirit of +Ref. [30], in order to estimate the intra-band contribu- +tion to the plasmon frequency. We will also propose a +simple and virtually zero-cost method to include an ap- +proximate treatment of the missing intra-band limit from +first-principles, without the need to resort to any add-on +model. +III. +RESULTS AND DISCUSSION +In the following, we present the results for three +bulk metallic systems highlighting different issues arising +when applying the GW approach to metals. We start +by studying the case of two simple metals, Al and Na +(see e.g. +Refs. [59–61] for a description of their band +structures). Next, we focus our attention on Cu, a more +challenging system whose electronic structure has been +DFT-PBE +GW-PPA +GW-MPA +Al +Γ1 +-11.12 +-10.79 +-10.94 +Γ25′ +12.71 +12.30 +12.48 +X4′ +-2.93 +-2.91 +-2.86 +W3 +-0.85 +-0.83 +-0.82 +Na +Γ1 +-3.27 +-2.85 +-2.97 +Γ25′ +11.76 +11.19 +10.81 +TABLE I. Al and Na quasi-particle energies (eV) with respect +the Fermi level computed within DFT-PBE, GW-PPA, and +GW-MPA using a 16 × 16 × 16 k-grid including the q → 0 +intra-band contribution through the CA method. +thoroughly studied, both experimentally [62, 63] and the- +oretically [20, 64–66]. The use of PPA for Cu has been +shown to be problematic [20] and, for this reason, copper +is not only an important test case for the application of +MPA and the description of intra-band effects, but also +provides a better understanding of the applicability of +PPA. +As a starting point for our GW simulations, we use +DFT calculations performed at the PBE [67] level using +scalar-relativistic optimized norm-conserving Vanderbilt +pseudopotentials [68], as implemented in the Quantum +ESPRESSO package [69, 70]. The kinetic energy cut-off +is set to 100, 70, and 150 Ry for Al, Na, and Cu, respec- +tively. The k-grids were determined by the convergence +requirements of the GW calculations, considering, in par- +ticular, the specific treatment of the intra-band limit. +When reporting quasi-particle energies, we use k-point +grids of 16 × 16 × 16 for Al and Na, and 12 × 12 × 12 +for Cu. Moreover, the GW correction to the Fermi level +is linearly interpolated from the corresponding correc- +tions to the closer quasi-particles present in the specific +k-mesh. +The DFT results are in good agreement with previ- +ous results obtained with the same method [65], and in +reasonable agreement with the results reported for Cu in +Ref. [20], performed using LDA [31]. In fact, the GW re- +sults for Cu have shown to be very sensitive to the choice +of the DFT starting point [65], though we will not address +this point here. The GW calculations were done using +the yambo [26, 27] code. The numerical convergence of +the GW results has been checked with care, and the re- +sulting parameters, being system dependent, are detailed +in the sections below when discussing the results. +A. +MPA for simple metals +We start by computing quasi-particle energies of Al +and Na using MPA. Here the frequency dependence of the +polarizability presents a structure with mainly one strong +plasmon peak, similar to that of silicon computed in +Ref. [25]. As expected, the double parallel sampling en- +sures convergence with a similar number of poles, np = 8. + +5 +80 +60 +40 +20 +0 +20 +40 +60 +Re[ ] (eV) +Al +40 +30 +20 +10 +0 +10 +Na +25 +20 +15 +10 +5 +0 +5 +EKS (eV) +0.0 +0.2 +0.4 +0.6 +0.8 +Im[G] (eV +1) +EKS +W10 = +0.88 eV +EKS +X10 = +2.96 eV +EKS +1 = +11.20 eV +15 +10 +5 +0 +EKS (eV) +0 +1 +2 +3 +4 +5 +6 +7 +EKS = +0.01 eV (nD) +EKS = +3.27 eV (nD) +EKS = +0.01 eV +EKS = +3.27 eV +0.5 +0.0 +0.5 +2 +0 +2 +1 +0 +1 +2.5 +0.0 +2.5 +b) +c) +d) +a) +FIG. 1. Frequency dependence of the real part of the self-energy (panels a) and c)) and spectral function (panels b) and d)) +computed with MPA for three quasi-particles of Al (panels a) and b)) and two of Na (panels c) and d)), including the intra-band +limit using CA (see text). In the case of Na, we also show the corresponding curves without any intra-band correction (nD). +The present results were obtained considering 300 bands +for both X and Σ and an energy cut-off for X of 20 and +15 Ry for Al and Na respectively. +In Table I we report the quasi-particle energies for Al +and Na, including Γ1 (the lowest QP peak at Γ, corre- +sponding to the valence bandwidth) and other reference +quasi-particles, computed using PPA and MPA. MPA +QPs are generally in very good agreement with FF val- +ues from the literature (see e.g. Ref. [19] and references +therein). According to our calculations, the computed +quasi-particles values for Al and Na with MPA are esti- +mate to differ by less than 8 meV from the correspond- +ing FF-RA results (comparison done using 10 Ry cutoff +to represent X0 for both MPA and FF-RA), as found +for semiconductors [25]. Instead, PPA QPs show devia- +tions that are systematically larger for states further from +Fermi. +Previous GW calculations for Al and Na [19] have +shown that PPA describes well the tail of the self-energy, +i.e. the frequency region around the Kohn-Sham ener- +gies, and gives reasonable QP solutions for both Al and +Na. However, if we consider the whole frequency range, +the agreement between PPA and FF-CD is less satisfac- +tory. +PPA shows sharp fluctuations in the self-energy +and spectral functions, that result in several spurious so- +lutions of the quasi-particle equation, evidenced by mul- +tiple small peaks in the spectral function (see e.g. Fig. 4 of +Ref [19]). In Fig. 1 we show the self-energy and spectral +function for Al and Na, this time computed with MPA. +The comparison with results obtained within FF-CD [19] +shows that MPA not only describes well the tail of the +X(ω) and Σ(ω) functions, but also correctly describes +the positions of the peaks and their relative intensities in +the whole frequency range. +In the left panels of Fig. 1 we focus on Al and +plot, as a function of the frequency, the MPA self- +energy, ⟨ψmk|Σ(ω)|ψmk⟩, and the spectral function, +⟨ψmk|Im[G(ω)]|ψmk⟩. +These quantities have been pro- +jected on three Al states, one corresponding to the bot- +tom of the valence band at Γ and two other Kohn-Sham +states closer to the Fermi level. +Comparing the three +self-energy functions, there is a more effective pole su- +perposition for states at energies further away from the +Fermi level. +Indeed, for the lowest energy state with +EKS = −11.2 eV, this leads to a frequency dependence +of Σ with an intense single pole (at about -15 eV with +respect to EKS) and consequently a very broad and shal- +low QP peak in the corresponding spectral function. At +the same time the satellite structure is enhanced to the +point of becoming a second peak, originating from a sec- +ond solution of the quasi-particle equation (intersections +of the dashed line with the self-energy function in the +upper panel). +This scenario is consistent with the so- +called ”plasmaron” peak, a sharp satellite feature emerg- +ing as an artefact of the G0W0 approximation to the self- +energy [2, 44, 71]. +The situation is similar for the two QPs computed for +Na shown in the rights panel of Fig. 1, with the lowest +state presenting again two solutions for the QP equation. + +6 +B. +Analysis of the intra-band contribution +In common GW implementations, especially those tar- +geting semiconductors, the intra-band contribution to the +dielectric function in the q → 0 limit, Eq. (6), is often +not included, as explained in Sec. II D. In the case of +Al, where a substantial part of the Fermi surface is very +close to the BZ boundary, one can expect [32] that many +of the metallic contributions are effectively inter- rather +than intra-band terms, resulting in a small error when +the intra-band term is neglected [32], while for Na the +intra-band terms are found to be more relevant. +For both Al and Na, in Fig. 2 we show how this af- +fects the frequency dependence of the YG=G’=0 matrix +elements computed for different q-vectors along an arbi- +trary direction. The curves in green shades correspond +to Y (ω) computed for finite but small q. +The orange +curve corresponds to the q → 0 limit evaluated only for +the inter-band term. There are two main differences be- +tween the green and orange curves. The first difference is +the limit of Re[Y ] as the frequency tends to zero (static +limit), that evolves smoothly for finite q but in general +tends to a value different from the one corresponding to +q = 0. As shown in the insets of Fig. 2, the smallest fi- +nite q provides a static limit very similar to the value for +q = 0 in the case of Al, while it is considerably larger in +the case of Na (both results in agreement with previous +studies [32]). +This difference has been commonly used as a measure +of the missing intra-band term [19, 32], since for metals +in the limit q → 0, ε−1 +G=G′=0(q, ω = 0) vanishes, meaning +that Y00(q, ω = 0) → −1, as apparent from the progres- +sion of the curves with finite q, that include intra-band +transitions. In fact, in the independent particle picture, +the q → 0 limit of Re[Y ] at ω = 0 is related to a non- +vanishing probability of vertical transitions within the +same band [30], and can therefore be used to estimate a +Drude frequency [74, 75]. However, this probability alone +does not determine the plasmon frequency (see Table II +for a summary of the nomenclature) or the position of +the pole of Re[Y ] for q → 0. +In fact, the second difference between the orange (q = +0, no intra-band contribution) and the green curves (fi- +nite q, intra-band included) in Fig. 2 is the position of +the main pole of Y (ω), here called Ωp, or in the case of +Na, to the apparent absence of poles for q = 0, whose +small amplitudes cannot be seen in the plot. If the whole +frequency range is considered, we see that the behaviour +of Re[Y (ω → 0)] depends on the position of Ωp. Follow- +ing the green curves at finite q, it is clear that YG=G′=0 +for both Al and Na change smoothly with q. The curves +present a pole, Ωp(q), of decreasing energy and increasing +amplitude, just above 0.5 Ha for Al and 0.2 Ha for Na. +As shown in Fig. 2(e), both the real and imaginary part +of this pole can be easily extrapolated to q = 0, by means +of the Lindhard plasmon dispersion [30, 32, 72, 73]. +In the same plot we show, as a reference, the Drude +frequency corresponding to the q → 0 limit of the intra- +band contributions, ωA (see Table II), as computed in +Ref. [19] for Al and Na, in addition to the experimental +plasmon frequency ωp of Al [53, 72, 76–79] and Na [72]. +In the simulations we can also extrapolate, already with +a 8×8×8 k-point mesh, the plasmon frequency at q → 0 +from the position of the main structure of the response +functions, namely ωp ≡ Re[Ωp]. This procedure provides +ωp = 0.55 Ha (15.01 eV) for Al, in excellent agreement +with the experimental value of 15.0 eV [30]. Similarly, +the value extrapolated for Na, ωp = 0.21 Ha (5.79 eV), +matches very well the experimental value of 5.9 eV [72] +and both compare well with the Drude intra-band fre- +quency computed in Ref. [19] (6.18 eV). The small dif- +ference between our theoretical result and Ref. [19] can be +attributed to methodological differences (e.g., the DFT +functional on top of which the GW calculations are per- +formed). +In contrast, the difference between ωp and +ωA for Al is larger than 2.5 eV since the plasmon fre- +quency ωp has non-negligible contributions from both +intra- and inter-band transitions, as previously reported +in Refs. [30, 53]. Note however that the inter-band con- +tributions are not included in the Drude frequency com- +puted in Ref. [19]. +In order to discriminate between the intra- and inter- +band contributions to the plasmon frequency, we have +used a simple expression based on the f-sum rule [2, 30, +80], but separating the two contributions: +Ω2 +A = lim +q→0 +2 +π +� ∞ +0 +dω ω Im[Y (q, ω) − YE(q, ω)], +(12) +where YE corresponds to inter-band transitions only, +while Y accounts for the complete response. +Within +MPA the integral is solved analytically (derivation in +Sec. I of the Supplemental Material [81]), leading to: +Ω2 +A = 2v(RpΩp − REΩE), +(13) +where ΩE and vRE are the position and the residue of +the most relevant pole of YE(q = 0), while Ωp and vRp +are the corresponding values for Y (q = 0). +In principle, the product vRpΩp should be computed in +the q → 0 limit. We have instead considered the extrap- +olation of Ω2 +p, which is equivalent in our model (see Sec. I +in the Supplemental Material [81]) and significantly more +stable. The values of vREΩE are taken directly from the +calculation at q = 0 (orange curves in Fig. 2a,b), since no +intra-band transitions are considered, as explained above. +For Al, the real part of ΩE is ωE = 0.37 Ha (10.08 eV) +and thus, applying Eq. (13), the real part of the intra- +band pole ΩA is ωA = 0.43 Ha (11.72 eV). For Na, ωp and +ωA are similar. The comparison of ωp and ωA confirms +that the experimental plasmon frequency, ωp, in the case +of Na corresponds mainly to intra-band contributions, +while for Al there is an important inter-band contribu- +tion [53], and its use as a Drude intra-band frequency +would result in an overestimation of the actual ωA. +Making use of the extrapolation procedures described +above in the context of the MPA framework, and of a + +7 +10 +5 +0 +5 +10 +Re[Y] +Al +10 +0 +10 Na +0.0 +0.5 +1.0 +1.5 +(Ha) +20 +15 +10 +5 +0 +Im[Y] +q4 +q3 +q2 +q1 +q0 +p +A +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +(Ha) +30 +20 +10 +0 +0 +Na +0.0 +0.5 +1.0 +1.5 +(Ha) +20 +15 +10 +5 +0 +Im[Y] +p +p +A +q4 +q3 +q2 +q1 +q0 +p +0 +1.0 +0.8 +10 +5 +0 +5 +10 +Re[Y] +Al +10 +0 +10 Na +0.0 +0.5 +1.0 +1.5 +(Ha) +20 +15 +10 +5 +0 +Im[Y] +q4 +q3 +q2 +q1 +q0 +p +A +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +(Ha) +30 +20 +10 +0 +q4 +q3 +q2 +q1 +q0 +p +0 +1.0 +0.8 +0 +1.0 +0.5 +10 +5 +0 +5 +10 +Re[Y] +Al +10 +0 +10 Na +15 +10 +5 +0 +Im[Y] +q4 +q3 +q2 +q1 +q0 +20 +10 +0 +q4 +q3 +q2 +q1 +0 +1.0 +0.8 +0 +1.0 +0.5 +10 +5 +0 +5 +10 +Re[Y] +Al +10 +0 +10 Na +15 +10 +5 +0 +Im[Y] +q4 +q3 +q2 +q1 +q0 +20 +10 +0 +q4 +q3 +q2 +q1 +0 +1.0 +0.8 +0 +1.0 +0.5 +10 +5 +0 +5 +10 +Re[Y] +Al +10 +0 +10 Na +15 +10 +5 +0 +Im[Y] +q4 +q3 +q2 +q1 +q0 +20 +10 +0 +q4 +q3 +q2 +q1 +0 +1.0 +0.8 +0 +1.0 +0.5 +10 +0 +10 Na +4 +3 +2 +1 +0 +20 +10 +0 +q4 +q3 +q2 +q1 +0 +1.0 +0.5 +a) +c) +e) +b) +d) +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +q (inv. crystal coord.) +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +p (Ha) +exp +Al +p +[19] +Al +A +[19] +Al +A +exp +Na +p +Na +A +Re +Al +p +Re +Na +p +Im +Al +p +Im +Na +p +FIG. 2. Frequency dependence of YG=G′=0 matrix elements computed with MPA for different q vectors of modulus q ≡ |q| +tending to 0, a) and b) for Al, and c) and d) for Na. For q = 0 (orange curves) the intra-band transitions are not included. +The insets in panels a) and c) show the region around ω = 0. Panel e) shows the q dispersion of the real and the imaginary +parts of the main pole of Y for q0–q4 (qn = n +8 in units of 2π/a, being a the respective Al and Na lattice parameters). The solid +lines show the corresponding parabolic fits consistent with a Lindhard (bulk) plasmon dispersion [30, 32, 72, 73]. The black +dashed lines correspond to the experimental plasmon frequency of Al [30] and Na [72]. Dash-dot purple lines correspond to +the values of the intra-band frequency, ωA, computed in Ref. [19] using the method described in Ref. [50], while the violet one +corresponds to our estimate for Al, computed by means of Eq. (13). +Contribution +Pole (complex) Frequency (real) +intra-band +ΩA +ωA = Re[ΩA] +inter-band +ΩE +ωE = Re[ΩE] +plasmon(intra+inter) +Ωp +ωp = Re[Ωp] +plasma +- +ωpl = √4πρe +Drude(model) +ωD + iγ +ωD +TABLE II. Summary of the notation concerning frequency +related quantities introduced in this work. The plasma fre- +quency is defined in terms of the electronic density, ρe. The +Drude pole/frequency are model parameters used to describe +the plasmon or only its intra-band contribution, as described +in Eq. (11). +simple f-sum rule, it is possible to determine not only +the real but also the imaginary part of both the plas- +mon and the intra-band pole, usually not considered in +other ab-initio methods. It is also worth noticing that the +extrapolation is done with points from a much coarser k- +grid (8 × 8 × 8 for both Al and Na), with respect to the +grids required to compute the intra-band frequency with +an independent particle formulation [31, 32]. +Despite the limited accuracy of the computed imagi- +nary values, they are meaningful and provide a qualita- +tive understanding of how intra- and inter-band terms, +linearly summed at the independent particle level, are +combined after the inversion of the Dyson equation. +While the Na case is trivial, since the inter-band con- +tribution is negligible, in the case of Al the small differ- +ence between ωA and ωE, comparable to their imaginary +parts, explains the presence of a single pole in Y (ω) lo- +cated roughly at ω2 +p ∼ ω2 +A+ω2 +E (see Sec. I of Supplemental +Material [81]). +C. +Modelling of the intra-band limit +Our analysis of the dressed response function Y (ω) +suggests that an alternative to the direct evaluation of +the intra-band limit, usually determined from X at the +independent particle level [31], can be obtained, either +by (1) including a complex Drude pole YD(ω), accord- +ing to Eq. (11), in the head (G = G′ = 0) of the in- +dependent particle dielectric function, with the Drude +frequency given by the computed intra-band pole; or (2) +approximating the full Y (q = 0) matrix element by its +nearest neighbour Y (q ̸= 0), i.e. with the q-vector clos- +est to 0 according to the adopted k-point grid. + +8 +The first method builds on using an estimate of the +Drude intra-band frequency, similar to the extrapolations +used in [75], but here considering the whole frequency +range and both intra- and inter-band contributions. The +second method, which we will call from now on constant +dielectric function approximation (CA), assumes that the +whole Y (q) matrix is constant in a small region around +q = 0. This approach is inspired by the leading term of +the Taylor expansion for small q of the Thomas-Fermi +distribution, and is corroborated by the small difference +of 0.006 Ha (0.17 eV) found for both, Al and Na, be- +tween the extrapolated value of Ωp and its value at the +first finite q, as shown in Fig. 2 e). Both methods simul- +taneously correct the position of the plasmon pole and +the limit of YG=G′=0 for ω = 0 and add virtually no +computational cost to the calculation. In addition, CA +also corrects other matrix elements for which the intra- +band limit may be important. +In Sec. II of Supplemental Material [81] we report +plots similar to the ones in Fig. 2 for Y matrix ele- +ments of Na other than the head, showing that after +the head (YG=G′=0), intra-band contributions are rele- +vant also for the so-called wing elements (YG=0̸=G′ and +YG̸=0=G′), while less important for the diagonal elements +(YG=G′̸=0), specially at increasing |G|. For finite |G| the +evolution of the Y (q) matrix elements when q → 0 is +less smooth and the position of the poles does not al- +ways change monotonously, meaning that an extrapola- +tion would require a denser k-point grid. Even if the con- +stant dielectric function approximation has limited accu- +racy for some of these matrix elements, it still provides +a significant overall improvement. In particular in ma- +terials such as Cu, as discussed below, the CA method +presents some clear advantages regarding the estimation +of ωA. +To assess the effect of this approximation in the QP +solution, in Fig. 3 we show Al and Na QP energies com- +puted without (nD) and with (CA) intra-band correc- +tions. +When the number of k-points is increased, the +weight of the Y (q = 0) element in the self-energy de- +creases and both methods eventually converge to the +same quasi-particle values, but only very slowly, as dis- +cussed above. In fact, Fig. 3 shows that for two selected +QPs of Na the intra-band term is fundamental due to the +importance of this contribution to the screening prop- +erties of the system. In contrast, for Al the difference +is small, and the convergence is governed by the inter- +rather than intra-band contributions for all the 4 QPs +considered. In the bottom panel of Fig. 3 one can see +the significant acceleration introduced by CA in the con- +vergence of the bandwidth of Na, where, besides a small +oscillation in the 20×20×20 grid (caused by oscillations +in the DFT eigenvalues), the first point corresponding to +the 8×8×8 mesh already provides very accurate results. +In the CA scheme the convergence benefits simultane- +ously from the decrease of the weight of the Y (q = 0) +contribution and from the fact that the correction itself +improves for denser grids in reciprocal space, since the +0 +100 +200 +300 +400 +QPCA +QPnD (meV) +Na: +25′ +Na: +1 +Al: +25′ +Al: +1 +Al: X4′ +Al: W3 +0 +2500 +5000 +7500 +10000 12500 +k-points +100 +0 +100 +200 +300 +- GW correction to +1 (meV) +Na (nD) +Na (CA) +Al (nD) +Al (CA) +FIG. 3. Top panel: Difference between GW-MPA corrections +computed with the (CA) intra-band term and without (nD) +as a function of the number of k-points, for 2 quasi-particles +of Na (light and dark yellow) and 4 of Al (green shades). +Bottom panel: Convergence of the GW correction for the QP +at Γ1 of Na (yellow) and Al (green) with CA (solid) and nD +(dashed). +first q ̸= 0 is closer to 0. +In Fig. 1 we show the frequency dependence of the real +part of the self-energy (top) and spectral function (bot- +tom) computed for two quasi-particles of Na, within MPA +with and without the intra-band correction. The correc- +tion does not change dramatically the shape of the self- +energy, but introduces an extra pole in the real part of +the self-energy at the intra-band frequency (∼-6 eV) and +renormalizes the peaks of the spectral function. The in- +clusion of this term promotes the pole overlapping around +the plasmon frequency, affecting the tail of the self-energy +and thus the QP solution as illustrated in the insets of +Fig. 1, differently for each quasi-particle. +In the case of Al and Na, the QP energies computed +with the Drude model, Eq. (11) using as input ωD = ωA, +and the CA schemes are very similar, with differences +below 20 meV when using the 8 × 8 × 8 k-grid. +This +leads us to conclude that the CA scheme could replace the + +9 +usual Drude correction, replacing a semiempirical scheme +by a simple ab-initio approximation. This is particularly +relevant when the Drude intra-band frequency is difficult +to estimate either from experiments or calculations, since +the CA scheme has virtually zero computational cost and, +as the extrapolation presented in the previous Section, +describes both the real and the imaginary part of Y . +To summarize this section, the inclusion of the intra- +band limit through the proposed CA scheme requires no +extra computational cost with respect to the standard +GW calculation and accelerates the k-grid convergence +of the QP energies for systems where the intra-band con- +tribution dominates, like Na, without resorting to semi- +empirical corrections such as the Drude model or com- +putationally costly ab-initio approaches. +D. +Frequency representation of the response +function of copper +As mentioned before, the case of copper presents sev- +eral challenges for an accurate GW description. The Cu +band structure features a series of flat d-bands around +2 eV below the Fermi level, leading to strong transitions +in YG,G′(q, ω) spread over a large energy range [20]. As +shown in Fig. 4 for q = 0, even for small values of G +and G′, YG,G′(q, ω) can behave very differently from a +single pole case, hindering the use of PPA but suggest- +ing that a multipole approach could prevent resorting to +more expensive FF methods. +When considering PPA or in general MPA with only +a few poles, one of the main issues is that the in- +terpolation of X or Y may give rise to non-physical +poles, posing representability problems. +Within the +Godby and Needs (GN) PPA scheme implemented in +yambo [11, 26, 82, 83], the condition used to identify +these so-called unfulfilled modes is the following: +Re +� +YGG′(q, 0) +YGG′(q, iϖpl) − 1 +� +< 0, +(14) +ϖpl being a frequency on the imaginary axis used to per- +form the GN interpolation, typically set to ϖpl = 1 Ha +or to a value of the order of the plasma frequency +(ϖpl ≳ ωpl), computed from the electronic density, ρe +(see Table II). As an example, for the diagonal elements +(G = G′), the polarizability evaluated on the imaginary +axis should be real and therefore unfulfilled modes are +those for which the resulting pole is instead imaginary. +In these cases, the position of the pole is typically set to +ΩGN +fail = 1 Ha. +Setting the pole at ΩGN +fail usually works well for simple +semiconductors [25, 82]. However, in more complex sys- +tems it can compromise the PPA approach. In fact, when +performing GW calculations using GN-PPA for Cu, we +found that no less than 48% of the matrix elements are +unfulfilled modes. This means that, for almost half of the +matrix elements, the position of the pole is spuriously set +to 1 Ha, severely affecting the self-energy and the quasi- +particle solution, as shown in the insets of Fig. 5. Within +MPA, increasing the number of poles in the description +of Y , together with the generalized condition to assign +the position of the poles of the unfulfilled modes, as de- +scribed in Ref. [25], leads to a significant improvement +in the representability of Y , as illustrated in Sec. III of +Supplemental Material [81]. +In Fig. 4 we compare selected Y matrix elements com- +puted within MPA with 1 and 12 poles, with the FF +results computed with a frequency grid of 1000 points +(all other convergence parameters being the same: k- +grid, number of empty bands, etc) At first glance, the en- +veloping structure of diagonal elements presents a strong +overall peak, as in the case of semiconductors such as +Si, hBN, and TiO2, which are well-described within PPA +and MPA [25]. +However, in the case of Cu, there are +other important peaks close to the origin not captured by +a single-pole model. In this case, PPA quasi-particle en- +ergies are not just numerically inaccurate, as in the case +of the discussed semiconductors, but PPA becomes an +inadequate model. Increasing the number of poles from +1 to 12 significantly improves the agreement between +Y computed with MPA and FF, reproducing the over- +all frequency dependence even if MPA presents a much +smoother shape. +While the rapid oscillations in the FF response func- +tion are enhanced by the discretization of the Brillouin +zone, the origin of such fluctuations can be related to +the topology of the flat d-bands of Cu [20], consistently +e.g. with the very structured W(ω) computed for Ni [84]. +In fact, regardless of the overall simple shape of X, nu- +merous inter-band transitions, close in energy and not +effectively overlapped, contribute to the fluctuations of +the polarizability X and of the inverse dielectric function +Y , when computed within FF. Nevertheless, as discussed +in the next Section, they do not significantly influence +the computed GW quasi-particle energies. +E. +Quasi-particles and spectral function of copper +In Fig. 5 (top panels) we show the frequency depen- +dence of the self-energy projected on three selected quasi- +particle states of Cu calculated within PPA, MPA, and +FF-RA. The details of Σ computed within the FF ap- +proach, better appreciated in Fig. 5 c), depend on the +fine structure of W, which requires a dense frequency grid +when computing the polarizability, as shown in Sec. V of +Supplemental Material [81]. Since these calculations are +very expensive, the curves shown in Fig. 5 were com- +puted including 200 bands for all the three methods, and +using a frequency grid with 1000 points for FF and no +intra-band correction. Fully converged MPA results and +intra-band corrections are discussed at the end of this +Section. +The FF self-energy presents a rather flat structure with +no dominant peaks. Since Σ is obtained from the con- + +10 +2 +1 +0 +1 +Re[Y] +G = G0= 0 +×1.0 +2 +1 +0 +1 +G = G0 +0 +×0.29 +2 +1 +0 +1 +G +G0 +0 +×0.07 +0 +5 +10 +15 +(Ha) +3 +2 +1 +0 +Im[Y] +×1.0 +Cu: +MP1 +MP12 +FF +0 +5 +10 +15 +(Ha) +3 +2 +1 +0 +×0.24 +0 +5 +10 +15 +(Ha) +2 +1 +0 +1 +×0.07 +0 +5 +10 +15 +(Ha) +3 +2 +1 +Im[Y] +Cu: +MP1 +MP12 +FF +2 +1 +0 +1 +[ ] +G = G0= 0 +2 +1 +0 +1 +G = G0 +0 +2 +1 +0 +1 +G +G0 +0 +0 +5 +10 +15 +(Ha) +3 +2 +1 +0 +[ ] +×1.0 +Cu: +MP1 +MP12 +FF +’ +’ +’ +a) +c) +e) +b) +d) +f) +FIG. 4. Selected Cu Y (q = 0) matrix elements computed within MPA with 1 and 12 poles compared with the corresponding +FF results. The y-axes are scaled with the factors indicated on top of each panel. +20 +0 +20 +40 +Re[ ] (eV) +EKS +EKS = +EKS +30 +20 +10 +0 +10 +EKS (eV) +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +Im[G] (eV +1) +Cu +40 +30 +20 +10 +0 +10 +EKS (eV) +40 +30 +20 +10 +0 +10 +EKS (eV) +2 +0 +2 +2 +0 +2 +2 +0 +2 +2 +0 +2 +2 +0 +2 +2 +0 +2 +20 +0 +20 +40 +Re[ ] (eV) +EKS +EF +pp +ff +mp +EKS = +12 +pp +ff +mp +EKS = +1 +pp +ff +mp +0 08 +0.10 +Cu +2 +0 +2 +2 +0 +2 +2 +0 +2 +2 +0 +2 +2 +20 +0 +20 +40 +Re[ ] (eV) +EKS +EF +pp +ff +mp +EKS = +12 +pp +ff +mp +EKS = +1 +pp +ff +mp +40 +30 +20 +10 +0 +10 +EKS (eV) +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +Im[G] (eV +1) +Cu +40 +30 +20 +10 +0 +10 +EKS (eV) +40 +30 +20 +10 +0 +10 +EKS (eV) +2 +0 +2 +2 +0 +2 +2 +0 +2 +2 +0 +2 +2 +0 +2 +2 +0 +2 +EKS +EF +pp +ff +mp +EKS = +12 +pp +ff +mp +EKS = +p +ff +m +Cu +2 +0 +2 +2 +0 +2 +2 +0 +2 +2 +0 +2 +EKS = +12 +pp +ff +mp +EKS = +1 +pp +ff +mp +2 +0 +2 +2 +0 +2 +2 +0 +2 +2 +0 +2 +EKS = +12 +pp +ff +mp +EKS = +1 +pp +ff +mp +2 +0 +2 +2 +0 +2 +2 +0 +2 +2 +0 +2 +2 +0 +2 +0 +2 +a) +c) +e) +b) +d) +f) +PPA +MPA +FF +FIG. 5. Frequency dependence of the real part of the self-energy (top) and spectral function (bottom) of three quasi-particle +states of Cu: one close to the Fermi energy (panels a) and b)), Γ12 (c) and d)) and Γ1 (e) and f)); computed with PPA, MPA +and FF. + +11 +QP(eV) +DFT/LDA +DFT/PBE +DFT/PBE +GW@LDA +GW@PBE +GW@PBE +Exp +Ref. [20] +Ref. [65] +(current work) +Ref. [20] +Ref. [65] +(current work) +Ref. [62] +Γ12 +-2.27 +-2.05 +-2.18 +-2.81 +-1.92 to -2.11 +-2.12 +-2.78 +Γ1 +-9.79 +-9.29 +-9.27 +-9.24 +-9.14 to -9.20 +-9.06 +-8.60 +X5 +-1.40 +-1.33 +-1.49 +-2.04 +-1.45 to -1.22 +-1.39 +-2.01 +L2′ +-1.12 +-0.92 +-0.99 +-0.57 +-0.98 to -1.02 +-1.05 +-0.85 +L3 +-1.63 +-1.47 +-1.63 +-2.24 +-1.58 to -1.36 +-1.57 +-2.25 +L gap +5.40 +4.80 +4.66 +4.76 +4.98 to 5.09 +4.88 +4.95 +TABLE III. DFT and GW quasi-particle energies of Cu computed with different methodologies by different groups and compared +with the experimental values. All the GW calculations correspond to FF approaches ran on top of LDA [20] and PBE [65]. +volution of G and W in Eq. (1), the oscillations of W +are attenuated, resulting in a much smoother function. +Nevertheless, the convergence of the QP solution is chal- +lenging, since it requires an accurate description of the +tail of the self-energy, as shown in the insets of Fig. 5. +This could explain, at least in part, the variety of results +present in the literature. +GW-PPA data (blue curves in Fig. 5) show that the +quasi-particle solution (insets of Fig. 5) obtained with a +single pole model for W deviates from the FF solution. +Besides the deviations at the tail of Σ, PPA fails to de- +scribe the frequency dependence of Σ and the spectral +function (bottom panels). On the other hand, the MPA +results, here obtained with 12 poles and the quadratic +sampling, are very accurate, not only in the tail region, +that determines the QP corrections, but also for the +whole frequency range of both the self-energy and the +spectral function. +Comparing the three selected quasi-particle states in +Fig. 5, the effect of the overlapping of the independent- +particle excitations (due to the inclusion of local field +effects via the Dyson equation for W) on the self-energy +of Cu is more relevant for Γ1 than for Γ12 and the QPs +around the Fermi energy. Indeed, as shown in the bot- +tom panels of Fig. 5, for the QPs closer to the Fermi +level, the shape of the spectral function has a very narrow +quasi-particle peak and three satellite. When compared +to the QPs close to Fermi, the QPs at deeper energies +(Γ12 and Γ1) present a broader quasi-particle peak and +more intense satellites. The shallower satellite (above - +10 eV) forms a shoulder structure for Γ12(central panel) +and eventually merges with the QP peak to form a single +broader peak for Γ1 (right panel). Despite its complexity, +the Cu states at different energies present similar trends +as the cases of Al and Na discussed in Sec. III A. +It is worth to emphasize the importance of the fre- +quency sampling in MPA. Since copper X and Y present +a rich structure at low frequencies, but the energy range, +ωm in Eq. (10) is still large, the quadratic sampling has +shown to be more efficient than the linear one. Specifi- +cally, it provides, with the same number of poles and the +same ωm, a larger density of points in the low frequency +region and therefore higher accuracy. The comparison +between the computational cost of MPA and the FF-RA +method can be done in a simplified way by comparing +the number of frequencies for which X is numerically +computed in each approach. Here, for MPA we use 24 +frequency points, corresponding to 12 poles, while the +FF-RA frequency grid has 1000 points, corresponding to +a 40 times gain in computational efficiency of MPA with +respect to FF-RA. +The convergence with respect to the number of bands +and the size of the X matrices is particularly challeng- +ing, as already reported for example for other systems +with d states [85–87], with a slow, non-monotone conver- +gence that hinders the use of extrapolations (more detail +in Sec. V of Supplemental Material [81]). For this rea- +son, the computational efficiency of MPA is particularly +beneficial as it allows for the use of fine GW convergence +parameters, thereby increasing the overall accuracy of +the results. +In Table III we show the MPA results obtained with +60 Ry of energy cut-off and 1000 bands for both, X and +Σ. These parameters are comparable to the largest ones +used within a static subspace approximation [66]. The +reported MPA quasi-particle energies are in good agree- +ment with previous calculations using different FF ap- +proaches, and summarized in Table III. The main differ- +ences can be explained by the use of different starting +points for the GW calculation, i.e. different exchange- +correlation functionals and/or pseudopotentials in the +DFT ground state, and possibly to an incomplete conver- +gence of some of the results. While the use of converged +parameters is essential when comparing the computed +QP energies with experiments, GW corrections do not +always improve over DFT/PBE results, as also observed +in Refs. [65, 66]. In the present case, GW significantly +improves Γ1, while for Γ12 and other QPs, the GW cor- +rection is rather small and slightly worsens the DFT re- +sults. The localized nature of the d states in Cu may +require methods beyond GW in order to further improve +the agreement with experiments [88–90]. + +12 +1.00 +0.75 +0.50 +0.25 +0.00 +0.25 +Re[Y] +Cu +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Re[ +n] (Ha) +estim +Cu +A +theo +Cu +A +Exp +1 +4 +0 +1 +2 +3 +4 +(Ha) +1.2 +1.0 +0.8 +0.6 +0.4 +0.2 +0.0 +Im[Y] +q4 +q3 +q2 +q1 +q0 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +q (inv. crystal coord.) +0.04 +0.06 +0.08 +0.10 +0.12 +0.14 +0.16 +Im[ +n] (Ha) +1 +2 +3 +4 +1.00 +0.75 +0.50 +0.25 +0.00 +0.25 +Cu +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Re[ +n] (Ha) +estim +Cu +A +theo +Cu +A +Exp +1 +4 +0 +1 +2 +3 +4 +(Ha) +1.2 +1.0 +0.8 +0.6 +0.4 +0.2 +0.0 +[ ] +q4 +q3 +q2 +q1 +q0 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +q (inv. crystal coord.) +0.04 +0.06 +0.08 +0.10 +0.12 +0.14 +0.16 +Im[ +n] (Ha) +1 +2 +3 +4 +1.00 +0.75 +0.50 +0.25 +Re[Y] +Cu +0.0 +0.2 +0.4 +0.6 +0.8 +Re[ +n] (Ha) +estim +Cu +A +Exp +1 +4 +0 +1 +2 +3 +4 +(Ha) +1.2 +1.0 +0.8 +0.6 +0.4 +0.2 +0.0 +Im[Y] +q4 +q3 +q2 +q1 +q0 +0.00 +0.05 +0.10 +q (inv. crysta +0.04 +0.06 +0.08 +0.10 +0.12 +0.14 +0.16 +Im[ +n] (Ha) +1 +2 +3 +4 +1.00 +0.75 +0.50 +0.25 +0.00 +Re[Y] +Cu +0.0 +0.2 +0.4 +0.6 +0.8 +Re[ +n] (Ha) +estim +Cu +A +Exp +1 +4 +0 +1 +2 +3 +4 +(Ha) +1.2 +1.0 +0.8 +0.6 +0.4 +0.2 +0.0 +Im[Y] +q4 +q3 +q2 +q1 +q0 +0.00 +0.05 +0.10 +q (inv. crysta +0.04 +0.06 +0.08 +0.10 +0.12 +0.14 +0.16 +Im[ +n] (Ha) +1 +2 +3 +4 +Cu +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Re[ +n] (Ha) +Cu +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Re[ +n] (Ha) +0.75 +0.50 +0.25 +0.00 +0.25 +Re[Y] +0.4 +0.6 +0.8 +1.0 +Re[ +n] (Ha) +ti +Cu +h +Cu +Exp +1 +4 +1.00 +0.75 +0.50 +0.25 +0.00 +0.25 +Re[Y] +Cu +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Re[ +n] (Ha) +estim +Cu +A +theo +Cu +A +Exp +1 +4 +0.8 +0.6 +0.4 +0.2 +0.0 +Im[Y] +q4 +q3 +q2 +0.08 +0.10 +0.12 +0.14 +0.16 +Im[ +n] (Ha) +1 +2 +3 +4 +1.00 +0.75 +0.50 +0.25 +0.00 +0.25 +Re[Y] +Cu +0 0 +0.2 +0.4 +0.6 +0.8 +1.0 +Re[ +n] (Ha) +estim +Cu +A +theo +Cu +A +Exp +1 +4 +1.00 +0.75 +0.50 +0.25 +0.00 +0.25 +Re[Y] +Cu +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Re[ +n] (Ha) +estim +Cu +A +theo +Cu +A +Exp +1 +4 +0 2 +0.0 +0.16 +1.00 +0.75 +0.50 +0.25 +0.00 +0.25 +Re[Y] +Cu +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Re[ +n] (Ha) +estim +Cu +A +theo +Cu +A +Exp +1 4 +1.00 +0.75 +0.50 +0.25 +0.00 +0.25 +Re[Y] +Cu +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Re[ +n] (Ha) +estim +Cu +A +Exp +1 +0 +1 +2 +3 +4 +1.2 +1.0 +0.8 +0.6 +0.4 +0.2 +0.0 +Im[Y] +q4 +q3 +q2 +q1 +q0 +0.00 +0.05 +0. +0.04 +0.06 +0.08 +0.10 +0.12 +0.14 +0.16 +Im[ +n] (Ha) +1 +2 +3 +4 +1.00 +0.75 +0.50 +0.25 +0.00 +0.25 +Re[Y] +Cu +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Re[ +n] (Ha) +estim +Cu +A +Exp +1 +1.00 +0.75 +0.50 +0.25 +0.00 +0.25 +Re[Y] +Cu +0 0 +0.2 +0.4 +0.6 +0.8 +1.0 +Re[ +n] (Ha) +estim +Cu +A +Exp +1 +1.00 +0.75 +0.50 +0.25 +0.00 +0.25 +Re[Y] +Cu +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Re[ +n] (Ha) +estim +Cu +A +Exp +1 +0.0 +0.16 +1.00 +0.75 +0.50 +0.25 +0.00 +0.25 +Re[Y] +Cu +0.2 +0.4 +0.6 +0.8 +1.0 +Re[ +n] (Ha) +estim +Cu +A +Exp +1 +A +1.00 +0.75 +0.50 +0.25 +0.00 +0.25 +Re[Y] +Cu +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Re[ +n] (Ha) +estim +Cu +A +theo +Cu +A +Exp +1 +4 +0 8 +0.6 +0.4 +0.2 +0.0 +Im[Y] +q4 +3 +0 08 +0.10 +0.12 +0.14 +0.16 +m[ +n] (Ha) +1 +2 +3 +A [31] +a) +c) +b) +d) +FIG. 6. Left panels: frequency dependence of the real (a)) and imaginary part (b)) of YG=G′=0 for Cu computed within MPA +for different q-values tending to 0 (qn = +n +16 in units of 2π/a, where a is the lattice parameter of Cu). For q = 0 (orange curves) +the intra-band term is not included. Right panels: real (c)) and imaginary part (d)) of the four most relevant poles at low +energies in the Y curves for different q values. The purple dashed lines lines correspond to the position of the poles extracted +from optical measurements collected in Ref. [24], as explained in the main text. The blue dashed lines correspond to the values +of the intra-band frequency, ωA, computed by means of Eq. (13), and reported in Ref. [31]. +F. +Intra-bands effects in copper +In order to investigate the intra-band contributions of +copper, in Fig. 6 we show the frequency dependence of +the YG=G′=0 matrix elements computed for for the small- +est q-vectors along one direction of a 16×16×16 k-grid. +Since Y (ω) of Cu is very structured at small frequen- +cies, where the effects of the intra-band contributions +are expected to be stronger, we have used MPA with +a quadratic sampling, Eq. (10) with α = 2 and np = 15, +a number of poles slightly larger than the value needed +to converge the quasi-particle energies. In contrast with +Na, the orange curve (q = 0, no intra-band contribu- +tion) presents a similar shape and scale with respect to +the green curves (small but finite q, with intra-band con- +tributions), even if with less intense peaks. +In the right panel of Fig. 6 we show the position of the +first 4 poles of Y (ω) as a function of q, which present a +rather flat dispersion, when compared with the plasmon +dispersion of Al in Fig. 2. As expected, for q = 0 the +position of some poles does not correspond exactly to +the limit given by the curves with finite q. However, the +main difference between the zero and finite q curves of +Y (ω) is not in the position but rather in the value of the +residues of the poles, which is reflected in the intensity +of some of the peaks, as shown in Fig. 6. +In order to compare the computed results with exper- +iments, we used electron energy loss data extracted from +a compilation of optical measurements found in Table 1 +of the Chapter Optical constants of metals of Ref. [24] +(see e.g. Fig. 8 of Ref. [31]), after interpolation with a +multipole model. For this, we chose 18 points of the spec- +tra, with a frequency distribution corresponding to the +quadratic sampling of Eq. (10) and used them to interpo- +late a 9 pole model. We then analysed the 4 poles with +the highest residues in the frequency interval we are in- +terested in. In the upper panel of Fig. 6 we show, as hor- +izontal lines, the corresponding experimental energies of +the poles. Interestingly, the experimental poles are very +similar to the poles computed at the RPA level within +MPA. This supports the interpretation that the MPA +poles of Y are not a mere mathematical construct aimed +at improving representability but indeed correspond to +physical collective excitations, each of them describing +the envelope of a set of single particle transitions, with a +finite imaginary part corresponding to the width of the +excitation. We emphasize that the agreement with the +experiment is achieved without resorting to any ad hoc +parameters such as the damping in the case of the FF-RA +method of Ref. [31]. +In simpler systems, the inclusion of the intra-band +limit, even with a simple Drude tail fitted from the ex- + +13 +perimental spectra, is expected to correct the residues +and thus the intensity of the peaks at q = 0. However, +in systems for which the intra- and inter-band contribu- +tions are superimposed in a more structured frequency +dependence, the description of the experimental spectra +with only the Drude term from Eq. (11) is not possi- +ble [52, 74, 91], and indeed models often resort to vari- +able or multiple relaxation frequencies [52, 77, 91]. In +fact, as shown in Fig 6, the q dependence of Y does not +allow one to discriminate between peaks with an intra- +or an inter-band character. In order to circumvent this +difficulty, in Ref. [31] the intra-band frequency is eval- +uated numerically as the limit of an intra-band integral +at the independent particle level, while in Ref. [51] it is +estimated within a non-interacting uniform-gas theory. +Here we use again the f-sum rule by integrating +Eq. (12), but generalizing Eq. (13) to the case where, +in contrast with Al and Na, more than one pole con- +tributes to the intra-band term (see Sec. I of Supplemen- +tal Material [81]). The resulting intra-band frequency, +ωA = 0.36 Ha (9.80 eV), compares well with the cor- +responding result of 0.34 Ha (9.27 eV) from Ref. [31] +and both values are very close in energy to the second +pole shown in Fig. 6. +We find that intra-band contri- +butions represent around the 25% of the corresponding +f-sum rule of this pole (RΩ product), being the largest +ratio among all the poles. However, as can be appreci- +ated in Fig. 6 from the change of intensity of the peaks, +the inter-band contributions are dominant. In fact, the +intra-band contributions to the total f-sum rule (sum of +all RΩ products) is rather small, less than 4%. +Using the frequency determined in Ref. [31] (9.27 eV) +and the relaxation frequency fixed to 0.1 eV as the in- +puts to the Drude correction of Eq. (11), in our MPA +calculations, we find that the Drude tail overlaps with +the several inter-band peaks of Y (ω), without affecting +the position of the poles whilst changing their residues +(Sec. IV of Supplemental Material [81]), similar to the +effect of the CA correction, Y (q = 0) ∼ Y (qmin), as +proposed in Sec. III B. In any case, CA is general and +independent of the complexity of the frequency structure +of the inverse dielectric function Y . It works well for Cu, +as confirmed by the comparison with the experimental +data, and constitutes a very simple procedure. Despite +these considerations, and similarly to the case of Al, the +intra-band correction has a small effect on the Cu QP +energies, that present differences of the order of 5 meV +when computed with and without CA in a 12 × 12 × 12 +k-grid. +IV. +SUMMARY AND CONCLUSIONS +In this work we address the accuracy of the MPA +scheme as applied to the full-frequency GW calculation +of metals. This approach, previously validated for semi- +conductors [25], is now applied to metals using Al, Na, +and Cu as prototype systems. Also in the case of metals, +MPA is shown to deliver results with an accuracy simi- +lar to other FF methods at a much lower computational +cost. +After presenting the MPA theoretical framework, we +have applied the approach to simple metals and discussed +the role of inter- and intra-band contributions to the di- +electric functions of bulk Al and Na. In order to eval- +uate the response function and the GW corrections in +metals, we have proposed two simple methods to include +the intra-band terms in the inverse dielectric function in +the q → 0 limit: (1) by extrapolating the position of +the main pole in Y00(q, ω), from small q to q = 0, and +computing the intra-band pole through the f-sum rule of +Eq. (13), which can then be used as an input value in a +Drude model to correct Y0. This approach is generalized +for a multipole structure of Y (q, ω) in the case of Cu. +And (2) by approximating Y (q = 0) by Y (qmin). The +second method, here called CA, is simpler and spares the +determination of the intra-band frequency. +Both methods significantly accelerate the convergence +of the QP energies with respect to the k-point grid. In ad- +dition, CA simultaneously corrects all Y matrix elements. +CA works equally within PPA, MPA and FF and can be +used independently of the dimensionality of the system +under study, even if the leading power of series expan- +sion of the inverse dielectric function in the q → 0 limit +depends on dimensionality. In fact, it can be thought of +as the most trivial case of a polynomial interpolation (a +constant) [92, 93]. A similar approach can be applied in +situations where the q → 0 limit of Y (or other many- +body operators, such as W) is difficult to evaluate. Even +if the proposed methodologies were exemplified for three +isotropic metals, the extension to non-isotropic systems +is straightforward. +Eventually, GW QP corrections for Na, Al and Cu +were evaluated, showing an excellent agreement with ex- +isting theoretical literature and experimental data, fur- +ther stressing the accuracy of the proposed approach. +Notably, the case of Cu was discussed with particular +detail, since PPA calculations present several drawbacks. +In fact, for Cu, the PPA quasi-particle solutions deviate +significantly from the FF results and completely fail to +describe the frequency dependence of Σ and the spectral +function. In contrast, MPA reproduces very accurately +the FF results, not only in the tail region that deter- +mines the quasi-particles corrections, but in the whole +frequency range for both the self-energy and the spectral +function. +The frequency representation of the polariz- +ability and the inverse dielectric function present strong +oscillations within FF. In contrast, MPA results are much +more stable, leading to a smooth frequency representa- +tion of X and Y . +Importantly, the smoother structure of the MPA di- +electric function does not necessarily result in a loss of +accuracy in the subsequent calculation of the self-energy, +the QP energies, and the spectral function. In fact, the +frequency dependence of Y given by MPA is meaning- +ful and reproduces the main peaks of the experimental + +14 +energy loss spectra. This leads us to conclude that the +MPA poles of Y may be seen not only as a mathemati- +cal tool, but also as an efficient description of collective +excitations, with each pole representing the envelope of +a set of single particle transitions. +In conclusion, MPA reproduces well the overall fre- +quency dependence of the polarizability, the inverse di- +electric function, the self-energy and the spectral func- +tion in metallic systems, and gives results for the quasi- +particle energies similar to those obtained within FF +methods. Moreover, the favourable computational per- +formance allows for the use of more stringent convergence +parameters such as denser k-grids and larger number of +bands and polarizability matrices. 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Leon1,2,3,∗ Andrea Ferretti2, Daniele Varsano2, Elisa Molinari1,2, and Claudia Cardoso2 +1 FIM Department, University of Modena & Reggio Emilia, 41125, Modena (Italy) +2S3 Centre, Istituto Nanoscienze, CNR, 41125, Modena (Italy) and +3Department of Mechanical Engineering and Technology Management, +Norwegian University of Life Sciences, 1430, ˚ +As (Norway) +I. +A SIMPLE INTRA + INTER-BAND MODEL +In this Section we analyze how two poles in the in- +dependent particle response function Y0, corresponding +to intra- and inter-band transitions, contribute to the +structure of its dressed counterpart, Y , as a result of the +Dyson equation. The Y0 and Y functions are related to +the non-interacting and dressed polarizability functions +as: +Y0 ≡ vX0, +Y ≡ vX. +(S1) +The Dyson equation for the inverse dielectric function is +then given by +Y = (1 − Y0)−1Y0 +(S2) +In the following we make use of the f-sum rule [1–3] in +the form +SY (q) = 2 +π +� ∞ +0 +dω ω Im[Y ](q, ω), +(S3) +where S is computed from the electronic density and the +above expression is valid for each (G, G′) components of +both Y0 and Y . In the case G = G′, one has SY (q) = ω2 +pl. +We then separate Y0 into intra (A) and inter-band (E) +contributions in the q → 0 limit, as +Y0(q = 0, ω) = Y A +0 (ω) + Y E +0 (ω), +(S4) +resulting in two different terms for the f-sum rule: +SY0(q = 0) = SA + SE, +(S5) +where SA is the power square of an intra-band frequency, +ΩA, and SE is obtained from the f-sum rule expression +for the polarizability X0. When computed from Kohn- +Sham (KS) states, as derived in Appendix B of Ref. [4], +one obtains: +SE = +� +n +2vRKS +n ΩKS +n . +(S6) +We now analyze the results of Eq. (S2) in different sce- +narios by means of a two poles model y0(ω). The first +∗ dario.alejandro.leon.valido@nmbu.no +pole of y0(ω) at ω = 0 corresponds to a Drude tail result- +ing from the intra-band transitions. The second broader +pole results from the superposition of the inter-band tran- +sitions. If we include both single particle contributions +in Eq. (S4), we obtain: +y0(ω) = Ω2 +A +ω2 + 2vREΩE +ω2 − Ω2 +E +. +(S7) +We consider the inversion of the Dyson equation (S2) +for y, disregarding the so called local field effects (i.e. y0 +and y are scalar functions instead of matrices). +Case SE → 0: +y(ω) = +Ω2 +A +ω2 − Ω2 +A +(S8) +In this case only the intra-band is relevant and we get a +pole at the Drude frequency, ΩA. +Case SA → 0: +y(ω) = +2vREΩE +ω2 − Ω2 +E − 2vREΩE +(S9) +In this case only the inter-band is relevant and the pole, +ΩE, is right-shifted by a factor +� +1 + 2vRE/ΩE, which +for many materials is close to 1 (no shift). +General case: +y(ω) = +S1 +ω2 − Ω2 +1 ++ +S2 +ω2 − Ω2 +2 +, +(S10) +where the poles are given by the expression: +Ω2 +1,2 = 1 +2 +� +Ω2 +E + 2vREΩE + Ω2 +A± +� +(Ω2 +E + 2vREΩE + Ω2 +A)2 − 4Ω2 +EΩ2 +A +� +, +(S11) +and S1 and S2 are compliant with the f-sum rule S1 + +S2 = SA + SE: +S1,2 = ±(SA + SE)Ω2 +1,2 − SAΩ2 +E +Ω2 +2 − Ω2 +1 +. +(S12) +arXiv:2301.02282v1 [cond-mat.mtrl-sci] 5 Jan 2023 + +2 +The two first cases can be obtained as limiting cases +of this general solution. But there are other situations +leading to a solution with a single plasmon pole: +y(ω) = 2vRpΩp +ω2 − Ω2p +. +(S13) +One case occurs when either S1 or S2 is much larger than +the other. A second possibility is when the two poles, Ω1 +and Ω2, are equal or very close to each other. There are +several ways to obtain this: the radicand in Eq. (S11) +could be small compared to the terms outside, the scales +of the poles could be very different (e.g. +ΩE ≫ ΩA), +2vRE/ΩE could be close to 0 or 1, or the intra- and +inter-band poles could be similar, ΩA ≈ ΩE. +If y(ω) has a single pole, the f-sum rule for y leads to +a simple relationship between the intra-band pole, ΩA, +the inter-band pole, ΩE and the plasmon pole, Ωp: +2vRpΩp = Ω2 +A + 2vREΩE, +(S14) +while a similar expression is obtained from the evaluation +of y(0): +Ω2 +p = Ω2 +A + 2vREΩE. +(S15) +From the previous two equations, vRp and Ωp are un- +equivocally determined, allowing one to discriminate be- +tween the intra- and inter-band contributions to the +structure of y. +For many systems, like the case of Cu addressed in +the main manuscript, the total Y (ω) presents a structure +with more than one pole. +In this situation the model +given in Eq. (S14) can be generalized to obtain the intra- +band frequency, again by means of the f-sum rule: +Ω2 +A = +� +2vRpΩp − +� +2vREΩE, +(S16) +where the sums of RΩ products run over the pole struc- +tures of the full and the inter-band part of Y (ω). +II. +ANALYSIS OF THE INTRA-BAND LIMIT +FOR DIFFERENT MATRIX ELEMENTS +In the main text we have shown that the YG=G′=0 +curves with finite q, computed for Al and Na, change +smoothly with decreasing q. The curves present a pole, +Ωp(q), of decreasing energy and increasing amplitude and +both the real and imaginary part of this pole can be +easily extrapolated to q = 0, by means of the Lindhard +plasmon dispersion. Here we have considered other Na +Y matrix elements (YG=G’̸=0 and YG̸=G’) with the same +not particularly dense grid of 8 × 8 × 8 k-points. +In Fig. S1 we show similar plots for wing (YG=0̸=G’ and +YG̸=0=G’) and diagonal (YG=G’̸=0) elements, where the +orange curves contain only inter-band terms. For some of +these elements the evolution of the Y curves when q → 0 +is less smooth than for the head, YG=G′=0, and the posi- +tion of the poles not always changes monotonously, thus +the extrapolation with a simple analytical form would re- +quire a denser k-point grid depending on the specific ma- +trix element. However, as can be seen from the compari- +son between green and orange curves, intra-band transi- +tions are more important for the wings than for diagonal +elements. Moreover, even if the constant dielectric func- +tion approximation (CA) scheme described in the main +manuscript, i.e, Y (q = 0) ∼ Y (qmin), has limited accu- +racy for some of these matrix elements, it still improves +the overall Y matrix. +III. +FREQUENCY REPRESENTABILITY OF +COPPER +As mentioned in the main manuscript, the nonlinear +interpolation of the polarizability, X, or the inverse di- +electric function, Y , may produce non-physical poles cor- +responding to the so-called unfulfilled modes. This is usu- +ally solved by reassigning the values of the poles. The +condition used to identify unfulfilled modes in the GN- +PPA scheme [4] is the following: +Re +� +YGG′(q, 0) +YGG′(q, iϖpl) − 1 +� +< 0, +(S17) +where the position of the pole is set to ΩGN +fail = 1 Ha in +case of failure. +The MPA scheme uses a generalized condition that +avoids reassigning the poles with a constant value [4]: +Ωn = +�� +Ω2n, +Re +� +Ω2 +n +� +≥ 0 +� +−(Ω2n)∗, +Re +� +Ω2 +n +� +< 0 +(S18) +It is possible to quantify the representability error re- +lated to this reassignment by computing the mean num- +ber of corrected X matrix elements, ⟨NF ⟩, and an av- +erage relative standard deviation of the extrapolated X +with respect to its sampling points, ⟨RSD⟩, as defined +in Ref. [4]. In Fig. S2 we show how these two quantities +evolve when increasing the number of poles used in the +description of X and Y . +When applying the PPA, we found that 48% of the po- +larizability matrix elements fail the plasmon pole condi- +tion (S17). These results have an averaged relative devi- +ation of ⟨RSD⟩ = 0.42. These large values lead to quasi- +particle solutions very different from FF, as described in +the main text. The MPA scheme, with only one pole, still +presents a larger percentage of corrected poles but con- +siderably improves the representability with respect to +PPA, lowering the average deviation to ⟨RSD⟩ = 0.35. +The reason for the larger number of corrected poles is +the use of a different sampling. +As mentioned in the +main manuscript, in the case of MPA with one pole the +frequency at the origin of coordinates is shifted along the +imaginary axis, which helps to reduce the numerical in- +stabilities found with the PPA sampling [5]. However, a +significant improvement happens only by increasing the + +3 +0.1 +0.0 +0.1 +Re[Y] (a.u.) +Al +Na +G = 0 +G′ +0.2 +0.1 +0.0 +Na +G = G′ +0 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +(Ha) +0.0 +0.1 +0.2 +0.3 +Im[Y] (a.u.) +q4 +q3 +q2 +q1 +q0 +0.00 0.25 0.50 0.75 1.00 1.25 +(Ha) +0.15 +0.10 +0.05 +0.00 +FIG. S1. Frequency dependency of Y matrix elements, other than the head (G = G′ = 0), computed with MPA for different +q vectors of modulus q ≡ |q| tending to 0, for Na. For q = 0 (orange curves) the intra-band term is not included. The chosen +values of q0–q4 correspond to qn = n +8 in units of 2π/a, with a the lattice parameter of Na. +2 +4 +6 +8 +10 +12 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +np +�RSD� +PPA +MPA +2 +4 +6 +8 +10 +12 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +np +�NF� +FIG. S2. Values calculated for Cu of (left) the mean number +of matrix elements, ⟨NF ⟩, for which the position of the poles +was corrected according to Eq. (S18) for MPA and Eq. (S17) +for PPA. +number of poles, as illustrated in Fig. S2, evidencing the +complexity of the frequency structure of the polarizabil- +ity of Cu and the efficiency of the multipole approxima- +tion in its description. +IV. +STANDARD DRUDE CORRECTION FOR +THE INTRA-BAND OF COPPER +In Fig. S3 we show the frequency dependence of Y +computed for Cu within MPA comparing three different +methods to include the intra-band corrections in the limit +q → 0. We show, Y computed without any intra-band +correction (nD), with the CA method and the Drude +model. For the latter, we considered an intra-band fre- +quency of 9.27 eV, as determined in Ref. [6]. We see that +the Drude model and CA give similar results, renormal- +izing the intensity of the peaks without shifting them in +the real frequency axis. +V. +CONVERGENCE OF GW PARAMETERS +FOR COPPER +As mentioned in the main text, the GW convergence +is very challenging for the case of Cu. In Fig. S4 we illus- +trate how the energy range of the transitions observed in +X rapidly increases with respect to the number of bands +included in the calculation. +The increasing number of +bands results in changes in the details of the frequency +structure of X, whose description requires, in the FF-RA +scheme, a large number of frequency points. +On the other hand, there is a complex relationship be- +tween the number of bands and the plane-wave energy + +4 +0 +2 +4 +6 +1.2 +1.0 +0.8 +0.6 +0.4 +0.2 +0.0 +0.2 +Re[Y] +0 +2 +4 +6 +(Ha) +1.2 +1.0 +0.8 +0.6 +0.4 +0.2 +0.0 +Im[Y] +MPA, Drude +MPA, CA +MPA, nD +FIG. S3. Frequency dependence of Y (q = 0) computed for Cu +with three different methods: without any intra-band correc- +tion (nD), with the CA correction and with the Drude model +using as input the intra-band frequency determined in Ref. [6]. +cut-off used to build the polarizability matrix. As shown +in Fig. S5, there is a change of monotony around 15 Ry +when increasing the number of bands from 200 to 500 +and smaller oscillation continue at higher cut-off ener- +gies, hindering the possibility to extrapolate the con- +verged QPs. For cut-off energies and number of bands +larger than 25 Ry and 500 respectively, the correction +changes sign, correcting the DFT in the right direction +with respect to the experimental data shown in the Table. +II of the main manuscript. +[1] G. Stefanucci and R. van Leeuwen, Nonequilibrium Many- +Body Theory of Quantum Systems: A Modern Introduc- +tion (Cambridge University Press, 2013). +[2] R. M. Martin, L. Reining, and D. M. Ceperley, Interacting +Electrons (Cambridge University Press, Cambridge, 2016). +[3] K.-H. Lee and K. J. Chang, Phys. Rev. B 49, 2362 (1994). +[4] D. A. Leon, C. Cardoso, T. Chiarotti, D. Varsano, E. Moli- +nari, and A. Ferretti, Phys. Rev. B 104, 115157 (2021). +[5] A. Marini, G. Onida, +and R. D. Sole, Phys. Rev. Lett. +88, 016403 (2002). +[6] A. Marini, G. Onida, and R. Del Sole, Phys. Rev. B 64, +195125 (2001). + +5 +0 +1 +2 +3 +2 +1 +0 +Re[X] +× a.u. +0 +1 +2 +3 +0.50 +0.25 +0.00 +× a.u. +0 +1 +2 +3 +0.1 +0.0 +× a.u. +Cu: +200 +500 +0 +10 +20 +30 +(Ha) +2 +1 +0 +Im[X] +× a.u. +0 +10 +20 +30 +(Ha) +0.6 +0.4 +0.2 +0.0 +× a.u. +0 +10 +20 +30 +(Ha) +0.10 +0.05 +0.00 +0.05 +× a.u. +FIG. S4. Selected Cu X matrix elements computed within FF with different number of bands (200 and 500). In order to show +both, the overall and the detailed behavior of the polarizability, we plotted the imaginary part of X in the bottom panels in +the full frequency interval determined by the number of bands, while the real part of X are plotted in the top panels in a +zoomed region. A similar scheme from Ref. [4] is used for the units of X, where we omitted the specific scales in order to avoid +confusion with the zoomed plots. +0 +10 +20 +30 +40 +50 +60 +Energy cut-off (Ry) +0.10 +0.05 +0.00 +0.05 +0.10 +0.15 +E +1 (eV) +200b +300b +400b +500b +800b +1000b +FIG. S5. G0W0 correction to the PBE quasi-particle energy of the Γ1 state of Cu. Convergence with respect to the number +of bands and the cut-off energy parameters used to build the polarizability matrix, X. The curves correspond to calculations +with different fixed number of bands. + diff --git a/dtE0T4oBgHgl3EQfWwBP/content/tmp_files/load_file.txt b/dtE0T4oBgHgl3EQfWwBP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..054158462b83ec6e3b7c9254a0620714c3125c1c --- /dev/null +++ b/dtE0T4oBgHgl3EQfWwBP/content/tmp_files/load_file.txt @@ -0,0 +1,2028 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf,len=2027 +page_content='Efficient full frequency GW for metals using a multipole approach for the dielectric screening Dario A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Leon1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='∗ Andrea Ferretti2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Daniele Varsano2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Elisa Molinari1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' and Claudia Cardoso2 1 FIM Department,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' University of Modena & Reggio Emilia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 41125,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Modena (Italy) 2S3 Centre,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Istituto Nanoscienze,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' CNR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 41125,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Modena (Italy) and 3Department of Mechanical Engineering and Technology Management,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Norwegian University of Life Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 1430,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' ˚ As (Norway) The properties of metallic systems with important and structured excitations at low energies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' such as Cu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' are challenging to describe with simple models like the plasmon pole approximation (PPA),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' and more accurate and sometimes prohibitive full frequency approaches are usually required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In this paper we propose a numerical approach to GW calculations on metals that takes into account the frequency dependence of the screening via the multipole approximation (MPA), an accurate and efficient alternative to current full-frequency methods that was recently developed and validated for semiconductors and overcomes several limitations of PPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' We now demonstrate that MPA can be successfully extended to metallic systems by optimizing the frequency sampling for this class of materials and introducing a simple method to include the q → 0 limit of the intra-band contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The good agreement between MPA and full frequency results for the calculations of quasi-particle energies, polarizability, self-energy and spectral functions in different metallic systems confirms the accuracy and computational efficiency of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Finally, we discuss the physical interpretation of the MPA poles through a comparison with experimental electron energy loss spectra for Cu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' INTRODUCTION Many-body perturbation theory provides accurate methods to study the spectroscopic properties of con- densed matter systems from first principles [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Cal- culations often adopt the so-called GW approxima- tion [2, 4–8], for which the frequency integration in the evaluation of the self-energy is crucial to the deploy- ment of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The frequency dependence of the screened potential, W, is often described within the plas- mon pole approximation (PPA) [9–14], successfully ap- plied to the calculation of quasi-particle energies of semi- conductors [9], the homogeneous electron gas [15] and simple metals as Al and Na [16–19], especially for quasi- particles with energies close to the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' However, the description of the self-energy and the spectral func- tions for the whole range of frequencies is still challenging and requires expensive full frequency (FF) approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Despite its success, the use of PPA is problematic when complex metals are concerned, even for the calculations of quasi-particle energies [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Its applicability for tran- sition and noble metals has often been disputed [6, 20], since the approximation is based on the homogeneous electron gas, for which PPA becomes exact in the long wave-length limit [4, 21, 22], while it is in principle not strictly valid in the presence of strongly localized d-bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In fact, these metals present complex screen- ing effects due to collective excitations [23, 24], which result in highly structured energy-loss spectra whose de- scription is unattainable with a single plasmon peak [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Moreover, metals with relevant excitations at low ener- gies, such as Cu, require a specially accurate description ∗ dario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='alejandro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='leon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='valido@nmbu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='no of the low frequency regime, which makes it difficult to determine the PPA parameters since it requires sampling the polarizability at zero frequency [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In this context, we have recently developed a multipole approach (MPA) that naturally bridges from PPA to FF treatments of the GW self-energy [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The method has been implemented in the yambo code [26, 27] and was validated for bulk semiconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' We have shown that, for semiconductors, MPA attains an accuracy compara- ble to that of FF methods at a much lower computational cost, while also circumventing several of the PPA short- comings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Here we extend the assessment of MPA valid- ity and performance to the case of metals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' We do so by computing quasi-particle energies, together with the full frequency dependence of the self-energy and the spectral function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The approach is similar to the one used for semiconductors [25], with only slight changes in the fre- quency sampling strategy used in the multipole interpo- lation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In the following, we show that MPA is accurate for metallic systems, even in cases in which the use of PPA is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In addition to MPA, we also propose a simple ab-initio method to include intra-band contri- butions [28–32] to the dielectric function in the q → 0 limit, absent in semiconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Despite its virtually zero computational cost, it significantly accelerates the convergence of quasi-particle energies with respect to the k-points grid, in systems where the intra-band contribu- tions are dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The paper is organized as follows: In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' II, we briefly summarize the GW approximation and the MPA ap- proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In the same Section, we further extend the strategy used in the frequency sampling for the multipole interpolation, with respect to the MPA implementation presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' [25] for semiconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' We also dis- cuss the relevance of the inclusion of the intra-band con- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='02282v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='mtrl-sci] 5 Jan 2023 2 tribution to the dielectric function in the limit q → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' III we first present MPA calculations for simple met- als and propose a simple way of including the aforemen- tioned intra-band limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' We then describe in detail the results obtained for Cu, a prototype challenging system for PPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Finally, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' IV we summarize and discuss the main conclusions of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' METHODS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Quasi-particle energies within GW We adopt the GW approximation [2, 4–8] for the eval- uation of the electron-electron self-energy, which is com- puted via a frequency convolution of the one-particle Green’s function G(ω) and the dynamical screened in- teraction potential W(ω): ΣGW (ω) = i 2π � +∞ −∞ dω′e−iω′ηG(ω − ω′)W(ω′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' (1) In the present work we limit ourselves to the G0W0 approximation, although MPA, the method we want to discuss here, can be exploited also within more advanced approaches such as different self-consistent GW schemes [33–39], or methods including vertex- corrections [35, 40–43] and cumulant expansions [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' A more comprehensive discussion of these aspects can be found e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The present implementation uses as a starting point single-particle energies and wave- functions computed within Kohn-Sham (KS) density fun- tional theory (DFT) to then build the non-interacting single-particle Green’s function G0(ω) and the irreducible polarizability, X0(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The dressed polarizability, X(ω), and the screened in- teraction, W(ω), are then numerically evaluated by solv- ing the Dyson equation for each given frequency: X(ω) = X0(ω) + X0(ω)vX(ω) (2) W(ω) = ε−1(ω)v = v + vX(ω)v, where v is the bare Coulomb potential, ε the dielectric function and, for simplicity, we have omitted the spatial, non-local, degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' All the quantities have to be thought as frequency dependent operators or ma- trices of the form X(ω) = X(r, r′, ω), or, when using a plane-wave basis set, XGG′(q, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The quasi-particle (QP) energies ϵQP m are then computed either by numeri- cally solving the exact QP equation, ϵQP m = ϵKS m + ⟨ψKS m |Σ(ϵQP m ) − vKS xc |ψKS m ⟩, (3) or its linearized form: ϵQP m ≈ ϵKS m + Zm⟨ψKS|Σ(ϵKS m ) − vKS xc |ψKS m ⟩, (4) with the renormalization factors Zm given by Zm = � 1 − ⟨ψKS m |∂Σ(ω) ∂ω ���� ω=ϵKS m |ψKS m ⟩ �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' (5) In the above equations we have made reference to the Kohn-Sham eigelvaues and eigenvectors, ϵKS m and |ψKS m ⟩, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' A key quantity in the above formulation is the dy- namical part of the inverse dielectric function, Y ≡ ε−1 − I = vX, which determines the correlation part of W, Wc ≡ W − v = Y v, and, through Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' (1), the cor- relation part of the self-energy, Σc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' With the purpose of avoiding the expensive numerical evaluation of the fre- quency convolution in Σc, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' (1), as required e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' by full frequency real axis (FF-RA) approaches [20, 45] or contour deformation (FF-CD) techniques [34, 46, 47], Y or X have been the target of several analytical simplifica- tions like the plasmon pole approximation (PPA) [9–13] or the multipole approach (MPA) [25], briefly sketched below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The multipole approach The multipole approximation is inspired by the Lehmann representation of the polarizability X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' At the independent particle level, X (equal to X0) is written in a compact way as a sum of poles with vanishing imag- inary part corresponding to all possible single particle transitions (here considered at the Kohn-Sham level for simplicity) of energy ΩKS and probability amplitude RKS: X0(ω) = NT � n 2RKS n ΩKS n ω2 − (ΩKS n )2 , (6) where Re[ΩKS n ] is positive defined and Im[ΩKS n ] → 0− to ensure the correct time ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The sum is truncated at a finite number of transitions (NT ) determined by the number of bands included in the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The MPA approach provides an analytic continuation for the dressed polarizability X to the complex frequency plane, z ≡ ω + iϖ, by representing it as a sum of a few complex poles np (usually of the order of 10 to 15), as XMP(z) = np � n 2RnΩn z2 − Ω2n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' (7) Note that this representation is applied to each matrix element in reciprocal space, XMP GG′(q, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' By considering Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' (7) and the Lehmann representa- tion for G0, the correlation part of the GW self-energy is then integrated analytically and reads: ΣMP c (ω) = NB � m np � n PmvRn � fm ω − Em + Ωn − iη + + (1 − fm) ω − Em − Ωn + iη � v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' (8) where Pm are projectors over KS states, Em their eigenenergies, and fm their occupations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The sum-over- states is truncated at the maximum number of bands, 3 NB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' This expression generalises the PPA solution to the case of a multipole expansion for X(z), and bridges be- tween PPA and an exact full-frequency approach by in- creasing the number of poles in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' More details about this procedure can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' MPA sampling for metals The poles and residues in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' (7) are obtained by nu- merically evaluating X for a number of frequencies equal to twice the number of poles and solving the resulting sys- tem of equations (see details in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' [25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Since the num- ber of poles used in the MPA model, np, is much smaller than the total number of electron-hole transitions of the target polarizability, NT , the representation, and there- fore the efficiency of the method, depends critically on the frequency sampling used in the interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' For semi- conductors, the so-called double parallel sampling proved to be the most robust and accurate with respect to FF calculations, with the fastest convergence with respect to the number of poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' It runs along two parallel lines above the real axis: sDP = � z1: z1 n = ωn + iϖ1 z2: z2 n = ωn + iϖ2, n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='., np (9) The first of the two branches is closer to the real axis (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' with ϖ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='1 Ha), except for the first point, set exactly at the origin of coordinates, z1 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The second branch is located further away, typically at ϖ2 = 1 Ha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In a simplified view, X sampled along the first line preserves some of the structure of X in a region close to its poles, while X sampled along the second line is simple enough to be described with a few poles, and accounts for the overall structure of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' A more detailed description can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In order to obtain a numerically stable and effective sampling for metals we found that, at variance with the semiconductor case [25], a small shift of the z1 1 point (in the origin) along the imaginary axis is needed, resulting in z1 1 = iϖ1, where ϖ1 = 10−5 Ha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The shift is done in order to avoid numerical instabilities due to intra-band transitions with energies close to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' This is similar to the PPA implementation for metals [26, 27], which adopts a 10−8 Ha shift, but in this case along the positive real axis instead of the imaginary axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' A second difference with respect the strategy used for semiconductors concerns the distribution of the fre- quency sampling of X along the real axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' For semi- conductors [25], the frequency sampling is done in non- uniform grids, in particular, a semi-homogeneous parti- tion in powers of 2 that ranges from 0 to ωm, called linear partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Here, we generalize it to any possible exponent α: {ωn}α : � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � (0) , np = 1 (0, 1) × ωm, np = 2 � 0, 1 2, 1 �α × ωm, np = 3 � 0, 1 4, 1 2, 1 �α × ωm, np = 4 � 0, 1 8, 1 4, 1 2, 1 �α × ωm, np = 5 � 0, 1 8, 1 4, 1 2, 3 4, 1 �α × ωm, np = 6 � 0, 1 8, 1 4, 3 8, 1 2, 3 4, 1 �α × ωm, np = 7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' (10) The distribution described on Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' [25] corresponds to α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' As discussed below, there are cases (see for exam- ple the case of copper in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 4), in which X presents a more complex structure at low frequencies and therefore a denser sampling grid in that region is convenient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The distribution corresponding to α = 2 concentrates more points at low frequencies than the linear case, α = 1, and permits to increase the accuracy of the X descrip- tion without changing the frequency range, ωm, or in- crease the number of poles used in MPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In this work, we adopt a quadratic partition, corresponding to α = 2, for Al and Cu, and a linear one, α = 1, for Na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Intra-band contributions Despite the success of the GW approximation, systems with metallic screening present specific methodological challenges, one being the inclusion of intra-band transi- tions [31, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Specifically, for partially filled bands, there is a non-vanishing probability that an electron is excited within the same band, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' within states with quantum numbers k, n and k − q, m, with n = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Notably, these transitions play an important role, for example, in noble metals [20, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Both inter- and intra-band transitions contribute to the irreducible polarizability as defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' However, the energy of the pole corresponding to intra-band transitions decreases with q until it van- ishes in the q → 0 limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Despite this behaviour, the contribution to the inverse dielectric function in the case of bulk metals is still finite, due to the divergence of the Coulomb potential, which makes Y = vX not vanish- ing for q → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' For this reason, in the case of metals it is important to properly take this term into account, since it cannot be simply evaluated as in the case of the inter-band contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In principle, it is possible to decrease the weight of the q = 0 element, that contains only inter-band terms, by systematically increasing the number of k-points in 4 the Brillouin zone (BZ) sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' However, the con- tributions from the Fermi surface can dramatically slow down the convergence with respect to the k-space sam- pling [29], resulting in spurious gaps at the Fermi level that vanish very slowly with increasing number of k- points [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Several approaches to include the intra- band limit have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The ones based on ex- plicit Fermi-surface integration [28, 30, 31] are, as ex- plained above, computationally expensive since they re- quire dense k-grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Alternatively, analytical models based on a Taylor expansion of the dielectric function in the small-q region, avoiding explicit Fermi-surface calcu- lations, are able to remove the spurious gap at the Fermi level with a limited number of k-points [32, 50, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Nev- ertheless, some of them may depend on ad hoc external parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' A common approach to include the missing intra-band contribution relies on the use of a phenomenological Drude-like term added to the head of the irreducible dielectric matrix in the q → 0 limit, YG=G′=0(q = 0, ω) [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In the long-wavelength limit, q → 0, the Drude term for the independent particle dielectric function can be written in the form [24, 28, 30, 52, 53] YD(ω) = ω2 D ω(ω + iγ) + O[q2], (11) where the Drude frequency, ωD (see Table II), is an input parameter of the model and the relaxation frequency γ is usually a free parameter set typically to γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='1 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In principle ωD can be determined fully ab-initio, resorting to very dense k-point grids [20, 30] or to an interpolation of the BZ, for instance with Wannier functions [54–57] or the tetrahedron method [29, 30, 39, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Alternatively, experimental values can also be used when available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In the next Sections we will discuss the possibility to extrapolate a complex plasmon frequency (see Ta- ble II) in the q → 0 limit from the frequency structure of Y (q, ω) at finite q, which in general is a superposi- tion of intra- and inter-band contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In a second step, we will use a f-sum rule [24] in the same spirit of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' [30], in order to estimate the intra-band contribu- tion to the plasmon frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' We will also propose a simple and virtually zero-cost method to include an ap- proximate treatment of the missing intra-band limit from first-principles, without the need to resort to any add-on model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' RESULTS AND DISCUSSION In the following, we present the results for three bulk metallic systems highlighting different issues arising when applying the GW approach to metals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' We start by studying the case of two simple metals, Al and Na (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' [59–61] for a description of their band structures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Next, we focus our attention on Cu, a more challenging system whose electronic structure has been DFT-PBE GW-PPA GW-MPA Al Γ1 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='12 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='79 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='94 Γ25′ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='71 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='30 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='48 X4′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='93 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='91 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='86 W3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='82 Na Γ1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='27 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='85 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='97 Γ25′ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='76 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='19 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='81 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Al and Na quasi-particle energies (eV) with respect the Fermi level computed within DFT-PBE, GW-PPA, and GW-MPA using a 16 × 16 × 16 k-grid including the q → 0 intra-band contribution through the CA method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' thoroughly studied, both experimentally [62, 63] and the- oretically [20, 64–66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The use of PPA for Cu has been shown to be problematic [20] and, for this reason, copper is not only an important test case for the application of MPA and the description of intra-band effects, but also provides a better understanding of the applicability of PPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' As a starting point for our GW simulations, we use DFT calculations performed at the PBE [67] level using scalar-relativistic optimized norm-conserving Vanderbilt pseudopotentials [68], as implemented in the Quantum ESPRESSO package [69, 70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The kinetic energy cut-off is set to 100, 70, and 150 Ry for Al, Na, and Cu, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The k-grids were determined by the convergence requirements of the GW calculations, considering, in par- ticular, the specific treatment of the intra-band limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' When reporting quasi-particle energies, we use k-point grids of 16 × 16 × 16 for Al and Na, and 12 × 12 × 12 for Cu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Moreover, the GW correction to the Fermi level is linearly interpolated from the corresponding correc- tions to the closer quasi-particles present in the specific k-mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The DFT results are in good agreement with previ- ous results obtained with the same method [65], and in reasonable agreement with the results reported for Cu in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' [20], performed using LDA [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In fact, the GW re- sults for Cu have shown to be very sensitive to the choice of the DFT starting point [65], though we will not address this point here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The GW calculations were done using the yambo [26, 27] code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The numerical convergence of the GW results has been checked with care, and the re- sulting parameters, being system dependent, are detailed in the sections below when discussing the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' MPA for simple metals We start by computing quasi-particle energies of Al and Na using MPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Here the frequency dependence of the polarizability presents a structure with mainly one strong plasmon peak, similar to that of silicon computed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' As expected, the double parallel sampling en- sures convergence with a similar number of poles, np = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 5 80 60 40 20 0 20 40 60 Re[ ] (eV) Al 40 30 20 10 0 10 Na 25 20 15 10 5 0 5 EKS (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='8 Im[G] (eV 1) EKS W10 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='88 eV EKS X10 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='96 eV EKS 1 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='20 eV 15 10 5 0 EKS (eV) 0 1 2 3 4 5 6 7 EKS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='01 eV (nD) EKS = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='27 eV (nD) EKS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='01 eV EKS = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='27 eV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='5 2 0 2 1 0 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='5 b) c) d) a) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Frequency dependence of the real part of the self-energy (panels a) and c)) and spectral function (panels b) and d)) computed with MPA for three quasi-particles of Al (panels a) and b)) and two of Na (panels c) and d)), including the intra-band limit using CA (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In the case of Na, we also show the corresponding curves without any intra-band correction (nD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The present results were obtained considering 300 bands for both X and Σ and an energy cut-off for X of 20 and 15 Ry for Al and Na respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In Table I we report the quasi-particle energies for Al and Na, including Γ1 (the lowest QP peak at Γ, corre- sponding to the valence bandwidth) and other reference quasi-particles, computed using PPA and MPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' MPA QPs are generally in very good agreement with FF val- ues from the literature (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' [19] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' According to our calculations, the computed quasi-particles values for Al and Na with MPA are esti- mate to differ by less than 8 meV from the correspond- ing FF-RA results (comparison done using 10 Ry cutoff to represent X0 for both MPA and FF-RA), as found for semiconductors [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Instead, PPA QPs show devia- tions that are systematically larger for states further from Fermi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Previous GW calculations for Al and Na [19] have shown that PPA describes well the tail of the self-energy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' the frequency region around the Kohn-Sham ener- gies, and gives reasonable QP solutions for both Al and Na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' However, if we consider the whole frequency range, the agreement between PPA and FF-CD is less satisfac- tory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' PPA shows sharp fluctuations in the self-energy and spectral functions, that result in several spurious so- lutions of the quasi-particle equation, evidenced by mul- tiple small peaks in the spectral function (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 4 of Ref [19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 1 we show the self-energy and spectral function for Al and Na, this time computed with MPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The comparison with results obtained within FF-CD [19] shows that MPA not only describes well the tail of the X(ω) and Σ(ω) functions, but also correctly describes the positions of the peaks and their relative intensities in the whole frequency range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In the left panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 1 we focus on Al and plot, as a function of the frequency, the MPA self- energy, ⟨ψmk|Σ(ω)|ψmk⟩, and the spectral function, ⟨ψmk|Im[G(ω)]|ψmk⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' These quantities have been pro- jected on three Al states, one corresponding to the bot- tom of the valence band at Γ and two other Kohn-Sham states closer to the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Comparing the three self-energy functions, there is a more effective pole su- perposition for states at energies further away from the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Indeed, for the lowest energy state with EKS = −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='2 eV, this leads to a frequency dependence of Σ with an intense single pole (at about -15 eV with respect to EKS) and consequently a very broad and shal- low QP peak in the corresponding spectral function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' At the same time the satellite structure is enhanced to the point of becoming a second peak, originating from a sec- ond solution of the quasi-particle equation (intersections of the dashed line with the self-energy function in the upper panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' This scenario is consistent with the so- called ”plasmaron” peak, a sharp satellite feature emerg- ing as an artefact of the G0W0 approximation to the self- energy [2, 44, 71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The situation is similar for the two QPs computed for Na shown in the rights panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 1, with the lowest state presenting again two solutions for the QP equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 6 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Analysis of the intra-band contribution In common GW implementations, especially those tar- geting semiconductors, the intra-band contribution to the dielectric function in the q → 0 limit, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' (6), is often not included, as explained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' II D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In the case of Al, where a substantial part of the Fermi surface is very close to the BZ boundary, one can expect [32] that many of the metallic contributions are effectively inter- rather than intra-band terms, resulting in a small error when the intra-band term is neglected [32], while for Na the intra-band terms are found to be more relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' For both Al and Na, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 2 we show how this af- fects the frequency dependence of the YG=G’=0 matrix elements computed for different q-vectors along an arbi- trary direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The curves in green shades correspond to Y (ω) computed for finite but small q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The orange curve corresponds to the q → 0 limit evaluated only for the inter-band term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' There are two main differences be- tween the green and orange curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The first difference is the limit of Re[Y ] as the frequency tends to zero (static limit), that evolves smoothly for finite q but in general tends to a value different from the one corresponding to q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' As shown in the insets of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 2, the smallest fi- nite q provides a static limit very similar to the value for q = 0 in the case of Al, while it is considerably larger in the case of Na (both results in agreement with previous studies [32]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' This difference has been commonly used as a measure of the missing intra-band term [19, 32], since for metals in the limit q → 0, ε−1 G=G′=0(q, ω = 0) vanishes, meaning that Y00(q, ω = 0) → −1, as apparent from the progres- sion of the curves with finite q, that include intra-band transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In fact, in the independent particle picture, the q → 0 limit of Re[Y ] at ω = 0 is related to a non- vanishing probability of vertical transitions within the same band [30], and can therefore be used to estimate a Drude frequency [74, 75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' However, this probability alone does not determine the plasmon frequency (see Table II for a summary of the nomenclature) or the position of the pole of Re[Y ] for q → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In fact, the second difference between the orange (q = 0, no intra-band contribution) and the green curves (fi- nite q, intra-band included) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 2 is the position of the main pole of Y (ω), here called Ωp, or in the case of Na, to the apparent absence of poles for q = 0, whose small amplitudes cannot be seen in the plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' If the whole frequency range is considered, we see that the behaviour of Re[Y (ω → 0)] depends on the position of Ωp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Follow- ing the green curves at finite q, it is clear that YG=G′=0 for both Al and Na change smoothly with q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The curves present a pole, Ωp(q), of decreasing energy and increasing amplitude, just above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='5 Ha for Al and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='2 Ha for Na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 2(e), both the real and imaginary part of this pole can be easily extrapolated to q = 0, by means of the Lindhard plasmon dispersion [30, 32, 72, 73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In the same plot we show, as a reference, the Drude frequency corresponding to the q → 0 limit of the intra- band contributions, ωA (see Table II), as computed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' [19] for Al and Na, in addition to the experimental plasmon frequency ωp of Al [53, 72, 76–79] and Na [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In the simulations we can also extrapolate, already with a 8×8×8 k-point mesh, the plasmon frequency at q → 0 from the position of the main structure of the response functions, namely ωp ≡ Re[Ωp].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' This procedure provides ωp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='55 Ha (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='01 eV) for Al, in excellent agreement with the experimental value of 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='0 eV [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Similarly, the value extrapolated for Na, ωp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='21 Ha (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='79 eV), matches very well the experimental value of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='9 eV [72] and both compare well with the Drude intra-band fre- quency computed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' [19] (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='18 eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The small dif- ference between our theoretical result and Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' [19] can be attributed to methodological differences (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=', the DFT functional on top of which the GW calculations are per- formed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In contrast, the difference between ωp and ωA for Al is larger than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='5 eV since the plasmon fre- quency ωp has non-negligible contributions from both intra- and inter-band transitions, as previously reported in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' [30, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Note however that the inter-band con- tributions are not included in the Drude frequency com- puted in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In order to discriminate between the intra- and inter- band contributions to the plasmon frequency, we have used a simple expression based on the f-sum rule [2, 30, 80], but separating the two contributions: Ω2 A = lim q→0 2 π � ∞ 0 dω ω Im[Y (q, ω) − YE(q, ω)], (12) where YE corresponds to inter-band transitions only, while Y accounts for the complete response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Within MPA the integral is solved analytically (derivation in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' I of the Supplemental Material [81]), leading to: Ω2 A = 2v(RpΩp − REΩE), (13) where ΩE and vRE are the position and the residue of the most relevant pole of YE(q = 0), while Ωp and vRp are the corresponding values for Y (q = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In principle, the product vRpΩp should be computed in the q → 0 limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' We have instead considered the extrap- olation of Ω2 p, which is equivalent in our model (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' I in the Supplemental Material [81]) and significantly more stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The values of vREΩE are taken directly from the calculation at q = 0 (orange curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 2a,b), since no intra-band transitions are considered, as explained above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' For Al, the real part of ΩE is ωE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='37 Ha (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='08 eV) and thus, applying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' (13), the real part of the intra- band pole ΩA is ωA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='43 Ha (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='72 eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' For Na, ωp and ωA are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The comparison of ωp and ωA confirms that the experimental plasmon frequency, ωp, in the case of Na corresponds mainly to intra-band contributions, while for Al there is an important inter-band contribu- tion [53], and its use as a Drude intra-band frequency would result in an overestimation of the actual ωA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Making use of the extrapolation procedures described above in the context of the MPA framework, and of a 7 10 5 0 5 10 Re[Y] Al 10 0 10 Na 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='5 (Ha) 20 15 10 5 0 Im[Y] q4 q3 q2 q1 q0 p A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='5 (Ha) 30 20 10 0 0 Na 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='5 (Ha) 20 15 10 5 0 Im[Y] p p A q4 q3 q2 q1 q0 p 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='8 10 5 0 5 10 Re[Y] Al 10 0 10 Na 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='5 (Ha) 20 15 10 5 0 Im[Y] q4 q3 q2 q1 q0 p A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='5 (Ha) 30 20 10 0 q4 q3 q2 q1 q0 p 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='8 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='5 10 5 0 5 10 Re[Y] Al 10 0 10 Na 15 10 5 0 Im[Y] q4 q3 q2 q1 q0 20 10 0 q4 q3 q2 q1 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='8 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='5 10 5 0 5 10 Re[Y] Al 10 0 10 Na 15 10 5 0 Im[Y] q4 q3 q2 q1 q0 20 10 0 q4 q3 q2 q1 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='8 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='5 10 5 0 5 10 Re[Y] Al 10 0 10 Na 15 10 5 0 Im[Y] q4 q3 q2 q1 q0 20 10 0 q4 q3 q2 q1 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='8 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='5 10 0 10 Na 4 3 2 1 0 20 10 0 q4 q3 q2 q1 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='5 a) c) e) b) d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='5 q (inv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' crystal coord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=') 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='8 p (Ha) exp Al p [19] Al A [19] Al A exp Na p Na A Re Al p Re Na p Im Al p Im Na p FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Frequency dependence of YG=G′=0 matrix elements computed with MPA for different q vectors of modulus q ≡ |q| tending to 0, a) and b) for Al, and c) and d) for Na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' For q = 0 (orange curves) the intra-band transitions are not included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The insets in panels a) and c) show the region around ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Panel e) shows the q dispersion of the real and the imaginary parts of the main pole of Y for q0–q4 (qn = n 8 in units of 2π/a, being a the respective Al and Na lattice parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The solid lines show the corresponding parabolic fits consistent with a Lindhard (bulk) plasmon dispersion [30, 32, 72, 73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The black dashed lines correspond to the experimental plasmon frequency of Al [30] and Na [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Dash-dot purple lines correspond to the values of the intra-band frequency, ωA, computed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' [19] using the method described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' [50], while the violet one corresponds to our estimate for Al, computed by means of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Contribution Pole (complex) Frequency (real) intra-band ΩA ωA = Re[ΩA] inter-band ΩE ωE = Re[ΩE] plasmon(intra+inter) Ωp ωp = Re[Ωp] plasma ωpl = √4πρe Drude(model) ωD + iγ ωD TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Summary of the notation concerning frequency related quantities introduced in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The plasma fre- quency is defined in terms of the electronic density, ρe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The Drude pole/frequency are model parameters used to describe the plasmon or only its intra-band contribution, as described in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' simple f-sum rule, it is possible to determine not only the real but also the imaginary part of both the plas- mon and the intra-band pole, usually not considered in other ab-initio methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' It is also worth noticing that the extrapolation is done with points from a much coarser k- grid (8 × 8 × 8 for both Al and Na), with respect to the grids required to compute the intra-band frequency with an independent particle formulation [31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Despite the limited accuracy of the computed imagi- nary values, they are meaningful and provide a qualita- tive understanding of how intra- and inter-band terms, linearly summed at the independent particle level, are combined after the inversion of the Dyson equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' While the Na case is trivial, since the inter-band con- tribution is negligible, in the case of Al the small differ- ence between ωA and ωE, comparable to their imaginary parts, explains the presence of a single pole in Y (ω) lo- cated roughly at ω2 p ∼ ω2 A+ω2 E (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' I of Supplemental Material [81]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Modelling of the intra-band limit Our analysis of the dressed response function Y (ω) suggests that an alternative to the direct evaluation of the intra-band limit, usually determined from X at the independent particle level [31], can be obtained, either by (1) including a complex Drude pole YD(ω), accord- ing to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' (11), in the head (G = G′ = 0) of the in- dependent particle dielectric function, with the Drude frequency given by the computed intra-band pole;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' or (2) approximating the full Y (q = 0) matrix element by its nearest neighbour Y (q ̸= 0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' with the q-vector clos- est to 0 according to the adopted k-point grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 8 The first method builds on using an estimate of the Drude intra-band frequency, similar to the extrapolations used in [75], but here considering the whole frequency range and both intra- and inter-band contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The second method, which we will call from now on constant dielectric function approximation (CA), assumes that the whole Y (q) matrix is constant in a small region around q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' This approach is inspired by the leading term of the Taylor expansion for small q of the Thomas-Fermi distribution, and is corroborated by the small difference of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='006 Ha (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='17 eV) found for both, Al and Na, be- tween the extrapolated value of Ωp and its value at the first finite q, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 2 e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Both methods simul- taneously correct the position of the plasmon pole and the limit of YG=G′=0 for ω = 0 and add virtually no computational cost to the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In addition, CA also corrects other matrix elements for which the intra- band limit may be important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' II of Supplemental Material [81] we report plots similar to the ones in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 2 for Y matrix ele- ments of Na other than the head, showing that after the head (YG=G′=0), intra-band contributions are rele- vant also for the so-called wing elements (YG=0̸=G′ and YG̸=0=G′), while less important for the diagonal elements (YG=G′̸=0), specially at increasing |G|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' For finite |G| the evolution of the Y (q) matrix elements when q → 0 is less smooth and the position of the poles does not al- ways change monotonously, meaning that an extrapola- tion would require a denser k-point grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Even if the con- stant dielectric function approximation has limited accu- racy for some of these matrix elements, it still provides a significant overall improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In particular in ma- terials such as Cu, as discussed below, the CA method presents some clear advantages regarding the estimation of ωA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' To assess the effect of this approximation in the QP solution, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 3 we show Al and Na QP energies com- puted without (nD) and with (CA) intra-band correc- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' When the number of k-points is increased, the weight of the Y (q = 0) element in the self-energy de- creases and both methods eventually converge to the same quasi-particle values, but only very slowly, as dis- cussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In fact, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 3 shows that for two selected QPs of Na the intra-band term is fundamental due to the importance of this contribution to the screening prop- erties of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In contrast, for Al the difference is small, and the convergence is governed by the inter- rather than intra-band contributions for all the 4 QPs considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In the bottom panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 3 one can see the significant acceleration introduced by CA in the con- vergence of the bandwidth of Na, where, besides a small oscillation in the 20×20×20 grid (caused by oscillations in the DFT eigenvalues), the first point corresponding to the 8×8×8 mesh already provides very accurate results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In the CA scheme the convergence benefits simultane- ously from the decrease of the weight of the Y (q = 0) contribution and from the fact that the correction itself improves for denser grids in reciprocal space, since the 0 100 200 300 400 QPCA QPnD (meV) Na: 25′ Na: 1 Al: 25′ Al: 1 Al: X4′ Al: W3 0 2500 5000 7500 10000 12500 k-points 100 0 100 200 300 GW correction to 1 (meV) Na (nD) Na (CA) Al (nD) Al (CA) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Top panel: Difference between GW-MPA corrections computed with the (CA) intra-band term and without (nD) as a function of the number of k-points, for 2 quasi-particles of Na (light and dark yellow) and 4 of Al (green shades).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Bottom panel: Convergence of the GW correction for the QP at Γ1 of Na (yellow) and Al (green) with CA (solid) and nD (dashed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' first q ̸= 0 is closer to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 1 we show the frequency dependence of the real part of the self-energy (top) and spectral function (bot- tom) computed for two quasi-particles of Na, within MPA with and without the intra-band correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The correc- tion does not change dramatically the shape of the self- energy, but introduces an extra pole in the real part of the self-energy at the intra-band frequency (∼-6 eV) and renormalizes the peaks of the spectral function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The in- clusion of this term promotes the pole overlapping around the plasmon frequency, affecting the tail of the self-energy and thus the QP solution as illustrated in the insets of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 1, differently for each quasi-particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In the case of Al and Na, the QP energies computed with the Drude model, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' (11) using as input ωD = ωA, and the CA schemes are very similar, with differences below 20 meV when using the 8 × 8 × 8 k-grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' This leads us to conclude that the CA scheme could replace the 9 usual Drude correction, replacing a semiempirical scheme by a simple ab-initio approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' This is particularly relevant when the Drude intra-band frequency is difficult to estimate either from experiments or calculations, since the CA scheme has virtually zero computational cost and, as the extrapolation presented in the previous Section, describes both the real and the imaginary part of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' To summarize this section, the inclusion of the intra- band limit through the proposed CA scheme requires no extra computational cost with respect to the standard GW calculation and accelerates the k-grid convergence of the QP energies for systems where the intra-band con- tribution dominates, like Na, without resorting to semi- empirical corrections such as the Drude model or com- putationally costly ab-initio approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Frequency representation of the response function of copper As mentioned before, the case of copper presents sev- eral challenges for an accurate GW description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The Cu band structure features a series of flat d-bands around 2 eV below the Fermi level, leading to strong transitions in YG,G′(q, ω) spread over a large energy range [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 4 for q = 0, even for small values of G and G′, YG,G′(q, ω) can behave very differently from a single pole case, hindering the use of PPA but suggest- ing that a multipole approach could prevent resorting to more expensive FF methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' When considering PPA or in general MPA with only a few poles, one of the main issues is that the in- terpolation of X or Y may give rise to non-physical poles, posing representability problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Within the Godby and Needs (GN) PPA scheme implemented in yambo [11, 26, 82, 83], the condition used to identify these so-called unfulfilled modes is the following: Re � YGG′(q, 0) YGG′(q, iϖpl) − 1 � < 0, (14) ϖpl being a frequency on the imaginary axis used to per- form the GN interpolation, typically set to ϖpl = 1 Ha or to a value of the order of the plasma frequency (ϖpl ≳ ωpl), computed from the electronic density, ρe (see Table II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' As an example, for the diagonal elements (G = G′), the polarizability evaluated on the imaginary axis should be real and therefore unfulfilled modes are those for which the resulting pole is instead imaginary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In these cases, the position of the pole is typically set to ΩGN fail = 1 Ha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Setting the pole at ΩGN fail usually works well for simple semiconductors [25, 82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' However, in more complex sys- tems it can compromise the PPA approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In fact, when performing GW calculations using GN-PPA for Cu, we found that no less than 48% of the matrix elements are unfulfilled modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' This means that, for almost half of the matrix elements, the position of the pole is spuriously set to 1 Ha, severely affecting the self-energy and the quasi- particle solution, as shown in the insets of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Within MPA, increasing the number of poles in the description of Y , together with the generalized condition to assign the position of the poles of the unfulfilled modes, as de- scribed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' [25], leads to a significant improvement in the representability of Y , as illustrated in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' III of Supplemental Material [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 4 we compare selected Y matrix elements com- puted within MPA with 1 and 12 poles, with the FF results computed with a frequency grid of 1000 points (all other convergence parameters being the same: k- grid, number of empty bands, etc) At first glance, the en- veloping structure of diagonal elements presents a strong overall peak, as in the case of semiconductors such as Si, hBN, and TiO2, which are well-described within PPA and MPA [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' However, in the case of Cu, there are other important peaks close to the origin not captured by a single-pole model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In this case, PPA quasi-particle en- ergies are not just numerically inaccurate, as in the case of the discussed semiconductors, but PPA becomes an inadequate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Increasing the number of poles from 1 to 12 significantly improves the agreement between Y computed with MPA and FF, reproducing the over- all frequency dependence even if MPA presents a much smoother shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' While the rapid oscillations in the FF response func- tion are enhanced by the discretization of the Brillouin zone, the origin of such fluctuations can be related to the topology of the flat d-bands of Cu [20], consistently e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' with the very structured W(ω) computed for Ni [84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In fact, regardless of the overall simple shape of X, nu- merous inter-band transitions, close in energy and not effectively overlapped, contribute to the fluctuations of the polarizability X and of the inverse dielectric function Y , when computed within FF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Nevertheless, as discussed in the next Section, they do not significantly influence the computed GW quasi-particle energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Quasi-particles and spectral function of copper In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 5 (top panels) we show the frequency depen- dence of the self-energy projected on three selected quasi- particle states of Cu calculated within PPA, MPA, and FF-RA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The details of Σ computed within the FF ap- proach, better appreciated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 5 c), depend on the fine structure of W, which requires a dense frequency grid when computing the polarizability, as shown in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' V of Supplemental Material [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Since these calculations are very expensive, the curves shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 5 were com- puted including 200 bands for all the three methods, and using a frequency grid with 1000 points for FF and no intra-band correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Fully converged MPA results and intra-band corrections are discussed at the end of this Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The FF self-energy presents a rather flat structure with no dominant peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Since Σ is obtained from the con- 10 2 1 0 1 Re[Y] G = G0= 0 ×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='0 2 1 0 1 G = G0 0 ×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='29 2 1 0 1 G G0 0 ×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='07 0 5 10 15 (Ha) 3 2 1 0 Im[Y] ×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='0 Cu: MP1 MP12 FF 0 5 10 15 (Ha) 3 2 1 0 ×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='24 0 5 10 15 (Ha) 2 1 0 1 ×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='07 0 5 10 15 (Ha) 3 2 1 Im[Y] Cu: MP1 MP12 FF 2 1 0 1 [ ] G = G0= 0 2 1 0 1 G = G0 0 2 1 0 1 G G0 0 0 5 10 15 (Ha) 3 2 1 0 [ ] ×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='0 Cu: MP1 MP12 FF ’ ’ ’ a) c) e) b) d) f) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Selected Cu Y (q = 0) matrix elements computed within MPA with 1 and 12 poles compared with the corresponding FF results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The y-axes are scaled with the factors indicated on top of each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 20 0 20 40 Re[ ] (eV) EKS EKS = EKS 30 20 10 0 10 EKS (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='10 Im[G] (eV 1) Cu 40 30 20 10 0 10 EKS (eV) 40 30 20 10 0 10 EKS (eV) 2 0 2 2 0 2 2 0 2 2 0 2 2 0 2 2 0 2 20 0 20 40 Re[ ] (eV) EKS EF pp ff mp EKS = 12 pp ff mp EKS = 1 pp ff mp 0 08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='10 Cu 2 0 2 2 0 2 2 0 2 2 0 2 2 20 0 20 40 Re[ ] (eV) EKS EF pp ff mp EKS = 12 pp ff mp EKS = 1 pp ff mp 40 30 20 10 0 10 EKS (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='10 Im[G] (eV 1) Cu 40 30 20 10 0 10 EKS (eV) 40 30 20 10 0 10 EKS (eV) 2 0 2 2 0 2 2 0 2 2 0 2 2 0 2 2 0 2 EKS EF pp ff mp EKS = 12 pp ff mp EKS = p ff m Cu 2 0 2 2 0 2 2 0 2 2 0 2 EKS = 12 pp ff mp EKS = 1 pp ff mp 2 0 2 2 0 2 2 0 2 2 0 2 EKS = 12 pp ff mp EKS = 1 pp ff mp 2 0 2 2 0 2 2 0 2 2 0 2 2 0 2 0 2 a) c) e) b) d) f) PPA MPA FF FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Frequency dependence of the real part of the self-energy (top) and spectral function (bottom) of three quasi-particle states of Cu: one close to the Fermi energy (panels a) and b)), Γ12 (c) and d)) and Γ1 (e) and f));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' computed with PPA, MPA and FF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 11 QP(eV) DFT/LDA DFT/PBE DFT/PBE GW@LDA GW@PBE GW@PBE Exp Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' [20] Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' [65] (current work) Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' [20] Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' [65] (current work) Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' [62] Γ12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='27 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='18 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='81 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='92 to -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='78 Γ1 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='79 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='29 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='27 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='24 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='14 to -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='20 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='06 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='60 X5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='40 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='33 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='49 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='45 to -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='22 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='39 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='01 L2′ 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='85 L3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='63 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='47 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='63 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='24 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='58 to -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='36 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='57 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='25 L gap 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='40 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='80 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='66 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='76 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='98 to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='09 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='88 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='95 TABLE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' DFT and GW quasi-particle energies of Cu computed with different methodologies by different groups and compared with the experimental values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' All the GW calculations correspond to FF approaches ran on top of LDA [20] and PBE [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' volution of G and W in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' (1), the oscillations of W are attenuated, resulting in a much smoother function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Nevertheless, the convergence of the QP solution is chal- lenging, since it requires an accurate description of the tail of the self-energy, as shown in the insets of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' This could explain, at least in part, the variety of results present in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' GW-PPA data (blue curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 5) show that the quasi-particle solution (insets of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 5) obtained with a single pole model for W deviates from the FF solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Besides the deviations at the tail of Σ, PPA fails to de- scribe the frequency dependence of Σ and the spectral function (bottom panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' On the other hand, the MPA results, here obtained with 12 poles and the quadratic sampling, are very accurate, not only in the tail region, that determines the QP corrections, but also for the whole frequency range of both the self-energy and the spectral function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Comparing the three selected quasi-particle states in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 5, the effect of the overlapping of the independent- particle excitations (due to the inclusion of local field effects via the Dyson equation for W) on the self-energy of Cu is more relevant for Γ1 than for Γ12 and the QPs around the Fermi energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Indeed, as shown in the bot- tom panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 5, for the QPs closer to the Fermi level, the shape of the spectral function has a very narrow quasi-particle peak and three satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' When compared to the QPs close to Fermi, the QPs at deeper energies (Γ12 and Γ1) present a broader quasi-particle peak and more intense satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The shallower satellite (above - 10 eV) forms a shoulder structure for Γ12(central panel) and eventually merges with the QP peak to form a single broader peak for Γ1 (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Despite its complexity, the Cu states at different energies present similar trends as the cases of Al and Na discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' III A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' It is worth to emphasize the importance of the fre- quency sampling in MPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Since copper X and Y present a rich structure at low frequencies, but the energy range, ωm in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' (10) is still large, the quadratic sampling has shown to be more efficient than the linear one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Specifi- cally, it provides, with the same number of poles and the same ωm, a larger density of points in the low frequency region and therefore higher accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The comparison between the computational cost of MPA and the FF-RA method can be done in a simplified way by comparing the number of frequencies for which X is numerically computed in each approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Here, for MPA we use 24 frequency points, corresponding to 12 poles, while the FF-RA frequency grid has 1000 points, corresponding to a 40 times gain in computational efficiency of MPA with respect to FF-RA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The convergence with respect to the number of bands and the size of the X matrices is particularly challeng- ing, as already reported for example for other systems with d states [85–87], with a slow, non-monotone conver- gence that hinders the use of extrapolations (more detail in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' V of Supplemental Material [81]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' For this rea- son, the computational efficiency of MPA is particularly beneficial as it allows for the use of fine GW convergence parameters, thereby increasing the overall accuracy of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In Table III we show the MPA results obtained with 60 Ry of energy cut-off and 1000 bands for both, X and Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' These parameters are comparable to the largest ones used within a static subspace approximation [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The reported MPA quasi-particle energies are in good agree- ment with previous calculations using different FF ap- proaches, and summarized in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The main differ- ences can be explained by the use of different starting points for the GW calculation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' different exchange- correlation functionals and/or pseudopotentials in the DFT ground state, and possibly to an incomplete conver- gence of some of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' While the use of converged parameters is essential when comparing the computed QP energies with experiments, GW corrections do not always improve over DFT/PBE results, as also observed in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' [65, 66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In the present case, GW significantly improves Γ1, while for Γ12 and other QPs, the GW cor- rection is rather small and slightly worsens the DFT re- sults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The localized nature of the d states in Cu may require methods beyond GW in order to further improve the agreement with experiments [88–90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='25 Re[Y] Cu 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='4 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='0 Re[ n] (Ha) estim Cu A Exp 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='25 Re[Y] Cu 0.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='25 Re[Y] Cu 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='0 Re[ n] (Ha) estim Cu A Exp 1 A 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='25 Re[Y] Cu 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='0 Re[ n] (Ha) estim Cu A theo Cu A Exp 1 4 0 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='0 Im[Y] q4 3 0 08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='16 m[ n] (Ha) 1 2 3 A [31] a) c) b) d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Left panels: frequency dependence of the real (a)) and imaginary part (b)) of YG=G′=0 for Cu computed within MPA for different q-values tending to 0 (qn = n 16 in units of 2π/a, where a is the lattice parameter of Cu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' For q = 0 (orange curves) the intra-band term is not included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Right panels: real (c)) and imaginary part (d)) of the four most relevant poles at low energies in the Y curves for different q values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The purple dashed lines lines correspond to the position of the poles extracted from optical measurements collected in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' [24], as explained in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The blue dashed lines correspond to the values of the intra-band frequency, ωA, computed by means of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' (13), and reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Intra-bands effects in copper In order to investigate the intra-band contributions of copper, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 6 we show the frequency dependence of the YG=G′=0 matrix elements computed for for the small- est q-vectors along one direction of a 16×16×16 k-grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Since Y (ω) of Cu is very structured at small frequen- cies, where the effects of the intra-band contributions are expected to be stronger, we have used MPA with a quadratic sampling, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' (10) with α = 2 and np = 15, a number of poles slightly larger than the value needed to converge the quasi-particle energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In contrast with Na, the orange curve (q = 0, no intra-band contribu- tion) presents a similar shape and scale with respect to the green curves (small but finite q, with intra-band con- tributions), even if with less intense peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 6 we show the position of the first 4 poles of Y (ω) as a function of q, which present a rather flat dispersion, when compared with the plasmon dispersion of Al in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' As expected, for q = 0 the position of some poles does not correspond exactly to the limit given by the curves with finite q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' However, the main difference between the zero and finite q curves of Y (ω) is not in the position but rather in the value of the residues of the poles, which is reflected in the intensity of some of the peaks, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In order to compare the computed results with exper- iments, we used electron energy loss data extracted from a compilation of optical measurements found in Table 1 of the Chapter Optical constants of metals of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' [24] (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 8 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' [31]), after interpolation with a multipole model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' For this, we chose 18 points of the spec- tra, with a frequency distribution corresponding to the quadratic sampling of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' (10) and used them to interpo- late a 9 pole model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' We then analysed the 4 poles with the highest residues in the frequency interval we are in- terested in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In the upper panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 6 we show, as hor- izontal lines, the corresponding experimental energies of the poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Interestingly, the experimental poles are very similar to the poles computed at the RPA level within MPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' This supports the interpretation that the MPA poles of Y are not a mere mathematical construct aimed at improving representability but indeed correspond to physical collective excitations, each of them describing the envelope of a set of single particle transitions, with a finite imaginary part corresponding to the width of the excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' We emphasize that the agreement with the experiment is achieved without resorting to any ad hoc parameters such as the damping in the case of the FF-RA method of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In simpler systems, the inclusion of the intra-band limit, even with a simple Drude tail fitted from the ex- 13 perimental spectra, is expected to correct the residues and thus the intensity of the peaks at q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' However, in systems for which the intra- and inter-band contribu- tions are superimposed in a more structured frequency dependence, the description of the experimental spectra with only the Drude term from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' (11) is not possi- ble [52, 74, 91], and indeed models often resort to vari- able or multiple relaxation frequencies [52, 77, 91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In fact, as shown in Fig 6, the q dependence of Y does not allow one to discriminate between peaks with an intra- or an inter-band character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In order to circumvent this difficulty, in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' [31] the intra-band frequency is eval- uated numerically as the limit of an intra-band integral at the independent particle level, while in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' [51] it is estimated within a non-interacting uniform-gas theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Here we use again the f-sum rule by integrating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' (12), but generalizing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' (13) to the case where, in contrast with Al and Na, more than one pole con- tributes to the intra-band term (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' I of Supplemen- tal Material [81]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The resulting intra-band frequency, ωA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='36 Ha (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='80 eV), compares well with the cor- responding result of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='34 Ha (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='27 eV) from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' [31] and both values are very close in energy to the second pole shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' We find that intra-band contri- butions represent around the 25% of the corresponding f-sum rule of this pole (RΩ product), being the largest ratio among all the poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' However, as can be appreci- ated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 6 from the change of intensity of the peaks, the inter-band contributions are dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In fact, the intra-band contributions to the total f-sum rule (sum of all RΩ products) is rather small, less than 4%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Using the frequency determined in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' [31] (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='27 eV) and the relaxation frequency fixed to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='1 eV as the in- puts to the Drude correction of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' (11), in our MPA calculations, we find that the Drude tail overlaps with the several inter-band peaks of Y (ω), without affecting the position of the poles whilst changing their residues (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' IV of Supplemental Material [81]), similar to the effect of the CA correction, Y (q = 0) ∼ Y (qmin), as proposed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' III B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In any case, CA is general and independent of the complexity of the frequency structure of the inverse dielectric function Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' It works well for Cu, as confirmed by the comparison with the experimental data, and constitutes a very simple procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Despite these considerations, and similarly to the case of Al, the intra-band correction has a small effect on the Cu QP energies, that present differences of the order of 5 meV when computed with and without CA in a 12 × 12 × 12 k-grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' SUMMARY AND CONCLUSIONS In this work we address the accuracy of the MPA scheme as applied to the full-frequency GW calculation of metals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' This approach, previously validated for semi- conductors [25], is now applied to metals using Al, Na, and Cu as prototype systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Also in the case of metals, MPA is shown to deliver results with an accuracy simi- lar to other FF methods at a much lower computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' After presenting the MPA theoretical framework, we have applied the approach to simple metals and discussed the role of inter- and intra-band contributions to the di- electric functions of bulk Al and Na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In order to eval- uate the response function and the GW corrections in metals, we have proposed two simple methods to include the intra-band terms in the inverse dielectric function in the q → 0 limit: (1) by extrapolating the position of the main pole in Y00(q, ω), from small q to q = 0, and computing the intra-band pole through the f-sum rule of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' (13), which can then be used as an input value in a Drude model to correct Y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' This approach is generalized for a multipole structure of Y (q, ω) in the case of Cu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' And (2) by approximating Y (q = 0) by Y (qmin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The second method, here called CA, is simpler and spares the determination of the intra-band frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Both methods significantly accelerate the convergence of the QP energies with respect to the k-point grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In ad- dition, CA simultaneously corrects all Y matrix elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' CA works equally within PPA, MPA and FF and can be used independently of the dimensionality of the system under study, even if the leading power of series expan- sion of the inverse dielectric function in the q → 0 limit depends on dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In fact, it can be thought of as the most trivial case of a polynomial interpolation (a constant) [92, 93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' A similar approach can be applied in situations where the q → 0 limit of Y (or other many- body operators, such as W) is difficult to evaluate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Even if the proposed methodologies were exemplified for three isotropic metals, the extension to non-isotropic systems is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Eventually, GW QP corrections for Na, Al and Cu were evaluated, showing an excellent agreement with ex- isting theoretical literature and experimental data, fur- ther stressing the accuracy of the proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Notably, the case of Cu was discussed with particular detail, since PPA calculations present several drawbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In fact, for Cu, the PPA quasi-particle solutions deviate significantly from the FF results and completely fail to describe the frequency dependence of Σ and the spectral function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In contrast, MPA reproduces very accurately the FF results, not only in the tail region that deter- mines the quasi-particles corrections, but in the whole frequency range for both the self-energy and the spectral function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The frequency representation of the polariz- ability and the inverse dielectric function present strong oscillations within FF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In contrast, MPA results are much more stable, leading to a smooth frequency representa- tion of X and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Importantly, the smoother structure of the MPA di- electric function does not necessarily result in a loss of accuracy in the subsequent calculation of the self-energy, the QP energies, and the spectral function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In fact, the frequency dependence of Y given by MPA is meaning- ful and reproduces the main peaks of the experimental 14 energy loss spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' This leads us to conclude that the MPA poles of Y may be seen not only as a mathemati- cal tool, but also as an efficient description of collective excitations, with each pole representing the envelope of a set of single particle transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In conclusion, MPA reproduces well the overall fre- quency dependence of the polarizability, the inverse di- electric function, the self-energy and the spectral func- tion in metallic systems, and gives results for the quasi- particle energies similar to those obtained within FF methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Moreover, the favourable computational per- formance allows for the use of more stringent convergence parameters such as denser k-grids and larger number of bands and polarizability matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The use of the pro- posed intra-band corrections further accelerates the con- vergence with the k-grid and the accuracy of the final results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' ACKNOWLEDGMENTS We acknowledge stimulating discussions with Massimo Rontani, Pino D’Amico, Alberto Guandalini and Gia- como Sesti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' This work was 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Fabris, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Fratesi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Gebauer, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Gerstmann, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Gougoussis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Kokalj, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Umari, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Wentz- covitch, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' : Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Matter 21, 395502 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Lazzeri, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Marsili, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Marzari, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Mauri, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Nguyen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='- V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Nguyen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' de-la Roza, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Paulatto, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Ponc´e, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Rocca, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Mat- ter 29, 465901 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' [71] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Caruso and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Giustino, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 183, 1269 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' [93] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Guandalini, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' D’Amico, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Ferretti, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Varsano, “Efficient gw calculations in two dimensional materials through a stochastic integration of the screened poten- tial,” https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='org/abs/2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='11946v2 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Efficient full frequency GW for metals using a multipole approach for the dielectric screening: Supplemental Material Dario A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Leon1,2,3,∗ Andrea Ferretti2, Daniele Varsano2, Elisa Molinari1,2, and Claudia Cardoso2 1 FIM Department, University of Modena & Reggio Emilia, 41125, Modena (Italy) 2S3 Centre, Istituto Nanoscienze, CNR, 41125, Modena (Italy) and 3Department of Mechanical Engineering and Technology Management, Norwegian University of Life Sciences, 1430, ˚ As (Norway) I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' A SIMPLE INTRA + INTER-BAND MODEL In this Section we analyze how two poles in the in- dependent particle response function Y0, corresponding to intra- and inter-band transitions, contribute to the structure of its dressed counterpart, Y , as a result of the Dyson equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The Y0 and Y functions are related to the non-interacting and dressed polarizability functions as: Y0 ≡ vX0, Y ≡ vX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' (S1) The Dyson equation for the inverse dielectric function is then given by Y = (1 − Y0)−1Y0 (S2) In the following we make use of the f-sum rule [1–3] in the form SY (q) = 2 π � ∞ 0 dω ω Im[Y ](q, ω), (S3) where S is computed from the electronic density and the above expression is valid for each (G, G′) components of both Y0 and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In the case G = G′, one has SY (q) = ω2 pl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' We then separate Y0 into intra (A) and inter-band (E) contributions in the q → 0 limit, as Y0(q = 0, ω) = Y A 0 (ω) + Y E 0 (ω), (S4) resulting in two different terms for the f-sum rule: SY0(q = 0) = SA + SE, (S5) where SA is the power square of an intra-band frequency, ΩA, and SE is obtained from the f-sum rule expression for the polarizability X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' When computed from Kohn- Sham (KS) states, as derived in Appendix B of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' [4], one obtains: SE = � n 2vRKS n ΩKS n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' (S6) We now analyze the results of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' (S2) in different sce- narios by means of a two poles model y0(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The first ∗ dario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='alejandro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='leon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='valido@nmbu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='no pole of y0(ω) at ω = 0 corresponds to a Drude tail result- ing from the intra-band transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The second broader pole results from the superposition of the inter-band tran- sitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' If we include both single particle contributions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' (S4), we obtain: y0(ω) = Ω2 A ω2 + 2vREΩE ω2 − Ω2 E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' (S7) We consider the inversion of the Dyson equation (S2) for y, disregarding the so called local field effects (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' y0 and y are scalar functions instead of matrices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Case SE → 0: y(ω) = Ω2 A ω2 − Ω2 A (S8) In this case only the intra-band is relevant and we get a pole at the Drude frequency, ΩA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Case SA → 0: y(ω) = 2vREΩE ω2 − Ω2 E − 2vREΩE (S9) In this case only the inter-band is relevant and the pole, ΩE, is right-shifted by a factor � 1 + 2vRE/ΩE, which for many materials is close to 1 (no shift).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' General case: y(ω) = S1 ω2 − Ω2 1 + S2 ω2 − Ω2 2 , (S10) where the poles are given by the expression: Ω2 1,2 = 1 2 � Ω2 E + 2vREΩE + Ω2 A± � (Ω2 E + 2vREΩE + Ω2 A)2 − 4Ω2 EΩ2 A � , (S11) and S1 and S2 are compliant with the f-sum rule S1 + S2 = SA + SE: S1,2 = ±(SA + SE)Ω2 1,2 − SAΩ2 E Ω2 2 − Ω2 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' (S12) arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='02282v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='mtrl-sci] 5 Jan 2023 2 The two first cases can be obtained as limiting cases of this general solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' But there are other situations leading to a solution with a single plasmon pole: y(ω) = 2vRpΩp ω2 − Ω2p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' (S13) One case occurs when either S1 or S2 is much larger than the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' A second possibility is when the two poles, Ω1 and Ω2, are equal or very close to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' There are several ways to obtain this: the radicand in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' (S11) could be small compared to the terms outside, the scales of the poles could be very different (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' ΩE ≫ ΩA), 2vRE/ΩE could be close to 0 or 1, or the intra- and inter-band poles could be similar, ΩA ≈ ΩE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' If y(ω) has a single pole, the f-sum rule for y leads to a simple relationship between the intra-band pole, ΩA, the inter-band pole, ΩE and the plasmon pole, Ωp: 2vRpΩp = Ω2 A + 2vREΩE, (S14) while a similar expression is obtained from the evaluation of y(0): Ω2 p = Ω2 A + 2vREΩE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' (S15) From the previous two equations, vRp and Ωp are un- equivocally determined, allowing one to discriminate be- tween the intra- and inter-band contributions to the structure of y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' For many systems, like the case of Cu addressed in the main manuscript, the total Y (ω) presents a structure with more than one pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In this situation the model given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' (S14) can be generalized to obtain the intra- band frequency, again by means of the f-sum rule: Ω2 A = � 2vRpΩp − � 2vREΩE, (S16) where the sums of RΩ products run over the pole struc- tures of the full and the inter-band part of Y (ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' ANALYSIS OF THE INTRA-BAND LIMIT FOR DIFFERENT MATRIX ELEMENTS In the main text we have shown that the YG=G′=0 curves with finite q, computed for Al and Na, change smoothly with decreasing q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The curves present a pole, Ωp(q), of decreasing energy and increasing amplitude and both the real and imaginary part of this pole can be easily extrapolated to q = 0, by means of the Lindhard plasmon dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Here we have considered other Na Y matrix elements (YG=G’̸=0 and YG̸=G’) with the same not particularly dense grid of 8 × 8 × 8 k-points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' S1 we show similar plots for wing (YG=0̸=G’ and YG̸=0=G’) and diagonal (YG=G’̸=0) elements, where the orange curves contain only inter-band terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' For some of these elements the evolution of the Y curves when q → 0 is less smooth than for the head, YG=G′=0, and the posi- tion of the poles not always changes monotonously, thus the extrapolation with a simple analytical form would re- quire a denser k-point grid depending on the specific ma- trix element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' However, as can be seen from the compari- son between green and orange curves, intra-band transi- tions are more important for the wings than for diagonal elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Moreover, even if the constant dielectric func- tion approximation (CA) scheme described in the main manuscript, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='e, Y (q = 0) ∼ Y (qmin), has limited accu- racy for some of these matrix elements, it still improves the overall Y matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' FREQUENCY REPRESENTABILITY OF COPPER As mentioned in the main manuscript, the nonlinear interpolation of the polarizability, X, or the inverse di- electric function, Y , may produce non-physical poles cor- responding to the so-called unfulfilled modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' This is usu- ally solved by reassigning the values of the poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The condition used to identify unfulfilled modes in the GN- PPA scheme [4] is the following: Re � YGG′(q, 0) YGG′(q, iϖpl) − 1 � < 0, (S17) where the position of the pole is set to ΩGN fail = 1 Ha in case of failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The MPA scheme uses a generalized condition that avoids reassigning the poles with a constant value [4]: Ωn = �� Ω2n, Re � Ω2 n � ≥ 0 � −(Ω2n)∗, Re � Ω2 n � < 0 (S18) It is possible to quantify the representability error re- lated to this reassignment by computing the mean num- ber of corrected X matrix elements, ⟨NF ⟩, and an av- erage relative standard deviation of the extrapolated X with respect to its sampling points, ⟨RSD⟩, as defined in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' S2 we show how these two quantities evolve when increasing the number of poles used in the description of X and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' When applying the PPA, we found that 48% of the po- larizability matrix elements fail the plasmon pole condi- tion (S17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' These results have an averaged relative devi- ation of ⟨RSD⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' These large values lead to quasi- particle solutions very different from FF, as described in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The MPA scheme, with only one pole, still presents a larger percentage of corrected poles but con- siderably improves the representability with respect to PPA, lowering the average deviation to ⟨RSD⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The reason for the larger number of corrected poles is the use of a different sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' As mentioned in the main manuscript, in the case of MPA with one pole the frequency at the origin of coordinates is shifted along the imaginary axis, which helps to reduce the numerical in- stabilities found with the PPA sampling [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' However, a significant improvement happens only by increasing the 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='1 Re[Y] (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=') Al Na G = 0 G′ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='0 Na G = G′ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='5 (Ha) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='3 Im[Y] (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=') q4 q3 q2 q1 q0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='25 (Ha) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='00 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Frequency dependency of Y matrix elements, other than the head (G = G′ = 0), computed with MPA for different q vectors of modulus q ≡ |q| tending to 0, for Na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' For q = 0 (orange curves) the intra-band term is not included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The chosen values of q0–q4 correspond to qn = n 8 in units of 2π/a, with a the lattice parameter of Na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 2 4 6 8 10 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='6 np �RSD� PPA MPA 2 4 6 8 10 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='6 np �NF� FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Values calculated for Cu of (left) the mean number of matrix elements, ⟨NF ⟩, for which the position of the poles was corrected according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' (S18) for MPA and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' (S17) for PPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' number of poles, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' S2, evidencing the complexity of the frequency structure of the polarizabil- ity of Cu and the efficiency of the multipole approxima- tion in its description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' STANDARD DRUDE CORRECTION FOR THE INTRA-BAND OF COPPER In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' S3 we show the frequency dependence of Y computed for Cu within MPA comparing three different methods to include the intra-band corrections in the limit q → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' We show, Y computed without any intra-band correction (nD), with the CA method and the Drude model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' For the latter, we considered an intra-band fre- quency of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='27 eV, as determined in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' We see that the Drude model and CA give similar results, renormal- izing the intensity of the peaks without shifting them in the real frequency axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' CONVERGENCE OF GW PARAMETERS FOR COPPER As mentioned in the main text, the GW convergence is very challenging for the case of Cu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' S4 we illus- trate how the energy range of the transitions observed in X rapidly increases with respect to the number of bands included in the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The increasing number of bands results in changes in the details of the frequency structure of X, whose description requires, in the FF-RA scheme, a large number of frequency points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' On the other hand, there is a complex relationship be- tween the number of bands and the plane-wave energy 4 0 2 4 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='2 Re[Y] 0 2 4 6 (Ha) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='0 Im[Y] MPA, Drude MPA, CA MPA, nD FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Frequency dependence of Y (q = 0) computed for Cu with three different methods: without any intra-band correc- tion (nD), with the CA correction and with the Drude model using as input the intra-band frequency determined in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' cut-off used to build the polarizability matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' S5, there is a change of monotony around 15 Ry when increasing the number of bands from 200 to 500 and smaller oscillation continue at higher cut-off ener- gies, hindering the possibility to extrapolate the con- verged QPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' For cut-off energies and number of bands larger than 25 Ry and 500 respectively, the correction changes sign, correcting the DFT in the right direction with respect to the experimental data shown in the Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' II of the main manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Stefanucci and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' van Leeuwen, Nonequilibrium Many- Body Theory of Quantum Systems: A Modern Introduc- tion (Cambridge University Press, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' [2] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Martin, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Reining, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Ceperley, Interacting Electrons (Cambridge University Press, Cambridge, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' [3] K.' metadata={'source': 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Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' B 104, 115157 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Marini, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Onida, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Sole, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 88, 016403 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' [6] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Marini, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Onida, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Del Sole, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' B 64, 195125 (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 5 0 1 2 3 2 1 0 Re[X] × a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 0 1 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='00 × a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 0 1 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='0 × a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Cu: 200 500 0 10 20 30 (Ha) 2 1 0 Im[X] × a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 0 10 20 30 (Ha) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='0 × a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 0 10 20 30 (Ha) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='05 × a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Selected Cu X matrix elements computed within FF with different number of bands (200 and 500).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' In order to show both, the overall and the detailed behavior of the polarizability, we plotted the imaginary part of X in the bottom panels in the full frequency interval determined by the number of bands, while the real part of X are plotted in the top panels in a zoomed region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' A similar scheme from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' [4] is used for the units of X, where we omitted the specific scales in order to avoid confusion with the zoomed plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' 0 10 20 30 40 50 60 Energy cut-off (Ry) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content='15 E 1 (eV) 200b 300b 400b 500b 800b 1000b FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' G0W0 correction to the PBE quasi-particle energy of the Γ1 state of Cu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' Convergence with respect to the number of bands and the cut-off energy parameters used to build the polarizability matrix, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} +page_content=' The curves correspond to calculations with different fixed number of bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE0T4oBgHgl3EQfWwBP/content/2301.02282v1.pdf'} diff --git a/fNAyT4oBgHgl3EQfjvgl/content/tmp_files/2301.00419v1.pdf.txt b/fNAyT4oBgHgl3EQfjvgl/content/tmp_files/2301.00419v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6dc410fd014bae0960fe9565800c67c4f87afc22 --- /dev/null +++ b/fNAyT4oBgHgl3EQfjvgl/content/tmp_files/2301.00419v1.pdf.txt @@ -0,0 +1,1751 @@ +arXiv:2301.00419v1 [math.OC] 1 Jan 2023 +POLICY ITERATION FOR THE DETERMINISTIC CONTROL PROBLEMS +– A VISCOSITY APPROACH +WENPIN TANG, HUNG VINH TRAN, YUMING PAUL ZHANG +Abstract. This paper is concerned with the convergence rate of policy iteration for (determin- +istic) optimal control problems in continuous time. To overcome the problem of ill-posedness +due to lack of regularity, we consider a semi-discrete scheme by adding a viscosity term via finite +differences in space. We prove that PI for the semi-discrete scheme converges exponentially +fast, and provide a bound on the error induced by the semi-discrete scheme. We also consider +the discrete space-time scheme, where both space and time are discretized. Convergence rate +of PI and the discretization error are studied. +1. Introduction +Optimal control is ubiquitous in science and engineering with a variety of applications includ- +ing aerospace engineering [6, 10], chemical engineering [31], economy [24], operations research +[36, 37] and robotics [2, 12]. Dynamic programming (DP) has proved to be an efficient tool to +solve multistage optimal control problems since its inception by Bellman [5]. In recent years, +reinforcement learning (RL) has shown great success in resolving complex decision making +problems, notably AlphaGo [38] and humanoid tasks [18]. Policy iteration (PI), as a class of +approximate or adaptive dynamic programming (ADP), is instrumental in many RL algorithms +[40]. +The idea of PI dates back to Howard [20] in a stochastic environment known as the Markov +decision process (MDP). Subsequent works [7, 33, 34] explored PI for MDPs in discrete time and +space; recently, [8, 30] considered PI for (deterministic) optimal control problems in discrete +time and continuous space. In these works, PIs are proved to converge to the optimal control +under suitable conditions on the model parameters. On the other hand, many real-world prob- +lems are modeled by dynamical systems evolving in continuous time, and it is known that DP +for optimal control in continuous time and space entails the Hamilton-Jacobi-Bellman (HJB) +partial differential equation (PDE). Despite its importance, PI for optimal control problems in +continuous time and space has not been rigorously studied until recently. [1, 43] proved the +convergence of PI for continuous-time linear quadratic optimal control problems; more general +cases were settled in [28] under a fixed point assumption. For the stochastic control problems, +[26, 35] showed that PI converges exponentially fast in the case where controls are only exer- +cised on the drift term of the state process. Similar results were derived for the corresponding +entropy-regularized problems [22, 41]. We also mention, in a closely related direction, [9, 45] +Date: January 3, 2023. +Key words and phrases. Finite differences, Hamilton-Jacobi-Bellman equations, optimal control, policy +iteration. +The work of W. Tang is supported by NSF grants DMS-2113779 and DMS-2206038, and a start-up grant +at Columbia University. The work of H. Tran is partially supported by NSF CAREER grant DMS-1843320, a +Simons Fellowship, and a Vilas Faculty Early-Career Investigator Award. +1 + +2 +W. TANG, H. V. TRAN, Y. P. ZHANG +studied value iteration for optimal control problems. See [29, 44] for recent progress on theory +and applications of ADP for optimal control and RL. +In this paper, we study the convergence rate of PI for optimal control problems in continuous +time and their discretization under fairly general conditions on the model parameters. Note +that the convergence analysis in [1, 43] relies on the linear quadratic structure of the problem, +while [28] assumed that the HJB operator enjoys a fixed point, or a contraction property +which is hard to verify. None of these works quantified the convergence of PI to the optimal +control. Moreover, PI for continuous-time control problems may even be ill-posed due to lack +of regularity. Our idea is to introduce a viscosity term “h∆h” in the policy evaluation, where +h is the mesh size and ∆h is the discrete Laplacian in space. We call it a semi-discrete scheme. +Essentially the viscosity term is of order 1, which assures that the finite difference scheme is +monotone. In fact, a monotone scheme is commonly desirable for numerical implementation so +the addition of the finite difference viscosity term is natural. On the other hand, the viscosity +term in the semi-discrete scheme mimics the vanishing viscosity approximation to first-order +PDEs [16], which forces PI to converge exponentially fast (Theorem 3.1) as for the stochastic +control problems. We also prove that the discrepancy between the optimal control problem +and its semi-discrete scheme is of order +√ +h as h → 0 (Theorem 3.3). Further we consider +the time-discretization, called a discrete space-time scheme. The same results hold for PI for +the discrete space-time scheme (Theorem 4.1 and Theorem 4.2). Our results echo recent work +[19], which asserts that noise enhances the convergence of finite-horizon RL algorithms. In +our setting, noise corresponds to the viscosity term, and the importance of finite-horizon is +seen from various bounds with exponential dependence in time. Our analysis relies on PDE +techniques, and may carry over to the study of differential games in solving Hamilton-Jacobi- +Bellman-Issacs (HJBI) equations. +The rest of the paper is organized as follows. In Section 2, we provide background, and +present the semi-discrete and the discrete space-time schemes. +In Section 3 we study the +semi-discrete scheme, and in Section 4 we analyze the discrete space-time scheme. We provide +further PDE perspectives in Section 5. We conclude with Section 6. +2. Setup and preliminary results +In this section, we present the semi-discrete and the discrete space-time schemes. Consider +a system whose state is governed by the ordinary differential equation: +dxt +dt = f(t, xt, αt), +(2.1) +where for 0 ≤ t ≤ T, xt ∈ Rd is the system state, and αt ∈ A ⊂ Rm is the control or policy. +Here, A is a given compact subset of Rm. The objective is +J(t, x, α) := +� T +t +c(s, xs, αs) ds + q(xT ) +given xt = x, +(2.2) +and the goal is to minimize this objective function. Denote by +v∗(t, x) := inf +α∈At J(t, x, α), +(2.3) +where At is the standard admissible policy defined as At = {α : [t, T] → A : α is measurable}. +It is known that under suitable conditions on c(·) and q(·) (see [17, Chapter 2] or [42, Chapter + +3 +2]), v∗ defined by (2.3) is the viscosity solution to +� +∂tv + H(t, x, ∇v) = 0 +in (0, T) × Rd, +v(T, x) = q(x) +on Rd, +(2.4) +where the Hamiltonian H is given by H(t, x, p) := infa∈A [c(t, x, a) + p · f(t, x, a)]. The optimal +policy is given by +α∗(t, x) = α(t, x, ∇v∗), +(2.5) +where +α(t, x, p) := arg mina∈A [c(t, x, a) + p · f(t, x, a)] , +(2.6) +is assumed for simplicity to be unique throughout this paper. +Policy iteration is an approximate dynamic programming, which alternates between policy +evaluation to get the value function with the current control and policy improvement to optimize +the value function. More precisely, for n = 0, 1, · · · , the iterative procedure is: +• Given αn(t, x), solve the PDE +� +∂tvn + c(t, x, αn) + ∇vn · f(t, x, αn) = 0 +in (0, T) × Rd, +vn(T, x) = q(x) +on Rd. +(2.7) +• Set +αn+1(t, x) = α(t, x, ∇vn) = arg mina∈A [c(t, x, a) + ∇vn(t, x) · f(t, x, a)] . +(2.8) +The key is to understand how the sequence {vn} approximates the optimal value v∗, and how +{αn} approximates the optimal policy α∗. +On the other hand, it is not clear whether the policy iteration scheme (2.7)–(2.8) is well- +posed. Intuitively, to make sense of αn+1 we need vn to be Lipschitz continuous, for which we +then need αn to be Lipschitz. This in turn requires ∇vn−1 to be Lipschitz. After iterations, +this means that we need v0 to be smooth which is in general not true. +2.1. Semi-discrete schemes. For T ≥ 1, h ∈ (0, 1), N ≥ max{1, ∥f∥∞/2} and a given +continuous function α0 : R × Rd → A, we solve for n = 0, 1, · · · +� +∂tvh +n + c(t, x, αn) + ∇hvh +n · f(t, x, αn) = −Nh∆hvh +n +in (0, T) × Rd +vh +n(T, x) = q(x) +on Rd. +(2.9) +Then set +αn+1(t, x) = α(t, x, ∇hvh +n) +in (0, T) × Rd. +(2.10) +Here, for any ϕ : Rd → R and h ∈ R \ {0}, we use the notations +∇hϕ(x) := +�ϕ(x + he1) − ϕ(x − he1) +2h +, · · · , ϕ(x + hed) − ϕ(x − hed) +2h +� +, +∆hϕ(x) := +d +� +i=1 +ϕ(x + hei) − 2ϕ(x) + ϕ(x − hei) +h2 +. +Later we will also write Dhϕ(x) := +� +ϕ(x+he1)−ϕ(x) +h +, · · · , ϕ(x+hed)−ϕ(x) +h +� +. It is clear that +∇hϕ(x) = 1 +2(Dhϕ(x) − D−hϕ(x)). +(2.11) + +4 +W. TANG, H. V. TRAN, Y. P. ZHANG +The assumption N ≥ ∥f∥∞/2 guarantees that the numerical Hamiltonian is monotone and, +as a consequence of this, the following comparison principle holds (see e.g., [13, 32, 42]). +Lemma 2.1. Let vh +0 and ˜vh +0 be, respectively, a bounded continuous super- and sub- solution +to (2.9) with n = 0, and satisfy ˜vh +0 ≤ vh +0 at t = T. Then ˜vh +0 ≤ vh +0 in [0, T] × Rd. Here by a +supersolution (resp. subsolution), we mean that it satisfies (2.9) with the first equality replaced +by ≤ (resp. ≥). +First, we show that the scheme (2.9)–(2.10) is well-posed. We need the following assumptions: +(A1) c(·, ·, ·), f(·, ·, ·), q(·) are uniformly bounded and Lipschitz continuous in all of their de- +pendences. +(A2) α(·, ·, ·) and α0(·, ·) are uniformly Lipschitz continuous in all of their dependences. +Throughout this paper, we write C as various universal constants that only depend on d, N, +and the constants in (A1)–(A2) unless otherwise stated. Specifically, since T is not a universal +constant, we keep track of the dependence on T in most estimates. The constants C might +vary from one line to another. By CX or C(X) we mean a constant that depend on universal +constants and X. +Proposition 2.2. Assume (A1)–(A2) and N ≥ max{1, ∥f∥∞/2}. Then the iterative process +(2.9)–(2.10) is well-defined, that is, there are Lipschitz continuous functions vh +n, αn satisfying +(2.9)–(2.10) and vh +n are uniformly bounded for all n ≥ 0 and h > 0. +Proof. Since α0 is Lipschitz continuous, the unique solvability of (2.9) for n = 0 follows from +[27, Theorem 2.4]. If one can show that vh +0 is uniformly bounded and Lipschitz continuous with +Lipschitz constant Ch, then α1 is Lipschitz continuous with Lipschitz constant Ch/h by the +assumption that α is Lipschitz. From the same argument, we obtain a unique bounded and +Lipschitz solution vh +1. The existence of solutions then follows from iterations. +First we prove the boundedness of vh +0 . Since c(·, ·, ·), q(·) are uniformly bounded, we have that +± [∥q∥∞ + ∥c∥∞(T − t)] are a supersolution and a subsolution to (2.9) with n = 0, respectively. +Hence +−∥q∥∞ − ∥c∥∞(T − t) ≤ vh +0(t, x) ≤ ∥q∥∞ + ∥c∥∞(T − t), +for all (t, x) ∈ [0, T] × Rd. Actually the same bound holds for all vh +n by this argument. +Next we show that vh +0 is Lipschitz continuous with Lipschitz constant independent of h when +T is sufficiently small depending only on the assumption (A1) (but not on q). The general +result follows immediately by iterations. For simplicity of notations, write +G(t, x, p) := c(t, x, α0(t, x)) + p · f(t, x, α0(t, x)). +Then for M := 2∥∇q∥∞ + 1, define +˜G(t, x, p) := +� +G(t, x, p) +if |p| ≤ M, +G(t, x, Mp/|p|) +if |p| > M. +It follows from (A1) and the choice of N that +| ˜Gt(t, x, p)|, | ˜Gx(t, x, p)| ≤ C(1 + M), +| ˜Gp(t, x, p)| ≤ 2N. +(2.12) + +5 +Now let ˜vh be the solution to +� +∂t˜vh + ˜G(t, x, ∇h˜vh) = −Nh∆h˜vh, +˜vh(T, x) = q(x). +The goal is to show that ˜vh is Lipschitz continuous, and ˜vh = vh +0 in [0, T] × Rd. +Note that for any e ∈ Sd−1 and s ∈ (0, 1), ps := ˜vh(t,x+se)−˜vh(t,x) +s +satisfies + + + +∂tps + G1(t, x) + G2(t, x) · ∇hps = −Nh∆hps +in (0, T) × Rd, +ps(T, x) = q(x + se) − q(x) +s +on Rd, +(2.13) +where +G1(t, x) := 1 +s +� s +0 +˜Gx +� +t, x + ze, ∇h˜vh(t, x + se) +� +· e dz, +G2(t, x) := +� 1 +0 +˜Gp +� +t, x, ∇h˜vh(t, x) + z(∇h˜vh(t, x + se) − ∇h˜vh(t, x)) +� +dz. +It is clear from (2.12) that |G1| ≤ C(1 + M) and |G2| ≤ 2N. This yields that the comparison +principle for (2.13) holds. Thus, by comparing ps with ±(∥∇q∥∞ +C(1+M)(T −t)), we obtain +|ps(t, x)| ≤ ∥∇q∥∞ + C(1 + M)(T − t). Sending s → 0 yields for some C depending only on +(A1), |∇e˜vh(t, x)| ≤ ∥∇q∥∞ + C(1+ M)(T − t). Thus, if t ≤ T ≤ (2C)−1, we have that ˜vh(t, x) +is Lipschitz and +sup +(t,x)∈[0,T]×Rd |∇˜vh(t, x)| ≤ ∥∇q∥∞ + 1/2 + M/2 = M. +From the definition of ∇h, we get sup(t,x)∈[0,T]×Rd |∇h˜vh(t, x)| ≤ M. Hence, ˜vh is a solution to +(2.9) for n = 0. The uniqueness of the solution to (2.9) yields that vh +0 ≡ ˜vh. So we obtain the +uniform Lipschitz continuity of vh +0 in space, with Lipschitz constant of the form C exp(CT). +The Lipschitz regularity in time follows from the equation. +□ +We point out that the Lipschitz constant of vh +n may depend on both n and h for n ≥ +1. Another consequence of the comparison principle is that the functions vh +n are monotone +decreasing in n. +Proposition 2.3. Under the assumptions of Proposition 2.2, we have for all n ≥ 0, +vh +n+1 ≤ vh +n +in [0, T] × Rd. +Proof. By the definition of αn, +c(t, x, αn+1(t, x)) + ∇hvh +n · f(t, x, αn+1(t, x)) +≤ c(t, x, αn(t, x)) + ∇hvh +n · f(t, x, αn(t, x)). +Thus vh +n is a supersolution to (2.9) with subscripts n + 1 as it satisfies +∂tvh +n + c(t, x, αn+1) + ∇hvh +n · f(t, x, αn+1) ≤ −Nh∆hvh +n +in (0, T) × Rd. +Therefore, the comparison principle yields vh +n ≤ vh +n+1 in [0, T] × Rd for each n ≥ 0. +□ + +6 +W. TANG, H. V. TRAN, Y. P. ZHANG +Since vh +n is uniformly bounded for all n ≥ 0, the monotonicity property yields that vh +n +converges locally uniformly as n → ∞. Let us denote the limit as vh. Then by the stability +property of viscosity solutions, vh solves +� +∂tvh + H(t, x, ∇hvh) = −Nh∆hvh +in (0, T) × Rd, +vh(T, x) = q(x) +on Rd, +(2.14) +where +H(t, x, p) := c(t, x, α(t, x, p)) + p · f(t, x, α(t, x, p)) += min +a∈A [c(t, x, a) + p · f(t, x, a)] . +(2.15) +Since α(t, x, p) is assumed to be uniformly Lipschitz continuous in all of its dependences, there +exists C > 0 such that for all (t, x, p) ∈ [0, T] × Rd × Rd, +|Ht(t, x, p)|, |Hx(t, x, p)| ≤ C(1 + |p|), +|Hp(t, x, p)| ≤ C. +(2.16) +Moreover, one can expect that vh converges as h → 0, and the limit v should be the unique +solution to (2.4). +It was proved in Proposition 2.2 that vh +0 is uniformly bounded and Lipschitz continuous in +[0, T]×Rd. By the same proof, we have that vh and v are also uniformly bounded and Lipschitz +continuous for all h > 0. +Lemma 2.4. Under the assumptions of Proposition 2.2, let vh +0, vh and v be, respectively, solu- +tions to (2.9) (for n = 0), (2.14) and (2.4). Then in [0, T] × Rd, vh +0 , vh and v are bounded by +C(1 + T) and are Lipschitz continuous with Lipschitz constant C exp(CT) for some universal +constant C > 0. +For a general class of first-order Hamilton-Jacobi (continuous) equations, we refer to [3, 4] +for the regularity results. +2.2. Discrete space-time schemes. Now we consider the scheme that is discrete in both +space and time. Let τ, h ∈ (0, 1), and N such that +max{1, ∥f∥∞/2} ≤ N ≤ h/(2τ). +(2.17) +Assuming that T/τ ∈ N+, we denote +Nτ +T := {0, τ, 2τ, · · · , T}, +Zd +h := hZd, +Ωτ,h +T +:= Nτ +T × Zd +h +and +Ω′ +T := (Nτ +T \{0}) × Zd +h. +Given a Lipschitz continuous function α0(t, x), let V τ,h +n +: Ωτ,h +T +→ R be defined iteratively for +n = 0, 1, · · · as follows: +� +∂τ +t V τ,h +n +(t, x) + c(t, x, αn) + ∇hV τ,h +n +· f(t, x, αn) = −Nh∆hV τ,h +n +in Ω′ +T , +V τ,h +n +(T, x) = q(x) +on Zd +h +(2.18) +with +αn+1(t, x) := α(t, x, ∇hV τ,h +n +) +in Ω′ +T . +(2.19) +Here we used the notation ∂τ +t V τ,h +n +(t, x) := V τ,h +n +(t,x)−V τ,h +n +(t−τ,x) +τ +. + +7 +We also consider the following equation +� +∂τ +t V τ,h + H(t, x, ∇hV τ,h) = −Nh∆hV τ,h +in Ω′ +T , +V τ,h(T, x) = q(x) +on Zd +h. +(2.20) +where H is given by (2.15). The goal is to show that V τ,h +n +converges to V τ,h as n → ∞, and +V τ,h converges to v as τ, h → 0, where v is given by (2.4). +Similarly as before, we write +c := c(t, x, α(t, x, ∇hV τ,h)) +and +f := f(t, x, α(t, x, ∇hV τ,h)) +cn := c(t, x, αn(t, x)) +and +fn := f(t, x, αn(t, x))) +for simplicity, when there is no confusion. +We will use the following operator. For each t ∈ Nτ +T , let Ft : L∞(Zd +h) → L∞(Zd +h) be defined +as +Ft(U)(x) := U(x) + τH(t, x, ∇hU(x)) + Nhτ∆hU(x). +(2.21) +Then the equation in (2.20) can be rewritten as V τ,h +n +(t − τ, x) = Ft(V τ,h +n +(t, ·))(x). We need +max{1, ∥Hp∥∞/2} ≤ N ≤ h/(2τ), +(which corresponds (2.17) as ∥Hp∥∞ = ∥f∥∞) to guarantee a monotonicity property of the +operator Ft. That is, for all t ∈ Nτ +T and U, V ∈ L∞(Zd +h) satisfying U ≤ V , we have Ft(U) ≤ +Ft(V ), see e.g., [13, 42]. +It is easy to see that the same holds if we replace H(t, x, p) by +cn(t, x) + p · fn(t, x). +The monotonicity property is important because it immediately implies the comparison prin- +ciple of (2.20) and the scheme (2.18)–(2.19), in the sense that is similar to Lemma 2.1. As a +consequence of this, one can show the following properties. +Proposition 2.5. Assume (A1)–(A2) and (2.17). Then in Ωτ,h +T , the solutions V τ,h +n +, V τ,h are +bounded by C(1+T), and are Lipschitz continuous with Lipschitz constant C exp(CT) for some +universal constant C > 0. Moreover for all n ≥ 0 we have V τ,h +n+1 ≤ V τ,h +n +in Ωτ,h +T . +The proof of Proposition 2.5 is similar to those of Propositions 2.2, 2.3 and Lemma 2.4, and +hence we skip it. +3. Analysis of semi-discrete schemes +3.1. Convergence of PI. We show that for each fixed h ∈ (0, 1), vh +n → vh as n → ∞ +exponentially fast in L2 +loc norm. +Theorem 3.1. Assume (A1)–(A2) and N ≥ 1. Let vh +n and vh be, respectively, continuous +solutions to (2.9) and (2.14). Then there exists a universal constant C > 0 such that for all +n ≥ 1, R ≥ 1 and t ∈ [0, T] we have +� +BR +���vh +n(t, x) − vh(t, x) +��� +2 +dx ≤ +h +2n+1 +� T +t +� +Rd exp +� +C(1 + ∥∇hvh∥2 +∞)(s − t)/h +� +× +���Dh(vh +0 (s, x) − vh(s, x)) +��� +2 +min +� +1, e−|x|+R+1� +dxds. +In particular, we have supt∈[0,T] +� +BR +��vh +n(t, x) − vh(t, x) +��2 dx ≤ C2−n exp [C exp(CT)/h] Rd. + +8 +W. TANG, H. V. TRAN, Y. P. ZHANG +Proof. In this proof, let us write vn := vh +n and v := vh, and assume T ≥ 1 for simplicity. For +any fixed R ≥ 1, let ϕ = ϕR : [0, ∞) → (0, 1] be C1 and satisfying +ϕ(r) = 1 +on [0, R], +ϕ(r) = e−r+R +on [R + 1, ∞), +− ϕ′(r) ∈ [0, 4ϕ(r)] +for all r > 0. +(3.1) +It is clear that such ϕ exists. +Next, for some A ≥ 1 to be determined, set +Et,n := 1 +2eAt +� +Rd |vn(t, x) − v(t, x)|2ϕ(|x|) dx +(3.2) +which is finite since vn, v are uniformly bounded. Direct computation yields +d +dtEt,n = AEt,n + eAt +� +Rd(vn(t, x) − v(t, x)) (∂tvn(t, x) − ∂tv(t, x)) ϕ(|x|) dx +� +�� +� +=:Xt,n +. +(3.3) +Recall from (2.15) that H(t, x, ∇hv) = c(t, x, α(t, x, ∇hv)) + ∇v · f(t, x, α(t, x, ∇hv)). Write +c := c(t, x, α(t, x, ∇hv)), f := f(t, x, α(t, x, ∇hv)), cn := c(t, x, αn(t, x)) and fn := f(t, x, αn(t, x)) +for simplicity. Then, using +� +Rd ∆hv vϕ dx = − +� +Rd |Dhv|2ϕ dx + 1 +h2 +d +� +i=1 +� +Rd v(t, x + hei)(v(t, x + hei) − v(t, x))ϕ(t, x) dx +− 1 +h2 +d +� +i=1 +� +Rd v(t, x)(v(t, x) − v(t, x − hei))ϕ(t, x) dx += − +� +Rd |Dhv|2ϕ dx − +� +Rd vD−hv · D−hϕ dx, +we obtain from the equation that +Xt,n = − +� +Rd(vn − v)(∇hvn · fn + cn + Nh∆hvn − ∇hv · f − c − Nh∆hv)ϕ dx +≥ Nh +� +Rd |Dh(vn − v)|2ϕ dx − Nh +� +Rd |vn − v||D−h(vn − v)||D−hϕ| dx +− +� +Rd |vn − v| +� +|∇h(vn − v)||fn| + |fn − f||∇hv| + |cn − c| +� +ϕ dx. +(3.4) +Due to (3.1), |D−hϕ(|x|)| ≤ Cϕ(|x|) for some constant C > 0. +Also, using ∥f∥∞ < ∞ +and (2.11), we have |∇h(vn − v)||fn| ≤ C(|Dh(vn − v)| + |D−h(vn − v)|). Since v is Lipschitz +continuous, |∇hv| ≤ M for some M ≥ 1. So, by (2.10) and the uniform Lipschitz continuity of +f, c and α, we have for some C > 0, +|fn − f||∇hv| + |cn − c| ≤ CM(|Dh(vn−1 − v)| + |D−h(vn−1 − v)|). +(3.5) +With all these, if denoting +Gh +t,n := +� +Rd |Dh(vn(t, x) − v(t, x))|2ϕ(|x|) dx, + +9 +it follows from (3.4) that for some C > 0, +Xt,n ≥ NhGh +t,n − C +� +Rd |vn − v|(|Dh(vn − v)| + |D−h(vn − v)|)ϕ dx +− CM +� +Rd |vn − v|(|Dh(vn−1 − v)| + |D−h(vn−1 − v)|)ϕ dx. +By the choice of ϕ, there exists C > 0 such that +G−h +t,n ≤ (1 + Ch)Gh +t,n +(3.6) +Then, using (3.2) and Young’s inequality, we get for any σ1, σ2 > 0, +Xt,n ≥ NhGh +t,n − +σ1 +2 + Ch +� +Rd(|Dh(vn − v)|2 + |D−h(vn − v)|2)ϕ dx +− +σ2 +2 + Ch +� +Rd(|Dh(vn−1 − v)|2 + |D−h(vn−1 − v)|2)ϕ dx +− C(2 + Ch)(σ−1 +1 ++ M2σ−1 +2 ) +� +Rd |vn − v|2ϕ dx +≥ (Nh − σ1)Gh +t,n − σ2Gh +t,n−1 − C(σ−1 +1 ++ M2σ−1 +2 )e−AtEt,n. +Using this and ET,n = 0, and integrating (3.3) over [t, T], we obtain for some universal C > 0, +−Et,n ≥ (A − Cσ−1 +1 +− CM2σ−1 +2 ) +� T +t +Es,n ds ++ (Nh − σ1) +� T +t +eAsGh +s,n ds − σ2 +� T +t +eAsGh +s,n−1 ds. +(3.7) +Now taking σ1 := h/2, σ2 := h/4 and A := 6CM2/h, then (3.7) and N ≥ 1 yield +� T +t +eAsGh +s,n ds ≤ 1 +2 +� T +t +eAsGh +s,n−1 ds ≤ . . . ≤ 2−n +� T +t +eAsGh +s,0 ds. +With this, (3.7) also shows that Et,n ≤ h +4 +� T +t eAsGh +s,n−1 ds ≤ +h +2n+1 +� T +t eAsGh +s,0 ds. Therefore, for +all n ≥ 0 and t ∈ [0, T], we obtain +� +BR +|vn(t, x) − v(t, x)|2 dx ≤ +h +2n+1 +� T +t +� +Rd eA(s−t)|Dh(v0(s, x) − v(s, x))|2ϕ(|x|) dxds, +which, combined with Lemma 2.4, concludes the proof. +□ +Remark 3.1. In the proof of Theorem 3.1, we only used the following: uniform Lipschitz +continuity of f, c and α, and uniform boundedness of f and |∇hvh|. In particular, the solutions +vh +n and vh are allowed to have certain growth at x = ∞, and the comparison principle is not +needed. +By Theorem 3.1, we immediately have the convergence of the policies. +Theorem 3.2. Assume (A1)–(A2) and N ≥ 1. Then there exists a universal constant C > 0 +such that for all n ≥ 0 and R ≥ 1 we have +sup +t∈[0,T] +� +BR +���α(t, x, ∇hvh +n(t, x)) − α(t, x, ∇hvh(t, x)) +��� +2 +dx ≤ C2−n exp [C exp(CT)/h] Rd. + +10 +W. TANG, H. V. TRAN, Y. P. ZHANG +Proof. Since α is Lipschitz continuous, +� +BR +���α(t, x, ∇hvh +n(t, x)) − α(t, x, ∇hvh(t, x)) +��� +2 +dx +≤ C +h2 +d +� +i=1 +� +BR +���vh +n(t, x + hei) − vh(t, x + hei) − vh +n(t, x − hei) + vh(t, x − hei) +��� +2 +dx. +We can then conclude the proof from Theorem 3.1. +□ +3.2. Convergence of vh as h → 0. Let vh and v be, respectively, solutions to (2.14) and +(2.4). We show |vh − v| ≤ CT +√ +h, where the rate is sharp (we refer to a simple example given +in [14]). +Theorem 3.3. Assume (A1)–(A2) and N ≥ max{1, ∥f∥∞/2}. Then there exists a universal +constant C > 0 such that +sup +(t,x)∈[0,T]×Rd |v(t, x) − vh(t, x)| ≤ C(1 + T)(1 + ∥∇v∥∞) +√ +h. +In particular, we have sup(t,x)∈[0,T]×Rd |v(t, x) − vh(t, x)| ≤ C exp(CT) +√ +h. +Remark 3.2. This rate was obtained in [15, 27] for a large class of parabolic Bellman equa- +tions with Lipschitz coefficients. We apply a different argument – the classical doubling variable +method that is used in [13] in which a discrete space-time homogeneous Hamilton-Jacobi equa- +tion is discussed. This argument allows us to obtain the same sharp estimate for the scheme +(2.18), while it seems that the method in [15, 27] cannot (see Remark 4.1). See also [11] for a +different proof of this convergence rate via the nonlinear adjoint method. +Proof. Let us assume that T ≥ 1. Suppose for some (t0, x0) ∈ [0, T] × Rd such that +8σ := v(t0, x0) − vh(t0, x0) ≥ 1 +2 +sup +(t,x)∈[0,T]×Rd +� +v(t, x) − vh(t, x) +� +> 0. +(3.8) +Below we will show σ ≤ CT(1 + ∥∇v∥∞) +√ +h. +Consider a smooth function g : Rd+1 → [0, 1] such that +(g1) g(t, x) = 1 − t2 − |x|2 if t2 + |x|2 < 1/2, +(g2) 0 ≤ g(t, x) ≤ 1/2 if t2 + |x|2 > 1/2, and g(t, x) = 0 if t2 + |x|2 > 1. +For ε > 0, denote gε(t, x) := g(t/ε, x/ε), and +L := sup +� +v(t, x), −vh(t, x) : (t, x) ∈ [0, T] × Rd� ++ 1 ≥ 1, +By Lemma 2.4, σ ≤ L ≤ CT for some universal constant C > 0. Next, for φ(x) := (1+ |x|2)1/2 +and R ≥ |x0| + T, we define Φh : [0, T]2 × R2d → R by +Φh(t, s, x, y) := v(t, x) − vh(s, y) − σ +T (2T − t − s) +− σ +R(φ(x) + φ(y)) + (8L + 2σ)gε(t − s, x − y). +Since v, vh are bounded, there exists (t1, s1, x1, y1) ∈ [0, T]2 × R2d such that +Φh(t1, s1, x1, y1) = +max +[0,T]2×R2d Φh(t, s, x, y). +(3.9) + +11 +Due to φ(x0) ≤ R, by (3.8), +Φh(t1, s1, x1, y1) ≥ Φh(t0, t0, x0, x0) ≥ 8L + 6σ. +(3.10) +Since max{v(t1, x1), −vh(s1, y1)} ≤ L, we deduce Φh(t1, s1, x1, y1) ≤ 2L + (8L + 2σ)gε(t1 − +s1, x1 − y1), which, together with (3.10), implies gε(t1 − s1, x1 − y1) ≥ 3/4. Then by (g1), we +get that for some C > 0, +gε(t − s, x − y) = 1 − ε−2|t − s|2 − ε−2|x − y|2, +(3.11) +whenever |t − t1|, |s − s1|, |x − x1|, |y − y1| ≤ ε/C. +Now, by (3.9), the mapping +(t, x) �→ v(t, x) + σ +T t − σ +Rφ(x) + (8L + 2σ)gε(t − s1, x − y1). +(3.12) +is maximized at (t, x) = (t1, x1). Together with the fact that v is Lipschitz continuous (taking +M := 1+∥∇v∥∞) and |∇φ| ≤ 1, we find that |∇x gε(t1−s1, x1 −y1)| ≤ (M +σR−1)(8L+2σ)−1 +and, |∂t gε(t1 − s1, x1 − y1)| ≤ (M + σT −1)(8L + 2σ)−1. By (3.11), σ ≤ L ≤ CT and R ≥ T, +these yield +|x1 − y1| ≤ Cε2(M + σR−1)(L + σ)−1 ≤ Cε2ML−1, +(3.13) +and +|t1 − s1| ≤ Cε2(M + σT −1)(L + σ)−1 ≤ Cε2ML−1. +(3.14) +Now, we firstly assume that t1, s1 < T. In view of (3.12), we apply the viscosity solution +test for v to get +− σ +T − (8L + 2σ) ∂tgε(t1 − s1, x1 − y1) ++ H +� +t1, x1, σ +R∇φ(x1) − (8L + 2σ)∇x gε(t1 − s1, x1 − y1)) +� +≥ 0. +(3.15) +Similarly, since (s, y) → vh(s, y) − σ +T s + σ +Rφ(y) − (8L + 2σ)gε(t1 − s, x1 − y) is minimized at +(s1, y1), the comparison principle yields +σ +T − (8L + 2σ) ∂tgε(t1 − s1, x1 − y1) ++ H +� +s1, y1, − σ +R∇hφ(y1) − (8L + 2σ)∇h +x gε(t1 − s1, x1 − y1) +� +− Nh∆h � σ +Rφ(y1) − (8L + 2σ)gε(t1 − s1, x1 − y1)) +� +≤ 0. +Thus we get +2σ +T ≤ H +� +t1, x1, σ +R∇φ(x1) − (8L + 2σ)∇x gε(t1 − s1, x1 − y1)) +� +− H +� +s1, y1, − σ +R∇φ(y1) − (8L + 2σ)∇h +x gε(t1 − s1, x1 − y1) +� ++ Nh∆h � σ +Rφ(y1) − (8L + 2σ)gε(t1 − s1, x1 − y1)) +� +. +(3.16) +It follows from (3.11) that for h ≪ ε, we have at point (t1 − s1, x1 − y1), +∇h +x gε = ∇x gε = 2ε−2(x1 − y1), +∆hgε = −2dε−2. +(3.17) +Due to |∇φ| ≤ 1 and ∆hφ ≤ C, we get +Nh∆h � σ +Rφ(y1) − (8L + 2σ)gε(t1 − s1, x1 − y1)) +� +≤ CLε−2h. +(3.18) + +12 +W. TANG, H. V. TRAN, Y. P. ZHANG +Using (3.16)–(3.18) and the regularity of H (see (2.16)), we obtain for some universal C, +2σT −1 ≤ CσR−1(|∇φ(x1)| + |∇φ(y1)|) + CLε−2h ++ C(|t1 − s1| + |x1 − y1|) [1 + (8L + 2σ)|∇x gε(t1 − s1, x1 − y1)|] +≤ CσR−1 + CLε−2h + C(|t1 − s1| + |x1 − y1|) +� +1 + Lε−2|x1 − y1| +� +which, by (3.13) and (3.14), yields σT −1 ≤ CσR−1 + CLε−2h + Cε2M2L−1. Now we take ε := +M−1/2L1/2h1/4 and pass R → ∞. Then when h is sufficiently small, we obtain σ ≤ CTM +√ +h +for some universal C > 0. This finishes the proof of the upper bound of sup[0,T]×Rd(v − vh) in +the case when t1, s1 < T. +Next, suppose that one of t1 and s1 equals to T. Let us only prove for the case when t1 = T. +By (3.10) and the definition of Φh, +8L + 6σ ≤ v(t1, x1) − vh(s1, y1) + (8L + 2σ)gε(t1 − s1, x1 − y1). +It follows from the proof Lemma 2.4 that vh is Lipschitz continuous with unit Lipschitz constant +when |T − t| ≤ C. Note that ε2ML−1 ≤ C. Hence (3.13) and (3.14) yield +8L + 6σ ≤ |v(T, x1) − q(y1)| + |q(y1) − vh(s1, y1)| + (8L + 2σ)gε(T − s1, x1 − y1) +≤ C(|x1 − y1| + |T − s1|) + 8L + 2σ ≤ Cε2ML−1 + 8L + 2σ. +This yields σ ≤ C +√ +h for some universal C > 0. +Finally, the upper bound estimate for sup[0,T]×BR(vh−v) follows by using the same argument +as the above. Applying Lemma 2.4 permits to conclude. +□ +3.3. Almost everywhere convergence of the policy. It was proved in [23] that the solution +v to (2.4) is semi-concave in space. From this, we are able to derive the almost everywhere +convergence of the policies. +Below we say that a function g : Rd → R is uniformly semi-concave if there exists C > 0 such +that for all x, y ∈ Rd we have g(x+y)+g(x−y)−2g(x) ≤ C|y|2. If g is uniformly bounded and +Lipschitz continuous, and both ±g are uniformly semi-concave, then g is bounded in W 2,∞(Rd). +We make the following assumption: +(A3) q(·) is uniformly semi-concave, and c(t, ·, a), f(t, ·, a) are bounded in W 2,∞(Rd) uni- +formly in t ∈ [0, T] and a ∈ A. +Theorem 3.4. Under the assumptions of Theorem 3.3, further assume (A3). Then v(t, ·) is +uniformly semi-concave for all t ∈ [0, T]. Moreover, for each t ∈ [0, T] we have for a.e. x ∈ Rd, +α(th, xh, ∇hvh(th, xh)) → α(t, x, ∇v(t, x)) +as h → 0 +where [0, T] × Rd ∋ (th, xh) → (t, x) as h → 0. +We next show a weak type of semi-concavity of vh. +Theorem 3.5. Under the assumptions of Theorem 3.4, there exists C > 0 (also depending on +(A3)) such that for all h ∈ (0, 1), t ∈ [0, T] and x, y ∈ Rd, +vh(t, x + y) + vh(t, x − y) − 2vh(t, x) ≤ C exp(CT) (|y|2 + +√ +h). +The proofs of the two theorems are similar to those of Theorem 4.3 and Theorem 4.4, and +we choose to write the full details down there (as it is slightly more complicated there). + +13 +4. Analysis of discrete space-time schemes +4.1. Convergence of PI. The parallel result of Theorem 3.1 on the convergence of V τ,h +n +→ +V τ,h holds the same. However the proof is more involved due to the discretization in the time +direction. In it, we will emphasize the difference. +Theorem 4.1. Assume (A1)–(A2) and N ≥ 1. Let Vn := V τ,h +n +and V := V τ,h be, respectively, +continuous solutions to (2.18) and (2.20). Then there exists a universal constant C > 0 such +that if C(1 + ∥∇hV ∥2 +∞)τ ≤ h, we have for all n ≥ 1, R ≥ 1 and t ∈ Nτ +T, +� +x∈Zd +h,|x|≤R +|Vn(t, x) − V (t, x)|2 ≤ +hτ +2n+1 +� +t≤s∈Nτ +T +� +x∈Zd +h +exp +� +C exp(1 + ∥∇hV ∥2 +∞)(s − t)/h +� ���Dh(V0(s, x) − V (s, x)) +��� +2 +min +� +1, e−|x|+R+1� +. +In particular, we have +max +t∈Nτ +T +� +x∈Zd +h,|x|≤R +|Vn(t, x) − V (t, x)|2 ≤ C2−n exp [C exp(CT)/h] Rd. +max +t∈Nτ +T +� +x∈Zd +h,|x|≤R +���α(t, x, ∇hVn(t, x)) − α(t, x, ∇hV (t, x)) +��� +2 +≤ C2−n exp [C exp(CT)/h] Rd. +Proof. Assume T ≥ 1. +Let ϕ = ϕR : [0, ∞) → [0, 1] be C1 and satisfying (3.1), and let +A := CT 2/h for some C > 0 to be determined. Then for t ∈ Nτ +T set +Et,n := 1 +2eAt � +x∈Zd +h +|Vn(t, x) − V (t, x)|2ϕ(|x|) +which is finite. Direct computation yields +Et,n − Et−τ,n +τ +≥ Ae−AτEt,n + 1 +2eA(t−τ) � +x∈Zd +h +(Vn(t, x) + Vn(t − τ, x) − V (t, x) − V (t − τ, x))× +(∂τ +t Vn(t, x) − ∂τ +t V (t, x))ϕ(|x|) += Ae−AτEt,n + eA(t−τ) � +x∈Zd +h +(Vn(t, x) − V (t, x))(∂τ +t Vn(t, x) − ∂τ +t V (t, x))ϕ(|x|) +− τ +2eA(t−τ) � +x∈Zd +h +|∂τ +t Vn(t, x) − ∂τ +t V (t, x)|2 ϕ(|x|) +=: Ae−AτEt,n + eA(t−τ)Xt,n − τ +2eA(t−τ)Yt,n. +(4.1) +First, we consider the term Yt,n (which does not appear in the semi-discretization problem +in Theorem 3.1). It follows from the equations (2.18) and (2.20) that +Yt,n = +� +x∈Zd +h +���cn + ∇hVn · fn + Nh∆hVn − H(t, x, ∇hV ) − Nh∆hV +��� +2 +ϕ(|x|) + +14 +W. TANG, H. V. TRAN, Y. P. ZHANG +Recall that α = α(t, x, ∇hV ), H(t, x, ∇hV ) = c(t, x, α) + f(t, x, α) · ∇hV , and |∇hV | ≤ M for +some M ≥ 1. Since αn = α(t, x, ∇hVn−1), the regularity assumptions and (2.11) yield +Yt,n ≤ C +� +x∈Zd +h +� +M2|DhVn−1 − DhV |2 + M2|D−hVn−1 − D−hV |2+ +|DhVn − DhV |2 + |D−hVn − D−hV |2� +ϕ(|x|) ≤ C +� +M2Gh +t,n−1 + Gh +t,n +� +, +where in the last inequality we used the notation Gh +t,n := � +x∈Zd +h |DhVn(t, x)−DhV (t, x)|2ϕ(|x|). +and (3.6) with the above defined Gh +t,n (which clearly holds the same). +Next, we consider the term Xt,n. Note that for any v ∈ L∞(Zd +h), � +x∈Zd +h ∆hv vϕ = − � +x∈Zd +h |Dhv|2ϕ− +� +x∈Zd +h vD−hv ·D−hϕ. So, similarly as before (also using the equation, (3.1), (3.6), the uniform +Lipschitz assumptions, and Young’s inequality), we have for any σ1, σ2 > 0, +Xt,n ≥ NhGh +t,n − C +� +x∈Zd +h +|Vn − V |(|Dh(Vn − V )| + |D−h(Vn − V )|)ϕ +− CM +� +x∈Zd +h +|Vn − V |(|Dh(Vn−1 − V )| + |D−h(Vn−1 − V )|)ϕ +≥ (Nh − σ1)Gh +t,n − σ2Gh +t,n−1 − C(σ−1 +1 ++ M2σ−1 +2 )e−AtEt,n. +Since ET,n ≡ 0, putting the above together and summing up (4.1) with respect to t yield +−Et,n/τ ≥ (Nh − σ1 − Cτ) +� +t+τ≤s∈Nτ +T +eA(s−τ)Gh +s,n − (σ2 + CM2τ) +� +t+τ≤s∈Nτ +T +eA(s−τ)Gh +s,n−1 ++ (A − Cσ−1 +1 +− CM2σ−1 +2 )e−Aτ +� +t+τ≤s∈Nτ +T +Eh +s,n +(4.2) +for some universal constant C > 0. +Finally we take σ1 := h/4, σ2 := h/8, A := 12CM2/h. Then if τ ≤ h/(8CM2), (4.2) yields +� +t+τ≤s∈Nτ +T +eA(s−τ)Gh +s,n ≤ 1 +2 +� +t+τ≤s∈Nτ +T +eA(s−τ)Gh +s,n−1 ≤ . . . ≤ 2−n +� +t+τ≤s∈Nτ +T +eA(s−τ)Gh +s,0, +and then Et,n ≤ hτ +4 +� +t+τ≤s∈Nτ +T eA(s−τ)Gh +s,n−1 ≤ +hτ +2n+1 +� +t+τ≤s∈Nτ +T eA(s−τ)Gh +s,0. This, together +with Proposition 2.5, concludes the proof of the first claim as before. +The second claim follows similarly as in Theorem 3.2. +□ +By shifting the solutions, we actually obtain uniform pointwise exponential convergence of +V τ,h +n +to V τ,h and α(·, ·, ∇hV τ,h +n +) to α(·, ·, ∇hV τ,h) as n → ∞, in Ωτ,h +T . +4.2. Convergence of V τ,h as τ, h → 0. Let V τ,h and v be, respectively, solutions to (2.20) +and (2.4). The following theorem proves that the difference between V τ,h and v is at most of +order +√ +h. The argument follows the idea of [13, Theorem 1], which considered the discrete +space-time scheme for the homogeneous Hamilton-Jacobi equation vt + H(Dv) = 0. + +15 +Theorem 4.2. Assume (A1)–(A2) and (2.17). Then there exists a universal C > 0 such that +sup +(t,x)∈Ωτ,h +T +|v(t, x) − V τ,h(t, x)| ≤ C(1 + T)(1 + ∥∇v∥∞) +√ +h. +In particular, we have sup(t,x)∈Ωτ,h +T +|v(t, x) − V τ,h(t, x)| ≤ C exp(CT) +√ +h. +Remark 4.1. It was shown in [14, 15, 27] that +sup +(t,x)∈Ωτ,h +T +|v(t, x) − V τ,h(t, x)| ≤ C(τ 1/4 + h1/2) +for some C = C(T) > 0, +where v solves a general degenerate parabolic Bellman equation and V τ,h is its space-time fi- +nite difference approximation. For the first order equations, our Theorem 4.2 obtains a better +convergence rate. +Proof. Assume T ≥ 1. And suppose for some (t0, x0) ∈ Ωτ,h +T +such that +8σ := v(t0, x0) − V τ,h(t0, x0) ≥ 1 +2 +sup +(t,x)∈Ωτ,h +T +� +v(t, x) − V τ,h(t, x) +� +> 0. +(4.3) +Let DT,τ,h := [0, T] × Nτ +T × Rd × Zd +h, and +L := sup +� +v(t, x), −V τ,h(t, x) : (t, x) ∈ Ωτ,h +T +� ++ 1. +Then σ ≤ L ≤ CT for some universal constant C > 0. Moreover, let R, g and gε with ε ∈ (0, 1), +and φ be from the proof of Theorem 3.3, and define Φh : DT,τ,h → R by +Φh(t, s, x, y) := v(t, x) − V τ,h(s, y) − σ +T (2T − t − s) +− σ +R(φ(x) + φ(y)) + (8L + 2σ)gε(t − s, x − y). +Suppose +Φh(t1, s1, x1, y1) = max +DT,τ,h +Φh(t, s, x, y). +(4.4) +It is clear that (3.10)–(3.14) hold the same. By (3.14) if τ ≪ ε2M/L, we get +|t1 − s1 − τ| ≤ Cε2M/L +with M = 1 + ∥∇v∥∞. +(4.5) +First, assume t1, s1 < T. The viscosity solution test for v shows (3.15) by (3.12). Next since +Ωτ,h +T +∋ (s, y) → V τ,h(s, y) − σ +T s + σ +Rφ(y) − (8L + 2σ)gε(t1 − s, x1 − y) is minimized at (s1, y1), +then for all (s, y) ∈ Ωτ,h +T , +V τ,h(s, y) ≥ V τ,h(s1, y1) − σ +T (s1 − s) + σ +R(φ(y1) − φ(y)) +− (8L + 2σ) [gε(t1 − s1, x1 − y1) − gε(t1 − s, x1 − y)] =: ˜V (s, y). +Recall that s1 + τ ≤ T and Ft from (2.21) satisfies the monotonicity property. We obtain +V τ,h(s1, y1) = Fs1+τ(V τ,h(s1 + τ, ·))(y1) ≥ Fs1+τ( ˜V (s1 + τ, ·))(y1), + +16 +W. TANG, H. V. TRAN, Y. P. ZHANG +which gives +0 ≥ σ +T − (8L + 2σ) ∂τ +t gε(t1 − s1, x1 − y1) ++ H +� +s1 + τ, y1, − σ +R∇hφ(y1) − (8L + 2σ)∇h +x gε(t1 − s1 − τ, x1 − y1) +� +− Nh∆h � σ +Rφ(y1) − (8L + 2σ)gε(t1 − s1 − τ, x1 − y1)) +� +. +(4.6) +By (3.11), if τ, h ≪ ε2, +|∂τ +t gε(t1 − s1, x1 − y1) − ∂tgε(t1 − s1, x1 − y1)| ≤ Cε−2τ, +(4.7) +∇h +x gε(t1 − s1 − τ, x1 − y1) = ∇x gε(t1 − s1, x1 − y1) = 2ε−2(x1 − y1). +(4.8) +Combining (4.6) with (3.15), and using (4.7) and (4.8) yield +2σ +T ≤ H +� +t1, x1, σ +R∇φ(x1) − (8L + 2σ)2ε−2(x1 − y1) +� +− H +� +s1 + τ, y1, − σ +R∇φ(y1) − (8L + 2σ)2ε−2(x1 − y1) +� ++ Nh∆h � σ +Rφ(y1) − (8L + 2σ)gε(t1 − s1 − τ, x1 − y1)) +� ++ CLε−2τ. +(4.9) +The definitions of φ and gε show (3.18). Then, applying (2.16) and (3.18) into (4.9), if (τ ≤ +) h ≪ ε2 we deduce for some C > 0 that +σT −1 ≤ CσR−1(|∇φ(x1)| + |∇φ(y1)|) + CLε−2h + CLε−2τ ++ C(|t1 − s1 − τ| + |x1 − y1|) +� +1 + (8L + 2σ)2ε−2|x1 − y1| +� +≤ CσR−1 + CLε−2h + Cε2M2L−1 +(4.10) +where in the second inequality we also used (3.13) and (4.5). +Now we take ε := M−1/2L1/2h1/4, and send R → ∞. It is clear that τ ≪ ε2M/L is satisfied +when h is small. We obtain from (4.10) that σ ≤ CTM +√ +h, which finishes the proof of the +upper bound of supΩτ,h +T (v − V τ,h) in the case when t1, s1 < T. +Next, if at least one of t1 and s1 equals to T, the argument of Theorem 3.3 applies the same +except that we need to use Proposition 2.5 in place of Lemma 2.4. Finally, the proof for the +upper bound of supΩτ,h +T (V τ,h − v) is the same. +□ +4.3. Almost everywhere convergence of the policy. We show the almost everywhere +convergence of the policy, and some semi-concavity property of the solution. +Theorem 4.3. Under the assumptions of Theorem 4.2, further assume (A3). Then v is uni- +formly semi-concave for all t ∈ [0, T]. Moreover, for each t ∈ [0, T] we have for a.e. x ∈ Rd, +α(th, xh, ∇hV τh,h(th, xh)) → α(t, x, ∇v(t, x)) +as h → 0 +where Ωτh,h +T +∋ (th, xh) → (t, x) as h → 0 and τh satisfies 0 < 2Nτh ≤ h. +Proof. The semi-concavity of v(t, ·) follows from [23]. +For the second claim, it suffices to prove that for a fixed t ∈ [0, T), and for a.e. x ∈ Rd, +∇hV τh,h(th, xh) → ∇v(t, x) +as h → 0. +(4.11) + +17 +For any function g : Rd → R, let us denote by D+g(x) the set of subdifferential of g: +D+g(x) := +� +p ∈ Rd �� lim sup +y→x +g(y) − g(x) − p · (y − x) +|y − x| +≤ 0 +� +. +Due to v(t, ·) is semi-concave, D+v(t, x) is non-empty for all x ∈ Rd. +Because v(t, ·) is Lipschitz continuous, ∇xv(t, x) exists for a.e. x ∈ Rd. We fix one such x. +Since V τh,h are Lipschitz continuous uniformly in h, after passing to a subsequence of h → 0, we +can assume that ∇hV τh,h(th, xh) → p for some p ∈ Rd. Since V τh,h(th, xh) → v(t, x) as h → 0, +the stability of subdifferential yields that p ∈ D+v(t, x). While because ∇xv(t, x) exists, we +get p = ∇xv(t, x). Note that this is for any convergent subsequence of ∇hV τh,h(th, xh), and so +we obtain (4.11). +□ +Below, we show a weak type of semi-concavity of V τ,h(t, ·). We adopt the “doubling variable” +method, see e.g., [23]. +Theorem 4.4. Under the assumptions of Theorem 4.3, there exists C > 0 (also depending on +(A3)) such that for all t ∈ Nτ +T and x, y ∈ Zd +h, +V τ,h(t, x + y) + V τ,h(t, x − y) − 2V τ,h(t, x) ≤ C exp(CT) (|y|2 + +√ +h). +Proof. It suffices to show that there exist CT , C′ +T > 0 depending on the assumptions such that +V τ,h(t, x) + V τ,h(t, z) − 2V τ,h(t, y) ≤ CT +� +|x − y|2 + |z − y|2 + |x + z − 2y| +� ++ C′ +T +√ +h +(4.12) +for all t ∈ Nτ +T and x, y, z ∈ Zd +h. By the assumption on q, the inequality holds for t = T with +CT = ∥q∥W 2,∞ =: C0, and C′ +T = 0. +Suppose for contradiction that (4.12) fails. Then we have for some C1 ≥ 1 to be determined, +and some C ≥ 2, +V τ,h(t, x) + V τ,h(t, z) − 2V τ,h(t, y) +− 2C0eC1(T−t) � +|x − y|4 + |z − y|4 + |x + z − 2y|2�1/2 ≥ CeC1(T−t)√ +h +(4.13) +for some (t, x, y, z) = (t′, x′, y′, z′) ∈ Nτ +T × Zd +h. Since V τ,h(t, ·) is Lipschitz continuous (with +Lipschitz constant bounded by C exp(C(T − t)) by Proposition 2.5 with a shift in time), after +possibly enlarging the constant C in (4.13), we can assume that +|x′ + z′ − 2y′| ≥ +√ +h. +(4.14) +Let us denote ψ(x, y, z) := |x−y|4+|z−y|4+|x+z−2y|2, and by (4.14), δ := ψ(x′, y′, z′)1/2 ≥ +√ +h. Then for all ε > 0 sufficiently small, we obtain from (4.13) that +Φ(t, x, y, z) := eC1t � +V τ,h(t, x) + V τ,h(t, z) − 2V τ,h(t, y) +� +− C0eC1T � +δ + δ−1ψ(x, y, z) +� +− ε|y|2 +satisfies Φ(t′, x′, y′, z′) ≥ eC1T √ +h. With the positive ε-term, Φ obtains its positive maximum +that is at least eC1T √ +h in Ωτ,h +T +at some point (t0, x0, y0, z0) ∈ Nτ +T × Zd +h, where (t0, x0, y0, z0) +depends on ε and δ. It is clear that t0 ≤ T − τ by the choice of C0. Moreover, for γ0 := +δ + δ−1ψ(x0, y0, z0), we have +V τ,h(t0, x0) + V τ,h(t0, z0) − 2V τ,h(t0, y0) ≥ C0eC1(T−t0)γ0 + eC1(T−t0)√ +h. +(4.15) +Due to uniform boundedness of V τ,h, by further taking ε to be small enough depending on C, T +and h, it is easy to get ε|y0| ≤ h. + +18 +W. TANG, H. V. TRAN, Y. P. ZHANG +Now since Ωτ,h +T +∋ (t, x) → eC1tV τ,h(t, x) − C0eC1T δ−1 � +|x − y0|4 + |x + z0 − 2y0|2� +is maxi- +mized at (t0, x0), we get for all (t, x) ∈ Ωτ,h +T +that +V τ,h(t, x) ≤ eC1(t0−t)V τ,h(t0, x0) + C0eC1(T−t)δ−1 � +|x − y0|4 + |x + z0 − 2y0|2� +− C0eC1(T−t0)δ−1 � +|x0 − y0|4 + |x0 + z0 − 2y0|2� +=: ˜V (t, x). +Due to the equation and the monotonicity property of Ft (defined in (2.21)), V τ,h(t0, x0) = +Ft0+τ(V τ,h(t0 + τ, ·))(x0) ≤ Ft0+τ( ˜V (t0 + τ, ·))(x0). By direct computation, +∇h +x(|x − y0|4 + |x + z0 − 2y0|2) = 4(|x − y0|2 + h2)(x − y0) + 2(x + z0 − 2y0), +∆h +x(|x − y0|4 + |x + z0 − 2y0|2) = (8 + 4d)|x − y0|2 + 2dh2 + 2d. +We then get +(1 − e−C1τ) +τ +V τ,h(t0, x0) ≤ H +� +t0 + τ, x0, ∇h +x ˜V (t0 + τ, x0) +� ++ Nh∆h +x ˜V (t0 + τ, x0) +≤ H (t0 + τ, x0, 2CT,δ(qx0 + p0)) + CCT,δh(|x0 − y0|2 + 1) +(4.16) +where qx0 := 2(|x0 − y0|2 + h2)(x0 − y0), +CT,δ := C0eC1(T−t0−τ)/δ +and +p0 := x0 + z0 − 2y0. +(4.17) +Similarly, since Ωτ,h +T +∋ (t, z) → eC1tV τ,h(t, z) − C0eC1T δ−1(|z − y0|4 + |x0 + z − 2y0|2) is +maximized at (t0, z0), we get +(1 − e−C1τ) +τ +V τ,h(t0, z0) ≤ H (t0 + τ, z0, 2CT,δ(qz0 + p0)) + CCT,δh(|z0 − y0|2 + 1). +(4.18) +where qz0 := 2(|z0 − y0|2 + h2)(z0 − y0). +Next, note that Ωτ,h +T +∋ (t, y) → 2eC1tV τ,h(t, y) + C0eC1T δ−1ψ(x0, y, z0) + ε|y|2 is minimized +at (t0, y0). Hence we get V τ,h(t0, y0) ≥ Ft0+τ( ˆV (t0 + τ, ·))(y0) where +ˆV (t, y) := eC1(t0−t)V τ,h(t0, y0) − (ε/2)|y|2 + (ε/2)|y0|2 +− (C0/2)eC1(T−t)δ−1ψ(x0, y, z0) + (C0/2)eC1(T−t0)δ−1ψ(x0, y0, z0). +From this we obtain +−(1 − e−C1τ) +τ +V τ,h(t0, y0) ≤ −H (t0 + τ, y0, 2CT,δ(qy0 + p0) − εy0) ++ CCT,δh(|x0 − y0|2 + |z0 − y0|2 + 1) + Chε +where qy0 := (|x0 − y0|2 + h2)(x0 − y0) + (|z0 − y0|2 + h2)(z0 − y0), and CT,δ and p0 are given +in (4.17). Using |Hp| ≤ C and ε|y0| ≤ h yields +−(1 − e−C1τ) +τ +V τ,h(t0, y0) ≤ −H (t0 + τ, y0, 2CT,δ(qy0 + p0)) ++ CCT,δh(|x0 − y0|2 + |z0 − y0|2 + 1) + Ch +(4.19) +Now let α ∈ A be such that +H (t0 + τ, y0, 2CT,δ(qy0 + p0)) = c(t0 + τ, y0, α) + 2CT,δf(t0 + τ, y0, α) · (qy0 + p0). + +19 +By (2.15), denoting cα(·) := c(t0 + τ, ·, α) and fα(·) := f(t0 + τ, ·, α), we have +H (t0 + τ, x0, 2CT,δ(qx0 + p0)) + H (t0 + τ, z0, 2CT,δ(qz0 + p0)) +− 2H (t0 + τ, y0, 2CT,δ(qy0 + p0)) +≤ cα(x0) + cα(z0) − 2cα(y0) + 2CT,δ [fα(x0) · (qx0 + p0) ++ fα(z0) · (qx0 + p0) − 2fα(y0) · (qy0 + p0)] += cα(x0) + cα(z0) − 2cα(y0) + 2CT,δ [(fα(x0) − fα(y0)) · qx0 + (fα(z0) − fα(y0)) · qz0+ ++ (fα(x0) + fα(z0) − 2fα(y0)) · p0] +≤ ∥cα∥W 2,∞(|x0 − y0|2 + |z0 − y0|2 + |x0 + z0 − 2y0|) ++ 2CT,δ∥fα∥Lip(|x0 − y0||qx0| + |z0 − y0||qz0|) ++ 2CT,δ∥fα∥W 2,∞(|x0 − y0|2 + |z0 − y0|2 + |x0 + z0 − 2y0|)|x0 + z0 − 2y0|, +(4.20) +where we used 2qy0 = qx0 + qz0 and that for any x, y, z ∈ Rd and g ∈ W 2,∞(Rd), |g(x) + g(z) − +2g(y)| ≤ ∥g∥W 2,∞(|x−y|2 +|z −y|2 +|x+z −2y|). By Young’s inequality, we get |x0 −y0||qx0|+ +|z0 − y0||qz0| ≤ 2|x0 − y0|4 + 2|z0 − y0|4 + h4. Also using the definitions of CT,δ and ψ, we get +the left-hand side of (4.20) ≤ CeC1(T−t0)(δ +δ−1ψ(x0, y0, z0)) = CeC1(T−t0)γ0 +CeC1(T−t0)h4/δ +with C > 0 only depending on ∥q∥W 2,∞, ∥cα∥W 2,∞ and ∥fα∥W 2,∞. +Now summing up (4.16), (4.18) and twice of (4.19), we get +(1 − e−C1τ) +τ +� +V τ,h(t0, x0) + V τ,h(t0, z0) − 2V τ,h(t0, y0) +� +≤ CeC1(T−t0)γ0 + CeC1(T−t0)h4/δ + CCT,δh(|x0 − y0|2 + |z0 − y0|2 + 1) + Ch +≤ CeC1(T−t0)γ0 + CeC1(T−t0)δ−1(|x0 − y0|4 + |z0 − y0|4) + CeC1(T−t0)√ +h +≤ CeC1(T−t0)γ0 + CeC1(T−t0)√ +h, +where in the second inequality, we used δ ≥ +√ +h. Finally, this and (4.15) yield +C1(C0eC1(T−t0)γ0 + eC1(T−t0)√ +h) ≤ CeC1(T−t0)γ0 + CeC1(T−t0)√ +h, +with C > 0 depending only on d, N and the regularity assumptions of q, c, f. Thus, if C1 is +sufficiently large depending only on the assumptions, we get a contradiction which finishes the +proof of (4.12), which finishes the proof. +□ +5. Generalization: a PDE perspective +In this section, we consider PI for HJB equations with a general Hamiltonian. For convenient +use of the Legendre transform, we write the system in the forward-in-time setting. It is easy +to carry over to the backward-in-time setting. +Suppose H : [0, T] × Rd × Rd → R is continuous such that H(t, x, p) is convex in p. Let +L(t, x, µ) be the Legendre transform of H, that is, +L(t, x, µ) := sup +p∈Rd [p · µ − H(t, x, p)] +for (t, x, µ) ∈ [0, T] × Rd × Rd. +We always have the following inequality L(t, x, µ)+H(t, x, p) ≥ p·µ, with equality holds if and +only if µ = ∇pH(t, x, p), and if and only if p = ∇µL(t, x, µ). + +20 +W. TANG, H. V. TRAN, Y. P. ZHANG +The HJB equation is +� +∂tv + H(t, x, ∇v) = 0 +in (0, T) × Rd, +v(0, x) = q(x) +on Rd. +(5.1) +Under some assumptions (see [3, 42]), it is a classical result that v is uniformly Lipschitz +continuous if q is Lipschitz continuous. So we can assume +∥∇v∥L∞([0,T]×Rd) ≤ M +for some M > 0. +(5.2) +Now we take m1 := min +|p|=2M, +t∈[0,T],x∈Rd +H(t, x, p) and m2 ≥ max +|p|=3M, +t∈[0,T],x∈Rd +[H(t, x, p) − m1]/M, +and we can assume that m2 ≥ 2. Then define +˜H(t, x, p) := + + + + + +H(t, x, p) +if |p| ≤ 2M, +max {H(t, x, p), m1 + m2(|p| − 2M)} +if 2M < |p| ≤ 3M, +m1 + m2(|p| − 2M) +if |p| > 3M. +It is not hard to verify that ˜H is continuous in all its dependencies, and is convex in p. Due +to (5.2), v is also a solution of (5.1) with H replaced by ˜H. Moreover for N := m2/2 ≥ 1, we +have +| ˜Hp(t, x, p)| ≤ 2N +in [0, T] × Rd × Rd. +(5.3) +We define ˜L as the Legendre transform of ˜H. Since the goal is to approximate v, it suffices to +study ˜H and ˜L instead of H and L. From now on, with a slight abuse of notation, we write H +and L as ˜H and ˜L, respectively. +With the modified operators, we can consider the semi-discretization. For h > 0, +� +∂tvh + H(t, x, ∇hvh) = Nh∆hvh +in (0, T) × Rd, +vh(0, x) = q(x) +on Rd. +(5.4) +As before, N ≥ ∥∇pH∥∞/2 guarantees that the finite difference scheme is monotone. Let us +also assume that there exists C > 0 such that for all t, x, p ∈ [0, T] × Rd × Rd, +|Ht(t, x, p)|, |Hx(t, x, p)| ≤ C(1 + |p|), +|H(t, x, 0)| ≤ C. +(5.5) +Actually, we can replace C(1 + |p|) by just C for the modified operator. We will not discuss +the space-time discretization of (5.1) since it is similar. +Now we present the iteration scheme for (5.4). Fixing small h > 0, we start with a uniformly +bounded and Lipschitz continuous function vh +0(t, x), and then iteratively compute vh +n as follows. +For n ≥ 1, let vh +n(t, x) be the solution to +� +∂tvh +n + ∇pH(t, x, ∇hvh +n−1) · ∇hvh +n − L(t, x, µh +n−1) = Nh∆hvh +n +in (0, T) × Rd, +vh +n(0, x) = q(x) +on Rd +(5.6) +where we denoted µh +n(t, x) := ∇pH(t, x, ∇hvh +n). +Note L(t, x, µh +n) is finite due to µh +n ≤ 2N. +Essentially, vh +n solves a linearized equation of (5.4). +Let vh +n (for each n ≥ 1 with given vh +0), vh and v be, respectively, Lipschitz continuous solutions +to (5.6), (5.4) and (5.1). We have the following monotonicity property. + +21 +Proposition 5.1. Suppose N ≥ max{1, ∥∇pH∥∞/2}, and H(t, x, p) is convex in p and satisfies +(5.3) and (5.5). Let q and vh +0 be uniformly bounded and Lipschitz continuous for all h > 0. +Then the solutions vh +n are uniformly bounded for all n ≥ 1 and h > 0. Moreover, we have for +all n ≥ 0, +vh +n+1 ≤ vh +n +in [0, T] × Rd. +We also have the following convergence results. +Theorem 5.2. Under the assumptions of Proposition 5.1, for all R ≥ 1 there exists a constant +C depending only on T and the assumptions such that we have for all t ∈ [0, T], +� +BR +���vh +n(t, x) − vh(t, x) +��� +2 +dx ≤ C2−nheCt/hRd, +� +BR +���α(t, x, ∇hvh +n(t, x)) − α(t, x, ∇hvh(t, x)) +��� +2 +dx ≤ C2−neCt/hRd/h. +Moreover, we have sup(t,x)∈[0,T]×Rd |vh(t, x) − v(t, x)| ≤ C +√ +h. +Next, let H take the form H(t, x, p) := supa∈A [c(t, x, a) + p · f(t, x, a)], where A is some set, +c : [0, T] × Rd × A → R and f : [0, T] × Rd × A → Rd. +Theorem 5.3. Under the assumptions of Theorem 5.2, assume that c(t, ·, a), f(t, ·, a) are +bounded in W 2,∞(Rd) uniformly for all t ∈ [0, T] and a ∈ A. Then for each t ∈ [0, T], we +have for a.e. x ∈ Rd, +α(th, xh, ∇hvh(th, xh)) → α(t, x, ∇v(t, x)) +as h → 0, +where [0, T] × Rd ∋ (th, xh) → (t, x) as h → 0. +Moreover, there exists C > 0 depending only on the assumptions such that for all h ∈ (0, 1), +t ∈ [0, T] and x, y ∈ Rd, vh(t, x + y) + vh(t, x − y) − 2vh(t, x) ≤ C exp(CT)(|y|2 + +√ +h). +6. Conclusion +In this paper, we study the convergence rate of PI for optimal control problems in continuous +time. To overcome the problem of ill-posedness, we consider a semi-discrete scheme by adding a +viscosity term using finite differences. We prove that PI for the semi-discrete scheme converges +exponentially fast, and provide a bound on the discrepancy between the semi-discrete scheme +and the optimal control. We also study the discrete space-time scheme, where both space and +time are discretized. +There are a few directions to extend this work. First, under what conditions on the model +parameters does PI (2.7)–(2.8) converge exponentially fast? For instance, for f(t, x, a) = a, +c(t, x, a) = 1 +2|a|2 and q ≡ 0, the HJB equation is ∂tv − 1 +2|∇v|2 = 0 and v(T, x) = 0, which +has the solution v∗ ≡ 0. On the other hand, PI yields vn(t, x) = cn(t)x2 with c1(t) = 1 +2 for +a suitable initialization. It is easy to check that cn(t) ≤ 2−n for n ≥ 1, and thus we get the +exponential convergence of vn to v∗ on any compact set. However, it is not clear what are the +right conditions to impose on the model parameters so that PI converges exponentially fast. +It is also interesting to adapt PI to the differential game setting and design efficient numerical +schemes (see e.g. [21]). We refer to [25, 39] for the use of PI to solve numerically fully nonlinear +HJB and HJBI equations. + +22 +W. TANG, H. V. TRAN, Y. P. ZHANG +References +[1] M. Abu-Khalaf and F. L. Lewis. 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Value iteration adaptive dynamic programming for optimal control of discrete- +time nonlinear systems. IEEE Trans. Cybern., 46(3):840–853, 2015. +(W. Tang) Department of Industrial Engineering and Operations Research, Columbia University, +S.W. Mudd Building, 500 W 120th St, New York, NY 10027 +Email address: wt2319@columbia.edu +(H. V. Tran) Department of Mathematics, University of Wisconsin Madison, Van Vleck Hall, 480 +Lincoln Drive, Madison, WI 53706 +Email address: hung@math.wisc.edu +(Y. P. Zhang) Department of Mathematics and Statistics, Auburn University, Parker Hall, 221 +Roosevelt Concourse, Auburn, AL 36849 +Email address: yzhangpaul@auburn.edu + diff --git a/fNAyT4oBgHgl3EQfjvgl/content/tmp_files/load_file.txt b/fNAyT4oBgHgl3EQfjvgl/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..aa0a0f11f0e688df2ae104553c59d7fb4729499f --- /dev/null +++ b/fNAyT4oBgHgl3EQfjvgl/content/tmp_files/load_file.txt @@ -0,0 +1,1177 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf,len=1176 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='00419v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='OC] 1 Jan 2023 POLICY ITERATION FOR THE DETERMINISTIC CONTROL PROBLEMS – A VISCOSITY APPROACH WENPIN TANG, HUNG VINH TRAN, YUMING PAUL ZHANG Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' This paper is concerned with the convergence rate of policy iteration for (determin- istic) optimal control problems in continuous time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' To overcome the problem of ill-posedness due to lack of regularity, we consider a semi-discrete scheme by adding a viscosity term via finite differences in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' We prove that PI for the semi-discrete scheme converges exponentially fast, and provide a bound on the error induced by the semi-discrete scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' We also consider the discrete space-time scheme, where both space and time are discretized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Convergence rate of PI and the discretization error are studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Introduction Optimal control is ubiquitous in science and engineering with a variety of applications includ- ing aerospace engineering [6, 10], chemical engineering [31], economy [24], operations research [36, 37] and robotics [2, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Dynamic programming (DP) has proved to be an efficient tool to solve multistage optimal control problems since its inception by Bellman [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' In recent years, reinforcement learning (RL) has shown great success in resolving complex decision making problems, notably AlphaGo [38] and humanoid tasks [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Policy iteration (PI), as a class of approximate or adaptive dynamic programming (ADP), is instrumental in many RL algorithms [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' The idea of PI dates back to Howard [20] in a stochastic environment known as the Markov decision process (MDP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Subsequent works [7, 33, 34] explored PI for MDPs in discrete time and space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' recently, [8, 30] considered PI for (deterministic) optimal control problems in discrete time and continuous space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' In these works, PIs are proved to converge to the optimal control under suitable conditions on the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' On the other hand, many real-world prob- lems are modeled by dynamical systems evolving in continuous time, and it is known that DP for optimal control in continuous time and space entails the Hamilton-Jacobi-Bellman (HJB) partial differential equation (PDE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Despite its importance, PI for optimal control problems in continuous time and space has not been rigorously studied until recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' [1, 43] proved the convergence of PI for continuous-time linear quadratic optimal control problems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' more general cases were settled in [28] under a fixed point assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' For the stochastic control problems, [26, 35] showed that PI converges exponentially fast in the case where controls are only exer- cised on the drift term of the state process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Similar results were derived for the corresponding entropy-regularized problems [22, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' We also mention, in a closely related direction, [9, 45] Date: January 3, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Finite differences, Hamilton-Jacobi-Bellman equations, optimal control, policy iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' The work of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Tang is supported by NSF grants DMS-2113779 and DMS-2206038, and a start-up grant at Columbia University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' The work of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Tran is partially supported by NSF CAREER grant DMS-1843320, a Simons Fellowship, and a Vilas Faculty Early-Career Investigator Award.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' 1 2 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' TANG, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' TRAN, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' ZHANG studied value iteration for optimal control problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' See [29, 44] for recent progress on theory and applications of ADP for optimal control and RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' In this paper, we study the convergence rate of PI for optimal control problems in continuous time and their discretization under fairly general conditions on the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Note that the convergence analysis in [1, 43] relies on the linear quadratic structure of the problem, while [28] assumed that the HJB operator enjoys a fixed point, or a contraction property which is hard to verify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' None of these works quantified the convergence of PI to the optimal control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Moreover, PI for continuous-time control problems may even be ill-posed due to lack of regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Our idea is to introduce a viscosity term “h∆h” in the policy evaluation, where h is the mesh size and ∆h is the discrete Laplacian in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' We call it a semi-discrete scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Essentially the viscosity term is of order 1, which assures that the finite difference scheme is monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' In fact, a monotone scheme is commonly desirable for numerical implementation so the addition of the finite difference viscosity term is natural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' On the other hand, the viscosity term in the semi-discrete scheme mimics the vanishing viscosity approximation to first-order PDEs [16], which forces PI to converge exponentially fast (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='1) as for the stochastic control problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' We also prove that the discrepancy between the optimal control problem and its semi-discrete scheme is of order √ h as h → 0 (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Further we consider the time-discretization, called a discrete space-time scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' The same results hold for PI for the discrete space-time scheme (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='1 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Our results echo recent work [19], which asserts that noise enhances the convergence of finite-horizon RL algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' In our setting, noise corresponds to the viscosity term, and the importance of finite-horizon is seen from various bounds with exponential dependence in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Our analysis relies on PDE techniques, and may carry over to the study of differential games in solving Hamilton-Jacobi- Bellman-Issacs (HJBI) equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' In Section 2, we provide background, and present the semi-discrete and the discrete space-time schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' In Section 3 we study the semi-discrete scheme, and in Section 4 we analyze the discrete space-time scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' We provide further PDE perspectives in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' We conclude with Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Setup and preliminary results In this section, we present the semi-discrete and the discrete space-time schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Consider a system whose state is governed by the ordinary differential equation: dxt dt = f(t, xt, αt), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='1) where for 0 ≤ t ≤ T, xt ∈ Rd is the system state, and αt ∈ A ⊂ Rm is the control or policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Here, A is a given compact subset of Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' The objective is J(t, x, α) := � T t c(s, xs, αs) ds + q(xT ) given xt = x, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='2) and the goal is to minimize this objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Denote by v∗(t, x) := inf α∈At J(t, x, α), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='3) where At is the standard admissible policy defined as At = {α : [t, T] → A : α is measurable}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' It is known that under suitable conditions on c(·) and q(·) (see [17, Chapter 2] or [42, Chapter 3 2]), v∗ defined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='3) is the viscosity solution to � ∂tv + H(t, x, ∇v) = 0 in (0, T) × Rd, v(T, x) = q(x) on Rd, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='4) where the Hamiltonian H is given by H(t, x, p) := infa∈A [c(t, x, a) + p · f(t, x, a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' The optimal policy is given by α∗(t, x) = α(t, x, ∇v∗), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='5) where α(t, x, p) := arg mina∈A [c(t, x, a) + p · f(t, x, a)] , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='6) is assumed for simplicity to be unique throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Policy iteration is an approximate dynamic programming, which alternates between policy evaluation to get the value function with the current control and policy improvement to optimize the value function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' More precisely, for n = 0, 1, · · · , the iterative procedure is: Given αn(t, x), solve the PDE � ∂tvn + c(t, x, αn) + ∇vn · f(t, x, αn) = 0 in (0, T) × Rd, vn(T, x) = q(x) on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='7) Set αn+1(t, x) = α(t, x, ∇vn) = arg mina∈A [c(t, x, a) + ∇vn(t, x) · f(t, x, a)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='8) The key is to understand how the sequence {vn} approximates the optimal value v∗, and how {αn} approximates the optimal policy α∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' On the other hand, it is not clear whether the policy iteration scheme (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='7)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='8) is well- posed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Intuitively, to make sense of αn+1 we need vn to be Lipschitz continuous, for which we then need αn to be Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' This in turn requires ∇vn−1 to be Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' After iterations, this means that we need v0 to be smooth which is in general not true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Semi-discrete schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' For T ≥ 1, h ∈ (0, 1), N ≥ max{1, ∥f∥∞/2} and a given continuous function α0 : R × Rd → A, we solve for n = 0, 1, · · · � ∂tvh n + c(t, x, αn) + ∇hvh n · f(t, x, αn) = −Nh∆hvh n in (0, T) × Rd vh n(T, x) = q(x) on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='9) Then set αn+1(t, x) = α(t, x, ∇hvh n) in (0, T) × Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='10) Here, for any ϕ : Rd → R and h ∈ R \\ {0}, we use the notations ∇hϕ(x) := �ϕ(x + he1) − ϕ(x − he1) 2h , · · · , ϕ(x + hed) − ϕ(x − hed) 2h � , ∆hϕ(x) := d � i=1 ϕ(x + hei) − 2ϕ(x) + ϕ(x − hei) h2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Later we will also write Dhϕ(x) := � ϕ(x+he1)−ϕ(x) h , · · · , ϕ(x+hed)−ϕ(x) h � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' It is clear that ∇hϕ(x) = 1 2(Dhϕ(x) − D−hϕ(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='11) 4 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' TANG, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' TRAN, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' ZHANG The assumption N ≥ ∥f∥∞/2 guarantees that the numerical Hamiltonian is monotone and, as a consequence of this, the following comparison principle holds (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=', [13, 32, 42]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Let vh 0 and ˜vh 0 be, respectively, a bounded continuous super- and sub- solution to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='9) with n = 0, and satisfy ˜vh 0 ≤ vh 0 at t = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Then ˜vh 0 ≤ vh 0 in [0, T] × Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Here by a supersolution (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' subsolution), we mean that it satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='9) with the first equality replaced by ≤ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' ≥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' First, we show that the scheme (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='9)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='10) is well-posed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' We need the following assumptions: (A1) c(·, ·, ·), f(·, ·, ·), q(·) are uniformly bounded and Lipschitz continuous in all of their de- pendences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' (A2) α(·, ·, ·) and α0(·, ·) are uniformly Lipschitz continuous in all of their dependences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Throughout this paper, we write C as various universal constants that only depend on d, N, and the constants in (A1)–(A2) unless otherwise stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Specifically, since T is not a universal constant, we keep track of the dependence on T in most estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' The constants C might vary from one line to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' By CX or C(X) we mean a constant that depend on universal constants and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Assume (A1)–(A2) and N ≥ max{1, ∥f∥∞/2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Then the iterative process (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='9)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='10) is well-defined, that is, there are Lipschitz continuous functions vh n, αn satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='9)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='10) and vh n are uniformly bounded for all n ≥ 0 and h > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Since α0 is Lipschitz continuous, the unique solvability of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='9) for n = 0 follows from [27, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' If one can show that vh 0 is uniformly bounded and Lipschitz continuous with Lipschitz constant Ch, then α1 is Lipschitz continuous with Lipschitz constant Ch/h by the assumption that α is Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' From the same argument, we obtain a unique bounded and Lipschitz solution vh 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' The existence of solutions then follows from iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' First we prove the boundedness of vh 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Since c(·, ·, ·), q(·) are uniformly bounded, we have that ± [∥q∥∞ + ∥c∥∞(T − t)] are a supersolution and a subsolution to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='9) with n = 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Hence −∥q∥∞ − ∥c∥∞(T − t) ≤ vh 0(t, x) ≤ ∥q∥∞ + ∥c∥∞(T − t), for all (t, x) ∈ [0, T] × Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Actually the same bound holds for all vh n by this argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Next we show that vh 0 is Lipschitz continuous with Lipschitz constant independent of h when T is sufficiently small depending only on the assumption (A1) (but not on q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' The general result follows immediately by iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' For simplicity of notations, write G(t, x, p) := c(t, x, α0(t, x)) + p · f(t, x, α0(t, x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Then for M := 2∥∇q∥∞ + 1, define ˜G(t, x, p) := � G(t, x, p) if |p| ≤ M, G(t, x, Mp/|p|) if |p| > M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' It follows from (A1) and the choice of N that | ˜Gt(t, x, p)|, | ˜Gx(t, x, p)| ≤ C(1 + M), | ˜Gp(t, x, p)| ≤ 2N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='12) 5 Now let ˜vh be the solution to � ∂t˜vh + ˜G(t, x, ∇h˜vh) = −Nh∆h˜vh, ˜vh(T, x) = q(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' The goal is to show that ˜vh is Lipschitz continuous, and ˜vh = vh 0 in [0, T] × Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Note that for any e ∈ Sd−1 and s ∈ (0, 1), ps := ˜vh(t,x+se)−˜vh(t,x) s satisfies \uf8f1 \uf8f2 \uf8f3 ∂tps + G1(t, x) + G2(t, x) · ∇hps = −Nh∆hps in (0, T) × Rd, ps(T, x) = q(x + se) − q(x) s on Rd, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='13) where G1(t, x) := 1 s � s 0 ˜Gx � t, x + ze, ∇h˜vh(t, x + se) � e dz, G2(t, x) := � 1 0 ˜Gp � t, x, ∇h˜vh(t, x) + z(∇h˜vh(t, x + se) − ∇h˜vh(t, x)) � dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' It is clear from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='12) that |G1| ≤ C(1 + M) and |G2| ≤ 2N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' This yields that the comparison principle for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='13) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Thus, by comparing ps with ±(∥∇q∥∞ +C(1+M)(T −t)), we obtain |ps(t, x)| ≤ ∥∇q∥∞ + C(1 + M)(T − t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Sending s → 0 yields for some C depending only on (A1), |∇e˜vh(t, x)| ≤ ∥∇q∥∞ + C(1+ M)(T − t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Thus, if t ≤ T ≤ (2C)−1, we have that ˜vh(t, x) is Lipschitz and sup (t,x)∈[0,T]×Rd |∇˜vh(t, x)| ≤ ∥∇q∥∞ + 1/2 + M/2 = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' From the definition of ∇h, we get sup(t,x)∈[0,T]×Rd |∇h˜vh(t, x)| ≤ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Hence, ˜vh is a solution to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='9) for n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' The uniqueness of the solution to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='9) yields that vh 0 ≡ ˜vh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' So we obtain the uniform Lipschitz continuity of vh 0 in space, with Lipschitz constant of the form C exp(CT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' The Lipschitz regularity in time follows from the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' □ We point out that the Lipschitz constant of vh n may depend on both n and h for n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Another consequence of the comparison principle is that the functions vh n are monotone decreasing in n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Under the assumptions of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='2, we have for all n ≥ 0, vh n+1 ≤ vh n in [0, T] × Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' By the definition of αn, c(t, x, αn+1(t, x)) + ∇hvh n · f(t, x, αn+1(t, x)) ≤ c(t, x, αn(t, x)) + ∇hvh n · f(t, x, αn(t, x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Thus vh n is a supersolution to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='9) with subscripts n + 1 as it satisfies ∂tvh n + c(t, x, αn+1) + ∇hvh n · f(t, x, αn+1) ≤ −Nh∆hvh n in (0, T) × Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Therefore, the comparison principle yields vh n ≤ vh n+1 in [0, T] × Rd for each n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' □ 6 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' TANG, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' TRAN, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' ZHANG Since vh n is uniformly bounded for all n ≥ 0, the monotonicity property yields that vh n converges locally uniformly as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Let us denote the limit as vh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Then by the stability property of viscosity solutions, vh solves � ∂tvh + H(t, x, ∇hvh) = −Nh∆hvh in (0, T) × Rd, vh(T, x) = q(x) on Rd, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='14) where H(t, x, p) := c(t, x, α(t, x, p)) + p · f(t, x, α(t, x, p)) = min a∈A [c(t, x, a) + p · f(t, x, a)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='15) Since α(t, x, p) is assumed to be uniformly Lipschitz continuous in all of its dependences, there exists C > 0 such that for all (t, x, p) ∈ [0, T] × Rd × Rd, |Ht(t, x, p)|, |Hx(t, x, p)| ≤ C(1 + |p|), |Hp(t, x, p)| ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='16) Moreover, one can expect that vh converges as h → 0, and the limit v should be the unique solution to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' It was proved in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='2 that vh 0 is uniformly bounded and Lipschitz continuous in [0, T]×Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' By the same proof, we have that vh and v are also uniformly bounded and Lipschitz continuous for all h > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Under the assumptions of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='2, let vh 0, vh and v be, respectively, solu- tions to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='9) (for n = 0), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='14) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Then in [0, T] × Rd, vh 0 , vh and v are bounded by C(1 + T) and are Lipschitz continuous with Lipschitz constant C exp(CT) for some universal constant C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' For a general class of first-order Hamilton-Jacobi (continuous) equations, we refer to [3, 4] for the regularity results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Discrete space-time schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Now we consider the scheme that is discrete in both space and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Let τ, h ∈ (0, 1), and N such that max{1, ∥f∥∞/2} ≤ N ≤ h/(2τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='17) Assuming that T/τ ∈ N+, we denote Nτ T := {0, τ, 2τ, · · · , T}, Zd h := hZd, Ωτ,h T := Nτ T × Zd h and Ω′ T := (Nτ T \\{0}) × Zd h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Given a Lipschitz continuous function α0(t, x), let V τ,h n : Ωτ,h T → R be defined iteratively for n = 0, 1, · · · as follows: � ∂τ t V τ,h n (t, x) + c(t, x, αn) + ∇hV τ,h n f(t, x, αn) = −Nh∆hV τ,h n in Ω′ T , V τ,h n (T, x) = q(x) on Zd h (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='18) with αn+1(t, x) := α(t, x, ∇hV τ,h n ) in Ω′ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='19) Here we used the notation ∂τ t V τ,h n (t, x) := V τ,h n (t,x)−V τ,h n (t−τ,x) τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' 7 We also consider the following equation � ∂τ t V τ,h + H(t, x, ∇hV τ,h) = −Nh∆hV τ,h in Ω′ T , V τ,h(T, x) = q(x) on Zd h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='20) where H is given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' The goal is to show that V τ,h n converges to V τ,h as n → ∞, and V τ,h converges to v as τ, h → 0, where v is given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Similarly as before, we write c := c(t, x, α(t, x, ∇hV τ,h)) and f := f(t, x, α(t, x, ∇hV τ,h)) cn := c(t, x, αn(t, x)) and fn := f(t, x, αn(t, x))) for simplicity, when there is no confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' We will use the following operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' For each t ∈ Nτ T , let Ft : L∞(Zd h) → L∞(Zd h) be defined as Ft(U)(x) := U(x) + τH(t, x, ∇hU(x)) + Nhτ∆hU(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='21) Then the equation in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='20) can be rewritten as V τ,h n (t − τ, x) = Ft(V τ,h n (t, ·))(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' We need max{1, ∥Hp∥∞/2} ≤ N ≤ h/(2τ), (which corresponds (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='17) as ∥Hp∥∞ = ∥f∥∞) to guarantee a monotonicity property of the operator Ft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' That is, for all t ∈ Nτ T and U, V ∈ L∞(Zd h) satisfying U ≤ V , we have Ft(U) ≤ Ft(V ), see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=', [13, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' It is easy to see that the same holds if we replace H(t, x, p) by cn(t, x) + p · fn(t, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' The monotonicity property is important because it immediately implies the comparison prin- ciple of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='20) and the scheme (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='18)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='19), in the sense that is similar to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' As a consequence of this, one can show the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Assume (A1)–(A2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Then in Ωτ,h T , the solutions V τ,h n , V τ,h are bounded by C(1+T), and are Lipschitz continuous with Lipschitz constant C exp(CT) for some universal constant C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Moreover for all n ≥ 0 we have V τ,h n+1 ≤ V τ,h n in Ωτ,h T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' The proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='5 is similar to those of Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='2, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='3 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='4, and hence we skip it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Analysis of semi-discrete schemes 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Convergence of PI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' We show that for each fixed h ∈ (0, 1), vh n → vh as n → ∞ exponentially fast in L2 loc norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Assume (A1)–(A2) and N ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Let vh n and vh be, respectively, continuous solutions to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='9) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Then there exists a universal constant C > 0 such that for all n ≥ 1, R ≥ 1 and t ∈ [0, T] we have � BR ���vh n(t, x) − vh(t, x) ��� 2 dx ≤ h 2n+1 � T t � Rd exp � C(1 + ∥∇hvh∥2 ∞)(s − t)/h � × ���Dh(vh 0 (s, x) − vh(s, x)) ��� 2 min � 1, e−|x|+R+1� dxds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' In particular, we have supt∈[0,T] � BR ��vh n(t, x) − vh(t, x) ��2 dx ≤ C2−n exp [C exp(CT)/h] Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' 8 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' TANG, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' TRAN, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' ZHANG Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' In this proof, let us write vn := vh n and v := vh, and assume T ≥ 1 for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' For any fixed R ≥ 1, let ϕ = ϕR : [0, ∞) → (0, 1] be C1 and satisfying ϕ(r) = 1 on [0, R], ϕ(r) = e−r+R on [R + 1, ∞), − ϕ′(r) ∈ [0, 4ϕ(r)] for all r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='1) It is clear that such ϕ exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Next, for some A ≥ 1 to be determined, set Et,n := 1 2eAt � Rd |vn(t, x) − v(t, x)|2ϕ(|x|) dx (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='2) which is finite since vn, v are uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Direct computation yields d dtEt,n = AEt,n + eAt � Rd(vn(t, x) − v(t, x)) (∂tvn(t, x) − ∂tv(t, x)) ϕ(|x|) dx � �� � =:Xt,n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='3) Recall from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='15) that H(t, x, ∇hv) = c(t, x, α(t, x, ∇hv)) + ∇v · f(t, x, α(t, x, ∇hv)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Write c := c(t, x, α(t, x, ∇hv)), f := f(t, x, α(t, x, ∇hv)), cn := c(t, x, αn(t, x)) and fn := f(t, x, αn(t, x)) for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' using � Rd ∆hv vϕ dx = − � Rd |Dhv|2ϕ dx + 1 h2 d � i=1 � Rd v(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' x + hei)(v(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' x + hei) − v(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' x))ϕ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' x) dx − 1 h2 d � i=1 � Rd v(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' x)(v(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' x) − v(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' x − hei))ϕ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' x) dx = − � Rd |Dhv|2ϕ dx − � Rd vD−hv · D−hϕ dx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' we obtain from the equation that Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='n = − � Rd(vn − v)(∇hvn · fn + cn + Nh∆hvn − ∇hv · f − c − Nh∆hv)ϕ dx ≥ Nh � Rd |Dh(vn − v)|2ϕ dx − Nh � Rd |vn − v||D−h(vn − v)||D−hϕ| dx − � Rd |vn − v| � |∇h(vn − v)||fn| + |fn − f||∇hv| + |cn − c| � ϕ dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='4) Due to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='1), |D−hϕ(|x|)| ≤ Cϕ(|x|) for some constant C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Also, using ∥f∥∞ < ∞ and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='11), we have |∇h(vn − v)||fn| ≤ C(|Dh(vn − v)| + |D−h(vn − v)|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Since v is Lipschitz continuous, |∇hv| ≤ M for some M ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' So, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='10) and the uniform Lipschitz continuity of f, c and α, we have for some C > 0, |fn − f||∇hv| + |cn − c| ≤ CM(|Dh(vn−1 − v)| + |D−h(vn−1 − v)|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='5) With all these, if denoting Gh t,n := � Rd |Dh(vn(t, x) − v(t, x))|2ϕ(|x|) dx, 9 it follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='4) that for some C > 0, Xt,n ≥ NhGh t,n − C � Rd |vn − v|(|Dh(vn − v)| + |D−h(vn − v)|)ϕ dx − CM � Rd |vn − v|(|Dh(vn−1 − v)| + |D−h(vn−1 − v)|)ϕ dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' By the choice of ϕ, there exists C > 0 such that G−h t,n ≤ (1 + Ch)Gh t,n (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='6) Then, using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='2) and Young’s inequality, we get for any σ1, σ2 > 0, Xt,n ≥ NhGh t,n − σ1 2 + Ch � Rd(|Dh(vn − v)|2 + |D−h(vn − v)|2)ϕ dx − σ2 2 + Ch � Rd(|Dh(vn−1 − v)|2 + |D−h(vn−1 − v)|2)ϕ dx − C(2 + Ch)(σ−1 1 + M2σ−1 2 ) � Rd |vn − v|2ϕ dx ≥ (Nh − σ1)Gh t,n − σ2Gh t,n−1 − C(σ−1 1 + M2σ−1 2 )e−AtEt,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Using this and ET,n = 0, and integrating (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='3) over [t, T], we obtain for some universal C > 0, −Et,n ≥ (A − Cσ−1 1 − CM2σ−1 2 ) � T t Es,n ds + (Nh − σ1) � T t eAsGh s,n ds − σ2 � T t eAsGh s,n−1 ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='7) Now taking σ1 := h/2, σ2 := h/4 and A := 6CM2/h, then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='7) and N ≥ 1 yield � T t eAsGh s,n ds ≤ 1 2 � T t eAsGh s,n−1 ds ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' ≤ 2−n � T t eAsGh s,0 ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' With this, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='7) also shows that Et,n ≤ h 4 � T t eAsGh s,n−1 ds ≤ h 2n+1 � T t eAsGh s,0 ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Therefore, for all n ≥ 0 and t ∈ [0, T], we obtain � BR |vn(t, x) − v(t, x)|2 dx ≤ h 2n+1 � T t � Rd eA(s−t)|Dh(v0(s, x) − v(s, x))|2ϕ(|x|) dxds, which, combined with Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='4, concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' In the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='1, we only used the following: uniform Lipschitz continuity of f, c and α, and uniform boundedness of f and |∇hvh|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' In particular, the solutions vh n and vh are allowed to have certain growth at x = ∞, and the comparison principle is not needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='1, we immediately have the convergence of the policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Assume (A1)–(A2) and N ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Then there exists a universal constant C > 0 such that for all n ≥ 0 and R ≥ 1 we have sup t∈[0,T] � BR ���α(t, x, ∇hvh n(t, x)) − α(t, x, ∇hvh(t, x)) ��� 2 dx ≤ C2−n exp [C exp(CT)/h] Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' 10 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' TANG, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' TRAN, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' ZHANG Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Since α is Lipschitz continuous, � BR ���α(t, x, ∇hvh n(t, x)) − α(t, x, ∇hvh(t, x)) ��� 2 dx ≤ C h2 d � i=1 � BR ���vh n(t, x + hei) − vh(t, x + hei) − vh n(t, x − hei) + vh(t, x − hei) ��� 2 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' We can then conclude the proof from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Convergence of vh as h → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Let vh and v be, respectively, solutions to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='14) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' We show |vh − v| ≤ CT √ h, where the rate is sharp (we refer to a simple example given in [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Assume (A1)–(A2) and N ≥ max{1, ∥f∥∞/2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Then there exists a universal constant C > 0 such that sup (t,x)∈[0,T]×Rd |v(t, x) − vh(t, x)| ≤ C(1 + T)(1 + ∥∇v∥∞) √ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' In particular, we have sup(t,x)∈[0,T]×Rd |v(t, x) − vh(t, x)| ≤ C exp(CT) √ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' This rate was obtained in [15, 27] for a large class of parabolic Bellman equa- tions with Lipschitz coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' We apply a different argument – the classical doubling variable method that is used in [13] in which a discrete space-time homogeneous Hamilton-Jacobi equa- tion is discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' This argument allows us to obtain the same sharp estimate for the scheme (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='18), while it seems that the method in [15, 27] cannot (see Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' See also [11] for a different proof of this convergence rate via the nonlinear adjoint method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Let us assume that T ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Suppose for some (t0, x0) ∈ [0, T] × Rd such that 8σ := v(t0, x0) − vh(t0, x0) ≥ 1 2 sup (t,x)∈[0,T]×Rd � v(t, x) − vh(t, x) � > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='8) Below we will show σ ≤ CT(1 + ∥∇v∥∞) √ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Consider a smooth function g : Rd+1 → [0, 1] such that (g1) g(t, x) = 1 − t2 − |x|2 if t2 + |x|2 < 1/2, (g2) 0 ≤ g(t, x) ≤ 1/2 if t2 + |x|2 > 1/2, and g(t, x) = 0 if t2 + |x|2 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' For ε > 0, denote gε(t, x) := g(t/ε, x/ε), and L := sup � v(t, x), −vh(t, x) : (t, x) ∈ [0, T] × Rd� + 1 ≥ 1, By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='4, σ ≤ L ≤ CT for some universal constant C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Next, for φ(x) := (1+ |x|2)1/2 and R ≥ |x0| + T, we define Φh : [0, T]2 × R2d → R by Φh(t, s, x, y) := v(t, x) − vh(s, y) − σ T (2T − t − s) − σ R(φ(x) + φ(y)) + (8L + 2σ)gε(t − s, x − y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Since v, vh are bounded, there exists (t1, s1, x1, y1) ∈ [0, T]2 × R2d such that Φh(t1, s1, x1, y1) = max [0,T]2×R2d Φh(t, s, x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='9) 11 Due to φ(x0) ≤ R, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='8), Φh(t1, s1, x1, y1) ≥ Φh(t0, t0, x0, x0) ≥ 8L + 6σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='10) Since max{v(t1, x1), −vh(s1, y1)} ≤ L, we deduce Φh(t1, s1, x1, y1) ≤ 2L + (8L + 2σ)gε(t1 − s1, x1 − y1), which, together with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='10), implies gε(t1 − s1, x1 − y1) ≥ 3/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Then by (g1), we get that for some C > 0, gε(t − s, x − y) = 1 − ε−2|t − s|2 − ε−2|x − y|2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='11) whenever |t − t1|, |s − s1|, |x − x1|, |y − y1| ≤ ε/C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Now, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='9), the mapping (t, x) �→ v(t, x) + σ T t − σ Rφ(x) + (8L + 2σ)gε(t − s1, x − y1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='12) is maximized at (t, x) = (t1, x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Together with the fact that v is Lipschitz continuous (taking M := 1+∥∇v∥∞) and |∇φ| ≤ 1, we find that |∇x gε(t1−s1, x1 −y1)| ≤ (M +σR−1)(8L+2σ)−1 and, |∂t gε(t1 − s1, x1 − y1)| ≤ (M + σT −1)(8L + 2σ)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='11), σ ≤ L ≤ CT and R ≥ T, these yield |x1 − y1| ≤ Cε2(M + σR−1)(L + σ)−1 ≤ Cε2ML−1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='13) and |t1 − s1| ≤ Cε2(M + σT −1)(L + σ)−1 ≤ Cε2ML−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='14) Now, we firstly assume that t1, s1 < T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' In view of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='12), we apply the viscosity solution test for v to get − σ T − (8L + 2σ) ∂tgε(t1 − s1, x1 − y1) + H � t1, x1, σ R∇φ(x1) − (8L + 2σ)∇x gε(t1 − s1, x1 − y1)) � ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='15) Similarly, since (s, y) → vh(s, y) − σ T s + σ Rφ(y) − (8L + 2σ)gε(t1 − s, x1 − y) is minimized at (s1, y1), the comparison principle yields σ T − (8L + 2σ) ∂tgε(t1 − s1, x1 − y1) + H � s1, y1, − σ R∇hφ(y1) − (8L + 2σ)∇h x gε(t1 − s1, x1 − y1) � − Nh∆h � σ Rφ(y1) − (8L + 2σ)gε(t1 − s1, x1 − y1)) � ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Thus we get 2σ T ≤ H � t1, x1, σ R∇φ(x1) − (8L + 2σ)∇x gε(t1 − s1, x1 − y1)) � − H � s1, y1, − σ R∇φ(y1) − (8L + 2σ)∇h x gε(t1 − s1, x1 − y1) � + Nh∆h � σ Rφ(y1) − (8L + 2σ)gε(t1 − s1, x1 − y1)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='16) It follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='11) that for h ≪ ε, we have at point (t1 − s1, x1 − y1), ∇h x gε = ∇x gε = 2ε−2(x1 − y1), ∆hgε = −2dε−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='17) Due to |∇φ| ≤ 1 and ∆hφ ≤ C, we get Nh∆h � σ Rφ(y1) − (8L + 2σ)gε(t1 − s1, x1 − y1)) � ≤ CLε−2h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='18) 12 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' TANG, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' TRAN, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' ZHANG Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='16)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='18) and the regularity of H (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='16)), we obtain for some universal C, 2σT −1 ≤ CσR−1(|∇φ(x1)| + |∇φ(y1)|) + CLε−2h + C(|t1 − s1| + |x1 − y1|) [1 + (8L + 2σ)|∇x gε(t1 − s1, x1 − y1)|] ≤ CσR−1 + CLε−2h + C(|t1 − s1| + |x1 − y1|) � 1 + Lε−2|x1 − y1| � which, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='13) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='14), yields σT −1 ≤ CσR−1 + CLε−2h + Cε2M2L−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Now we take ε := M−1/2L1/2h1/4 and pass R → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Then when h is sufficiently small, we obtain σ ≤ CTM √ h for some universal C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' This finishes the proof of the upper bound of sup[0,T]×Rd(v − vh) in the case when t1, s1 < T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Next, suppose that one of t1 and s1 equals to T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Let us only prove for the case when t1 = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='10) and the definition of Φh, 8L + 6σ ≤ v(t1, x1) − vh(s1, y1) + (8L + 2σ)gε(t1 − s1, x1 − y1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' It follows from the proof Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='4 that vh is Lipschitz continuous with unit Lipschitz constant when |T − t| ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Note that ε2ML−1 ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Hence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='13) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='14) yield 8L + 6σ ≤ |v(T, x1) − q(y1)| + |q(y1) − vh(s1, y1)| + (8L + 2σ)gε(T − s1, x1 − y1) ≤ C(|x1 − y1| + |T − s1|) + 8L + 2σ ≤ Cε2ML−1 + 8L + 2σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' This yields σ ≤ C √ h for some universal C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Finally, the upper bound estimate for sup[0,T]×BR(vh−v) follows by using the same argument as the above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='4 permits to conclude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Almost everywhere convergence of the policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' It was proved in [23] that the solution v to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='4) is semi-concave in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' From this, we are able to derive the almost everywhere convergence of the policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Below we say that a function g : Rd → R is uniformly semi-concave if there exists C > 0 such that for all x, y ∈ Rd we have g(x+y)+g(x−y)−2g(x) ≤ C|y|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' If g is uniformly bounded and Lipschitz continuous, and both ±g are uniformly semi-concave, then g is bounded in W 2,∞(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' We make the following assumption: (A3) q(·) is uniformly semi-concave, and c(t, ·, a), f(t, ·, a) are bounded in W 2,∞(Rd) uni- formly in t ∈ [0, T] and a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Under the assumptions of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='3, further assume (A3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Then v(t, ·) is uniformly semi-concave for all t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Moreover, for each t ∈ [0, T] we have for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' x ∈ Rd, α(th, xh, ∇hvh(th, xh)) → α(t, x, ∇v(t, x)) as h → 0 where [0, T] × Rd ∋ (th, xh) → (t, x) as h → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' We next show a weak type of semi-concavity of vh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Under the assumptions of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='4, there exists C > 0 (also depending on (A3)) such that for all h ∈ (0, 1), t ∈ [0, T] and x, y ∈ Rd, vh(t, x + y) + vh(t, x − y) − 2vh(t, x) ≤ C exp(CT) (|y|2 + √ h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' The proofs of the two theorems are similar to those of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='3 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='4, and we choose to write the full details down there (as it is slightly more complicated there).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' 13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Analysis of discrete space-time schemes 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Convergence of PI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' The parallel result of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='1 on the convergence of V τ,h n → V τ,h holds the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' However the proof is more involved due to the discretization in the time direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' In it, we will emphasize the difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Assume (A1)–(A2) and N ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Let Vn := V τ,h n and V := V τ,h be, respectively, continuous solutions to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='18) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Then there exists a universal constant C > 0 such that if C(1 + ∥∇hV ∥2 ∞)τ ≤ h, we have for all n ≥ 1, R ≥ 1 and t ∈ Nτ T, � x∈Zd h,|x|≤R |Vn(t, x) − V (t, x)|2 ≤ hτ 2n+1 � t≤s∈Nτ T � x∈Zd h exp � C exp(1 + ∥∇hV ∥2 ∞)(s − t)/h � ���Dh(V0(s, x) − V (s, x)) ��� 2 min � 1, e−|x|+R+1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' In particular, we have max t∈Nτ T � x∈Zd h,|x|≤R |Vn(t, x) − V (t, x)|2 ≤ C2−n exp [C exp(CT)/h] Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' max t∈Nτ T � x∈Zd h,|x|≤R ���α(t, x, ∇hVn(t, x)) − α(t, x, ∇hV (t, x)) ��� 2 ≤ C2−n exp [C exp(CT)/h] Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Assume T ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Let ϕ = ϕR : [0, ∞) → [0, 1] be C1 and satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='1), and let A := CT 2/h for some C > 0 to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Then for t ∈ Nτ T set Et,n := 1 2eAt � x∈Zd h |Vn(t, x) − V (t, x)|2ϕ(|x|) which is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Direct computation yields Et,n − Et−τ,n τ ≥ Ae−AτEt,n + 1 2eA(t−τ) � x∈Zd h (Vn(t, x) + Vn(t − τ, x) − V (t, x) − V (t − τ, x))× (∂τ t Vn(t, x) − ∂τ t V (t, x))ϕ(|x|) = Ae−AτEt,n + eA(t−τ) � x∈Zd h (Vn(t, x) − V (t, x))(∂τ t Vn(t, x) − ∂τ t V (t, x))ϕ(|x|) − τ 2eA(t−τ) � x∈Zd h |∂τ t Vn(t, x) − ∂τ t V (t, x)|2 ϕ(|x|) =: Ae−AτEt,n + eA(t−τ)Xt,n − τ 2eA(t−τ)Yt,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='1) First, we consider the term Yt,n (which does not appear in the semi-discretization problem in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' It follows from the equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='18) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='20) that Yt,n = � x∈Zd h ���cn + ∇hVn · fn + Nh∆hVn − H(t, x, ∇hV ) − Nh∆hV ��� 2 ϕ(|x|) 14 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' TANG, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' TRAN, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' ZHANG Recall that α = α(t, x, ∇hV ), H(t, x, ∇hV ) = c(t, x, α) + f(t, x, α) · ∇hV , and |∇hV | ≤ M for some M ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Since αn = α(t, x, ∇hVn−1), the regularity assumptions and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='11) yield Yt,n ≤ C � x∈Zd h � M2|DhVn−1 − DhV |2 + M2|D−hVn−1 − D−hV |2+ |DhVn − DhV |2 + |D−hVn − D−hV |2� ϕ(|x|) ≤ C � M2Gh t,n−1 + Gh t,n � , where in the last inequality we used the notation Gh t,n := � x∈Zd h |DhVn(t, x)−DhV (t, x)|2ϕ(|x|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='6) with the above defined Gh t,n (which clearly holds the same).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Next, we consider the term Xt,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Note that for any v ∈ L∞(Zd h), � x∈Zd h ∆hv vϕ = − � x∈Zd h |Dhv|2ϕ− � x∈Zd h vD−hv ·D−hϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' So, similarly as before (also using the equation, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='1), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='6), the uniform Lipschitz assumptions, and Young’s inequality), we have for any σ1, σ2 > 0, Xt,n ≥ NhGh t,n − C � x∈Zd h |Vn − V |(|Dh(Vn − V )| + |D−h(Vn − V )|)ϕ − CM � x∈Zd h |Vn − V |(|Dh(Vn−1 − V )| + |D−h(Vn−1 − V )|)ϕ ≥ (Nh − σ1)Gh t,n − σ2Gh t,n−1 − C(σ−1 1 + M2σ−1 2 )e−AtEt,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Since ET,n ≡ 0, putting the above together and summing up (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='1) with respect to t yield −Et,n/τ ≥ (Nh − σ1 − Cτ) � t+τ≤s∈Nτ T eA(s−τ)Gh s,n − (σ2 + CM2τ) � t+τ≤s∈Nτ T eA(s−τ)Gh s,n−1 + (A − Cσ−1 1 − CM2σ−1 2 )e−Aτ � t+τ≤s∈Nτ T Eh s,n (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='2) for some universal constant C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Finally we take σ1 := h/4, σ2 := h/8, A := 12CM2/h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Then if τ ≤ h/(8CM2), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='2) yields � t+τ≤s∈Nτ T eA(s−τ)Gh s,n ≤ 1 2 � t+τ≤s∈Nτ T eA(s−τ)Gh s,n−1 ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' ≤ 2−n � t+τ≤s∈Nτ T eA(s−τ)Gh s,0, and then Et,n ≤ hτ 4 � t+τ≤s∈Nτ T eA(s−τ)Gh s,n−1 ≤ hτ 2n+1 � t+τ≤s∈Nτ T eA(s−τ)Gh s,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' This, together with Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='5, concludes the proof of the first claim as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' The second claim follows similarly as in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' □ By shifting the solutions, we actually obtain uniform pointwise exponential convergence of V τ,h n to V τ,h and α(·, ·, ∇hV τ,h n ) to α(·, ·, ∇hV τ,h) as n → ∞, in Ωτ,h T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Convergence of V τ,h as τ, h → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Let V τ,h and v be, respectively, solutions to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='20) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' The following theorem proves that the difference between V τ,h and v is at most of order √ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' The argument follows the idea of [13, Theorem 1], which considered the discrete space-time scheme for the homogeneous Hamilton-Jacobi equation vt + H(Dv) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' 15 Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Assume (A1)–(A2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Then there exists a universal C > 0 such that sup (t,x)∈Ωτ,h T |v(t, x) − V τ,h(t, x)| ≤ C(1 + T)(1 + ∥∇v∥∞) √ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' In particular, we have sup(t,x)∈Ωτ,h T |v(t, x) − V τ,h(t, x)| ≤ C exp(CT) √ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' It was shown in [14, 15, 27] that sup (t,x)∈Ωτ,h T |v(t, x) − V τ,h(t, x)| ≤ C(τ 1/4 + h1/2) for some C = C(T) > 0, where v solves a general degenerate parabolic Bellman equation and V τ,h is its space-time fi- nite difference approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' For the first order equations, our Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='2 obtains a better convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Assume T ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' And suppose for some (t0, x0) ∈ Ωτ,h T such that 8σ := v(t0, x0) − V τ,h(t0, x0) ≥ 1 2 sup (t,x)∈Ωτ,h T � v(t, x) − V τ,h(t, x) � > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='3) Let DT,τ,h := [0, T] × Nτ T × Rd × Zd h, and L := sup � v(t, x), −V τ,h(t, x) : (t, x) ∈ Ωτ,h T � + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Then σ ≤ L ≤ CT for some universal constant C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Moreover, let R, g and gε with ε ∈ (0, 1), and φ be from the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='3, and define Φh : DT,τ,h → R by Φh(t, s, x, y) := v(t, x) − V τ,h(s, y) − σ T (2T − t − s) − σ R(φ(x) + φ(y)) + (8L + 2σ)gε(t − s, x − y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Suppose Φh(t1, s1, x1, y1) = max DT,τ,h Φh(t, s, x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='4) It is clear that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='10)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='14) hold the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='14) if τ ≪ ε2M/L, we get |t1 − s1 − τ| ≤ Cε2M/L with M = 1 + ∥∇v∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='5) First, assume t1, s1 < T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' The viscosity solution test for v shows (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='15) by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Next since Ωτ,h T ∋ (s, y) → V τ,h(s, y) − σ T s + σ Rφ(y) − (8L + 2σ)gε(t1 − s, x1 − y) is minimized at (s1, y1), then for all (s, y) ∈ Ωτ,h T , V τ,h(s, y) ≥ V τ,h(s1, y1) − σ T (s1 − s) + σ R(φ(y1) − φ(y)) − (8L + 2σ) [gε(t1 − s1, x1 − y1) − gε(t1 − s, x1 − y)] =: ˜V (s, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Recall that s1 + τ ≤ T and Ft from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='21) satisfies the monotonicity property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' We obtain V τ,h(s1, y1) = Fs1+τ(V τ,h(s1 + τ, ·))(y1) ≥ Fs1+τ( ˜V (s1 + τ, ·))(y1), 16 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' TANG, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' TRAN, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' ZHANG which gives 0 ≥ σ T − (8L + 2σ) ∂τ t gε(t1 − s1, x1 − y1) + H � s1 + τ, y1, − σ R∇hφ(y1) − (8L + 2σ)∇h x gε(t1 − s1 − τ, x1 − y1) � − Nh∆h � σ Rφ(y1) − (8L + 2σ)gε(t1 − s1 − τ, x1 − y1)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='6) By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='11), if τ, h ≪ ε2, |∂τ t gε(t1 − s1, x1 − y1) − ∂tgε(t1 − s1, x1 − y1)| ≤ Cε−2τ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='7) ∇h x gε(t1 − s1 − τ, x1 − y1) = ∇x gε(t1 − s1, x1 − y1) = 2ε−2(x1 − y1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='8) Combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='6) with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='15), and using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='7) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='8) yield 2σ T ≤ H � t1, x1, σ R∇φ(x1) − (8L + 2σ)2ε−2(x1 − y1) � − H � s1 + τ, y1, − σ R∇φ(y1) − (8L + 2σ)2ε−2(x1 − y1) � + Nh∆h � σ Rφ(y1) − (8L + 2σ)gε(t1 − s1 − τ, x1 − y1)) � + CLε−2τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='9) The definitions of φ and gε show (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Then, applying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='16) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='18) into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='9), if (τ ≤ ) h ≪ ε2 we deduce for some C > 0 that σT −1 ≤ CσR−1(|∇φ(x1)| + |∇φ(y1)|) + CLε−2h + CLε−2τ + C(|t1 − s1 − τ| + |x1 − y1|) � 1 + (8L + 2σ)2ε−2|x1 − y1| � ≤ CσR−1 + CLε−2h + Cε2M2L−1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='10) where in the second inequality we also used (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='13) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Now we take ε := M−1/2L1/2h1/4, and send R → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' It is clear that τ ≪ ε2M/L is satisfied when h is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' We obtain from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='10) that σ ≤ CTM √ h, which finishes the proof of the upper bound of supΩτ,h T (v − V τ,h) in the case when t1, s1 < T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Next, if at least one of t1 and s1 equals to T, the argument of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='3 applies the same except that we need to use Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='5 in place of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Finally, the proof for the upper bound of supΩτ,h T (V τ,h − v) is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Almost everywhere convergence of the policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' We show the almost everywhere convergence of the policy, and some semi-concavity property of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Under the assumptions of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='2, further assume (A3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Then v is uni- formly semi-concave for all t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Moreover, for each t ∈ [0, T] we have for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' x ∈ Rd, α(th, xh, ∇hV τh,h(th, xh)) → α(t, x, ∇v(t, x)) as h → 0 where Ωτh,h T ∋ (th, xh) → (t, x) as h → 0 and τh satisfies 0 < 2Nτh ≤ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' The semi-concavity of v(t, ·) follows from [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' For the second claim, it suffices to prove that for a fixed t ∈ [0, T), and for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' x ∈ Rd, ∇hV τh,h(th, xh) → ∇v(t, x) as h → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='11) 17 For any function g : Rd → R, let us denote by D+g(x) the set of subdifferential of g: D+g(x) := � p ∈ Rd �� lim sup y→x g(y) − g(x) − p · (y − x) |y − x| ≤ 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Due to v(t, ·) is semi-concave, D+v(t, x) is non-empty for all x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Because v(t, ·) is Lipschitz continuous, ∇xv(t, x) exists for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' We fix one such x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Since V τh,h are Lipschitz continuous uniformly in h, after passing to a subsequence of h → 0, we can assume that ∇hV τh,h(th, xh) → p for some p ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Since V τh,h(th, xh) → v(t, x) as h → 0, the stability of subdifferential yields that p ∈ D+v(t, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' While because ∇xv(t, x) exists, we get p = ∇xv(t, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Note that this is for any convergent subsequence of ∇hV τh,h(th, xh), and so we obtain (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' □ Below, we show a weak type of semi-concavity of V τ,h(t, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' We adopt the “doubling variable” method, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=', [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Under the assumptions of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='3, there exists C > 0 (also depending on (A3)) such that for all t ∈ Nτ T and x, y ∈ Zd h, V τ,h(t, x + y) + V τ,h(t, x − y) − 2V τ,h(t, x) ≤ C exp(CT) (|y|2 + √ h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' It suffices to show that there exist CT , C′ T > 0 depending on the assumptions such that V τ,h(t, x) + V τ,h(t, z) − 2V τ,h(t, y) ≤ CT � |x − y|2 + |z − y|2 + |x + z − 2y| � + C′ T √ h (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='12) for all t ∈ Nτ T and x, y, z ∈ Zd h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' By the assumption on q, the inequality holds for t = T with CT = ∥q∥W 2,∞ =: C0, and C′ T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Suppose for contradiction that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='12) fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Then we have for some C1 ≥ 1 to be determined, and some C ≥ 2, V τ,h(t, x) + V τ,h(t, z) − 2V τ,h(t, y) − 2C0eC1(T−t) � |x − y|4 + |z − y|4 + |x + z − 2y|2�1/2 ≥ CeC1(T−t)√ h (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='13) for some (t, x, y, z) = (t′, x′, y′, z′) ∈ Nτ T × Zd h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Since V τ,h(t, ·) is Lipschitz continuous (with Lipschitz constant bounded by C exp(C(T − t)) by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='5 with a shift in time), after possibly enlarging the constant C in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='13), we can assume that |x′ + z′ − 2y′| ≥ √ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='14) Let us denote ψ(x, y, z) := |x−y|4+|z−y|4+|x+z−2y|2, and by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='14), δ := ψ(x′, y′, z′)1/2 ≥ √ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Then for all ε > 0 sufficiently small, we obtain from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='13) that Φ(t, x, y, z) := eC1t � V τ,h(t, x) + V τ,h(t, z) − 2V τ,h(t, y) � − C0eC1T � δ + δ−1ψ(x, y, z) � − ε|y|2 satisfies Φ(t′, x′, y′, z′) ≥ eC1T √ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' With the positive ε-term, Φ obtains its positive maximum that is at least eC1T √ h in Ωτ,h T at some point (t0, x0, y0, z0) ∈ Nτ T × Zd h, where (t0, x0, y0, z0) depends on ε and δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' It is clear that t0 ≤ T − τ by the choice of C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Moreover, for γ0 := δ + δ−1ψ(x0, y0, z0), we have V τ,h(t0, x0) + V τ,h(t0, z0) − 2V τ,h(t0, y0) ≥ C0eC1(T−t0)γ0 + eC1(T−t0)√ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='15) Due to uniform boundedness of V τ,h, by further taking ε to be small enough depending on C, T and h, it is easy to get ε|y0| ≤ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' 18 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' TANG, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' TRAN, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' ZHANG Now since Ωτ,h T ∋ (t, x) → eC1tV τ,h(t, x) − C0eC1T δ−1 � |x − y0|4 + |x + z0 − 2y0|2� is maxi- mized at (t0, x0), we get for all (t, x) ∈ Ωτ,h T that V τ,h(t, x) ≤ eC1(t0−t)V τ,h(t0, x0) + C0eC1(T−t)δ−1 � |x − y0|4 + |x + z0 − 2y0|2� − C0eC1(T−t0)δ−1 � |x0 − y0|4 + |x0 + z0 − 2y0|2� =: ˜V (t, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Due to the equation and the monotonicity property of Ft (defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='21)), V τ,h(t0, x0) = Ft0+τ(V τ,h(t0 + τ, ·))(x0) ≤ Ft0+τ( ˜V (t0 + τ, ·))(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' By direct computation, ∇h x(|x − y0|4 + |x + z0 − 2y0|2) = 4(|x − y0|2 + h2)(x − y0) + 2(x + z0 − 2y0), ∆h x(|x − y0|4 + |x + z0 − 2y0|2) = (8 + 4d)|x − y0|2 + 2dh2 + 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' We then get (1 − e−C1τ) τ V τ,h(t0, x0) ≤ H � t0 + τ, x0, ∇h x ˜V (t0 + τ, x0) � + Nh∆h x ˜V (t0 + τ, x0) ≤ H (t0 + τ, x0, 2CT,δ(qx0 + p0)) + CCT,δh(|x0 − y0|2 + 1) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='16) where qx0 := 2(|x0 − y0|2 + h2)(x0 − y0), CT,δ := C0eC1(T−t0−τ)/δ and p0 := x0 + z0 − 2y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='17) Similarly, since Ωτ,h T ∋ (t, z) → eC1tV τ,h(t, z) − C0eC1T δ−1(|z − y0|4 + |x0 + z − 2y0|2) is maximized at (t0, z0), we get (1 − e−C1τ) τ V τ,h(t0, z0) ≤ H (t0 + τ, z0, 2CT,δ(qz0 + p0)) + CCT,δh(|z0 − y0|2 + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='18) where qz0 := 2(|z0 − y0|2 + h2)(z0 − y0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Next, note that Ωτ,h T ∋ (t, y) → 2eC1tV τ,h(t, y) + C0eC1T δ−1ψ(x0, y, z0) + ε|y|2 is minimized at (t0, y0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Hence we get V τ,h(t0, y0) ≥ Ft0+τ( ˆV (t0 + τ, ·))(y0) where ˆV (t, y) := eC1(t0−t)V τ,h(t0, y0) − (ε/2)|y|2 + (ε/2)|y0|2 − (C0/2)eC1(T−t)δ−1ψ(x0, y, z0) + (C0/2)eC1(T−t0)δ−1ψ(x0, y0, z0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' From this we obtain −(1 − e−C1τ) τ V τ,h(t0, y0) ≤ −H (t0 + τ, y0, 2CT,δ(qy0 + p0) − εy0) + CCT,δh(|x0 − y0|2 + |z0 − y0|2 + 1) + Chε where qy0 := (|x0 − y0|2 + h2)(x0 − y0) + (|z0 − y0|2 + h2)(z0 − y0), and CT,δ and p0 are given in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Using |Hp| ≤ C and ε|y0| ≤ h yields −(1 − e−C1τ) τ V τ,h(t0, y0) ≤ −H (t0 + τ, y0, 2CT,δ(qy0 + p0)) + CCT,δh(|x0 − y0|2 + |z0 − y0|2 + 1) + Ch (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='19) Now let α ∈ A be such that H (t0 + τ, y0, 2CT,δ(qy0 + p0)) = c(t0 + τ, y0, α) + 2CT,δf(t0 + τ, y0, α) · (qy0 + p0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' 19 By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='15),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' denoting cα(·) := c(t0 + τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' ·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' α) and fα(·) := f(t0 + τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' ·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' α),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' we have H (t0 + τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' 2CT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='δ(qx0 + p0)) + H (t0 + τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' z0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' 2CT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='δ(qz0 + p0)) − 2H (t0 + τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' y0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' 2CT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='δ(qy0 + p0)) ≤ cα(x0) + cα(z0) − 2cα(y0) + 2CT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='δ [fα(x0) · (qx0 + p0) + fα(z0) · (qx0 + p0) − 2fα(y0) · (qy0 + p0)] = cα(x0) + cα(z0) − 2cα(y0) + 2CT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='δ [(fα(x0) − fα(y0)) · qx0 + (fα(z0) − fα(y0)) · qz0+ + (fα(x0) + fα(z0) − 2fα(y0)) · p0] ≤ ∥cα∥W 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='∞(|x0 − y0|2 + |z0 − y0|2 + |x0 + z0 − 2y0|) + 2CT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='δ∥fα∥Lip(|x0 − y0||qx0| + |z0 − y0||qz0|) + 2CT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='δ∥fα∥W 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='∞(|x0 − y0|2 + |z0 − y0|2 + |x0 + z0 − 2y0|)|x0 + z0 − 2y0|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='20) where we used 2qy0 = qx0 + qz0 and that for any x, y, z ∈ Rd and g ∈ W 2,∞(Rd), |g(x) + g(z) − 2g(y)| ≤ ∥g∥W 2,∞(|x−y|2 +|z −y|2 +|x+z −2y|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' By Young’s inequality, we get |x0 −y0||qx0|+ |z0 − y0||qz0| ≤ 2|x0 − y0|4 + 2|z0 − y0|4 + h4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Also using the definitions of CT,δ and ψ, we get the left-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='20) ≤ CeC1(T−t0)(δ +δ−1ψ(x0, y0, z0)) = CeC1(T−t0)γ0 +CeC1(T−t0)h4/δ with C > 0 only depending on ∥q∥W 2,∞, ∥cα∥W 2,∞ and ∥fα∥W 2,∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Now summing up (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='16), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='18) and twice of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='19), we get (1 − e−C1τ) τ � V τ,h(t0, x0) + V τ,h(t0, z0) − 2V τ,h(t0, y0) � ≤ CeC1(T−t0)γ0 + CeC1(T−t0)h4/δ + CCT,δh(|x0 − y0|2 + |z0 − y0|2 + 1) + Ch ≤ CeC1(T−t0)γ0 + CeC1(T−t0)δ−1(|x0 − y0|4 + |z0 − y0|4) + CeC1(T−t0)√ h ≤ CeC1(T−t0)γ0 + CeC1(T−t0)√ h, where in the second inequality, we used δ ≥ √ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Finally, this and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='15) yield C1(C0eC1(T−t0)γ0 + eC1(T−t0)√ h) ≤ CeC1(T−t0)γ0 + CeC1(T−t0)√ h, with C > 0 depending only on d, N and the regularity assumptions of q, c, f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Thus, if C1 is sufficiently large depending only on the assumptions, we get a contradiction which finishes the proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='12), which finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Generalization: a PDE perspective In this section, we consider PI for HJB equations with a general Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' For convenient use of the Legendre transform, we write the system in the forward-in-time setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' It is easy to carry over to the backward-in-time setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Suppose H : [0, T] × Rd × Rd → R is continuous such that H(t, x, p) is convex in p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Let L(t, x, µ) be the Legendre transform of H, that is, L(t, x, µ) := sup p∈Rd [p · µ − H(t, x, p)] for (t, x, µ) ∈ [0, T] × Rd × Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' We always have the following inequality L(t, x, µ)+H(t, x, p) ≥ p·µ, with equality holds if and only if µ = ∇pH(t, x, p), and if and only if p = ∇µL(t, x, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' 20 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' TANG, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' TRAN, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' ZHANG The HJB equation is � ∂tv + H(t, x, ∇v) = 0 in (0, T) × Rd, v(0, x) = q(x) on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='1) Under some assumptions (see [3, 42]), it is a classical result that v is uniformly Lipschitz continuous if q is Lipschitz continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' So we can assume ∥∇v∥L∞([0,T]×Rd) ≤ M for some M > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='2) Now we take m1 := min |p|=2M, t∈[0,T],x∈Rd H(t, x, p) and m2 ≥ max |p|=3M, t∈[0,T],x∈Rd [H(t, x, p) − m1]/M, and we can assume that m2 ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Then define ˜H(t, x, p) := \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 H(t, x, p) if |p| ≤ 2M, max {H(t, x, p), m1 + m2(|p| − 2M)} if 2M < |p| ≤ 3M, m1 + m2(|p| − 2M) if |p| > 3M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' It is not hard to verify that ˜H is continuous in all its dependencies, and is convex in p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Due to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='2), v is also a solution of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='1) with H replaced by ˜H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Moreover for N := m2/2 ≥ 1, we have | ˜Hp(t, x, p)| ≤ 2N in [0, T] × Rd × Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='3) We define ˜L as the Legendre transform of ˜H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Since the goal is to approximate v, it suffices to study ˜H and ˜L instead of H and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' From now on, with a slight abuse of notation, we write H and L as ˜H and ˜L, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' With the modified operators, we can consider the semi-discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' For h > 0, � ∂tvh + H(t, x, ∇hvh) = Nh∆hvh in (0, T) × Rd, vh(0, x) = q(x) on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='4) As before, N ≥ ∥∇pH∥∞/2 guarantees that the finite difference scheme is monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Let us also assume that there exists C > 0 such that for all t, x, p ∈ [0, T] × Rd × Rd, |Ht(t, x, p)|, |Hx(t, x, p)| ≤ C(1 + |p|), |H(t, x, 0)| ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='5) Actually, we can replace C(1 + |p|) by just C for the modified operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' We will not discuss the space-time discretization of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='1) since it is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Now we present the iteration scheme for (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Fixing small h > 0, we start with a uniformly bounded and Lipschitz continuous function vh 0(t, x), and then iteratively compute vh n as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' For n ≥ 1, let vh n(t, x) be the solution to � ∂tvh n + ∇pH(t, x, ∇hvh n−1) · ∇hvh n − L(t, x, µh n−1) = Nh∆hvh n in (0, T) × Rd, vh n(0, x) = q(x) on Rd (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='6) where we denoted µh n(t, x) := ∇pH(t, x, ∇hvh n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Note L(t, x, µh n) is finite due to µh n ≤ 2N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Essentially, vh n solves a linearized equation of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Let vh n (for each n ≥ 1 with given vh 0), vh and v be, respectively, Lipschitz continuous solutions to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='6), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='4) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' We have the following monotonicity property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' 21 Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Suppose N ≥ max{1, ∥∇pH∥∞/2}, and H(t, x, p) is convex in p and satisfies (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='3) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Let q and vh 0 be uniformly bounded and Lipschitz continuous for all h > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Then the solutions vh n are uniformly bounded for all n ≥ 1 and h > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Moreover, we have for all n ≥ 0, vh n+1 ≤ vh n in [0, T] × Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' We also have the following convergence results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Under the assumptions of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='1, for all R ≥ 1 there exists a constant C depending only on T and the assumptions such that we have for all t ∈ [0, T], � BR ���vh n(t, x) − vh(t, x) ��� 2 dx ≤ C2−nheCt/hRd, � BR ���α(t, x, ∇hvh n(t, x)) − α(t, x, ∇hvh(t, x)) ��� 2 dx ≤ C2−neCt/hRd/h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Moreover, we have sup(t,x)∈[0,T]×Rd |vh(t, x) − v(t, x)| ≤ C √ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Next, let H take the form H(t, x, p) := supa∈A [c(t, x, a) + p · f(t, x, a)], where A is some set, c : [0, T] × Rd × A → R and f : [0, T] × Rd × A → Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Under the assumptions of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='2, assume that c(t, ·, a), f(t, ·, a) are bounded in W 2,∞(Rd) uniformly for all t ∈ [0, T] and a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Then for each t ∈ [0, T], we have for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' x ∈ Rd, α(th, xh, ∇hvh(th, xh)) → α(t, x, ∇v(t, x)) as h → 0, where [0, T] × Rd ∋ (th, xh) → (t, x) as h → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Moreover, there exists C > 0 depending only on the assumptions such that for all h ∈ (0, 1), t ∈ [0, T] and x, y ∈ Rd, vh(t, x + y) + vh(t, x − y) − 2vh(t, x) ≤ C exp(CT)(|y|2 + √ h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Conclusion In this paper, we study the convergence rate of PI for optimal control problems in continuous time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' To overcome the problem of ill-posedness, we consider a semi-discrete scheme by adding a viscosity term using finite differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' We prove that PI for the semi-discrete scheme converges exponentially fast, and provide a bound on the discrepancy between the semi-discrete scheme and the optimal control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' We also study the discrete space-time scheme, where both space and time are discretized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' There are a few directions to extend this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' First, under what conditions on the model parameters does PI (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='7)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='8) converge exponentially fast?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' For instance, for f(t, x, a) = a, c(t, x, a) = 1 2|a|2 and q ≡ 0, the HJB equation is ∂tv − 1 2|∇v|2 = 0 and v(T, x) = 0, which has the solution v∗ ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' On the other hand, PI yields vn(t, x) = cn(t)x2 with c1(t) = 1 2 for a suitable initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' It is easy to check that cn(t) ≤ 2−n for n ≥ 1, and thus we get the exponential convergence of vn to v∗ on any compact set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' However, it is not clear what are the right conditions to impose on the model parameters so that PI converges exponentially fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' It is also interesting to adapt PI to the differential game setting and design efficient numerical schemes (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='g.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' (W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Tang) Department of Industrial Engineering and Operations Research, Columbia University, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Mudd Building, 500 W 120th St, New York, NY 10027 Email address: wt2319@columbia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='edu (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Tran) Department of Mathematics, University of Wisconsin Madison, Van Vleck Hall, 480 Lincoln Drive, Madison, WI 53706 Email address: hung@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='wisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='edu (Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content=' Zhang) Department of Mathematics and Statistics, Auburn University, Parker Hall, 221 Roosevelt Concourse, Auburn, AL 36849 Email address: yzhangpaul@auburn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} +page_content='edu' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAyT4oBgHgl3EQfjvgl/content/2301.00419v1.pdf'} diff --git a/g9E2T4oBgHgl3EQfHQbG/content/tmp_files/2301.03667v1.pdf.txt b/g9E2T4oBgHgl3EQfHQbG/content/tmp_files/2301.03667v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1c73b74b944abca87b83c3a6cb0f26365e675ba0 --- /dev/null +++ b/g9E2T4oBgHgl3EQfHQbG/content/tmp_files/2301.03667v1.pdf.txt @@ -0,0 +1,911 @@ +Implementations of two Algorithms for the +Threshold Synthesis Problem +Jan-Georg Smaus +IRIT +Universit´e Paul Sabatier Toulouse +France +Christian Schilling +Fabian Wenzelmann +Institut f¨ur Informatik +Albert-Ludwigs-Universit¨at Freiburg +Germany +Abstract +A linear pseudo-Boolean constraint (LPB) is an expres- +sion of the form +a1 · ℓ1 + . . . + am · ℓm ≥ d, +where each ℓi is a literal (it assumes the value 1 or +0 depending on whether a propositional variable xi is +true or false) and a1, . . . , am, d are natural numbers. An +LPB represents a Boolean function, and those Boolean +functions that can be represented by exactly one LPB +are called threshold functions. The problem of finding +an LPB representation of a Boolean function if possi- +ble is called threshold recognition problem or threshold +synthesis problem. The problem has an O(m7t5) algo- +rithm using linear programming, where m is the di- +mension and t the number of terms in the DNF input. +It has been an open question whether one can recog- +nise threshold functions through an entirely combina- +torial procedure. Smaus has developed such a proce- +dure for doing this, which works by decomposing the +DNF and “counting” the variable occurrences in it. We +have implemented both algorithms as a thesis project. +We report here on this experience. The most important +insight was that the algorithm by Smaus is, unfortu- +nately, incomplete. +1 +Introduction +A linear pseudo-Boolean constraint (LPB) (Dixon and +Ginsberg 2000; Fr¨anzle and Herde 2007) is an expres- +sion of the form a1ℓ1 + . . . + amℓm ≥ d. Here each ℓi is +a literal of the form xi or ¯xi ≡ 1 − xi, i.e. xi becomes +0 if xi is false and 1 if xi is true, and vice versa for ¯xi. +Moreover, a1, . . . , am, d are natural numbers. +An LPB can be used to represent a Boolean1 func- +tion; e.g. x1 + ¯x2 + x3 ≥ 3 represents the same func- +tion as the propositional formula x1 ∧ ¬x2 ∧ x3. It has +been observed that a function can be often represented +more compactly as a set of LPBs than as a conjunc- +tive or disjunctive normal form (CNF or DNF) (Dixon +and Ginsberg 2000; Fr¨anzle and Herde 2007). E.g. the +LPB 2x1 + ¯x2 + x3 + x4 ≥ 2 corresponds to the DNF +x1 ∨ (¬x2 ∧ x3) ∨ (¬x2 ∧ x4) ∨ (x3 ∧ x4), which has four +clauses. +1Whenever we say “function” we mean “Boolean func- +tion”. +In this work we are concerned with functions that can +be represented by a single LPB, the so-called threshold +functions. The problem of recognising a Boolean func- +tion given in DNF as threshold function and computing +the LPB representation if possible, is called threshold +recognition problem or threshold synthesis problem. The +problem is known to have an O(m7t5) algorithm using +linear programming, where m is the dimension and t +the number of terms in the DNF (Crama and Hammer +2011). +It has been an open question for decades whether it +is possible to recognise threshold functions through an +entirely combinatorial procedure, i.e., without resorting +to the equivalent linear program. Smaus has developed +a procedure, which works by decomposing the DNF and +“counting” its variable occurrences in an appropriate +way (Smaus 2007a). +Schilling and Wenzelmann, students of Freiburg Uni- +versity, have implemented the classical linear pro- +gramming algorithm and the more recent combinato- +rial algorithm, respectively, as Bachelor thesis projects +(Schilling 2011; Wenzelmann 2011). We report here on +this experience. The most important insight was that +the algorithm by Smaus is, unfortunately, incomplete. +This paper is organised as follows. We continue with +some preliminaries. Sec. 3 describes the linear program- +ming algorithm, Sec. 4 the combinatorial procedure, +Sec. 5 the implementation, and Sec. 6 concludes and +discusses future work. +2 +Preliminaries +We assume the reader to be familiar with the basic no- +tions of propositional logic. +An m-dimensional Boolean function f is a func- +tion Boolm → Bool. A linear pseudo-Boolean con- +straint (LPB) is an inequality of the form +a1ℓ1 + . . . + amℓm ≥ d +ai ∈ N, d ∈ Z, ℓi ∈ {xi, ¯xi}. +(1) +We call the ai coefficients and d the degree (Hooker +1992). An occurrence of a literal xi (resp., ¯xi) is called +an occurrence of xi in positive (resp., negative) polar- +ity. Note that if d ≤ 0, then the LPB is a tautology. The +reason for allowing for negative d will become apparent +in Subsec. 4.2. +arXiv:2301.03667v1 [cs.LO] 9 Jan 2023 + +A DNF is a formula of the form c1 ∨ . . . ∨ cn where +each clause cj is a conjunction of literals. Formally, a +DNF is a set of sets of literals, i.e., the order of clauses +and the order of literals within a clause are insignificant. +For DNFs, we assume without loss of generality that +no clause is a subset of another clause (the latter clause +would be redundant since it is absorbed). We call a DNF +prime irredundant if every clause is a prime implicant, +i.e., if for clause c1 there is no clause c2 ̸= c1 such that +c1 ∨ c2 = c2. Any Boolean function can be represented +by a DNF (Wegener 1987). +It is easy to see that an LPB can only represent mono- +tone functions, i.e., functions represented by a DNF +where each variable occurs in only one polarity. Hence +any DNF containing a variable in different polarities is +immediately uninteresting for us. Without loss of gen- +erality, we assume that this polarity is positive. +3 +The linear programming algorithm +We shortly summarise the solution via linear program- +ming, established by Peled & Simeone (Peled and Sime- +one 1985; Crama and Hammer 2011). +For some DNFs, it is possible to establish a complete +order ⪰ on the variables which, intuitively speaking, +has the following meaning: xi ⪰ xj iff starting from +any given input tuple X∗ ∈ Boolm, setting x∗ +i to true is +more likely to make the DNF true than setting x∗ +j true. +The functions represented by such a DNF are called +regular. +The algorithm first tests the input DNF for the regu- +larity property. The property is weaker than the thresh- +old property, and so if a DNF is not regular, then it is +not convertible and we must give up. +The order is established by counting the variables in +a special way. Intuitively, a variable is “important” if it +occurs in many clauses and if it occurs in short clauses. +This is formalised as the so-called occurrence pattern of +a variable x in φ, written OP(φ, x). For space reasons, +we do not give the formal definition and refer the reader +to (Smaus 2007a). +Computing the set of occurrence patterns for all vari- +ables in φ can be done in time linear in the size of φ +as it can be done in a single pass over φ. In fact, the +number of elements of all occurrence patterns is exactly +the number of literals in φ. Thus sorting the variables +w.r.t. the occurrence patterns can be done in time poly- +nomial in |φ|. +The notion of occurrence patterns is equivalent to the +so-called Winder matrix (Winder 1962). We will need +the concept again in the next section. +Provided the DNF is regular, we make use of the +minimal true points of the DNF, i.e. the true tuples +where we cannot set any 1-value to 0 without making +the point false. We also use the maximal false points de- +fined analogously. Note that these together characterise +the DNF uniquely. In general, no polynomial algorithm +is known to find these points (which is no surprise since +the general task is NP-complete (Peled and Simeone +1985)), but for the special case that the input DNF is +prime irredundant this is possible. The reason is that +the true points can be read directly from the clauses. It +is for this reason that we require the input DNF to be +in prime irredundant form. +Having these, there exists a polynomial time proce- +dure to find the maximal false points. Then we can for- +mulate the following linear program where the minimal +true points are x1, . . . , xk and the maximal false points +are y1, . . . , yl: +m +� +i=1 +aixj +i +≥ +d +(1 ≤ j ≤ k) +m +� +i=1 +aiyj +i +< +d +(1 ≤ j ≤ l) +ai +≥ +0 +(1 ≤ i ≤ m) +Note that the weights ai are the variables in the LP +formulation and the threshold is d. Finally, the linear +program is passed to an LP solver. The reason for the +complexity blow-up (O(m7t5) where m is the dimension +and t the number of terms in the DNF) is mainly due to +the linear programming. The other parts run in O(m2t), +so the whole procedure gains from future improvements +of linear programming. It should be mentioned that for +most inputs the well-known simplex method for solving +linear programs runs in linear time. +4 +The combinatorial algorithm +In this section we recall the results from our previous +work (Smaus 2007a) and present an algorithm for the +problem of converting a DNF to an equivalent LPB if +possible. +4.1 +Determining the order of coefficients +Given a DNF φ, if φ can be represented as an LPB +at all, then the coefficients must respect the order ⪰ +introduced in the previous section, i.e., OP(φ, xi) ⪰ +OP(φ, xk) implies that ai ≥ ak in the resulting LPB: +Lemma 4.1 Let φ be a DNF represented by the LPB +�m +i=1 aixi ≥ d. Then ai ≥ ak implies OP(φ, xi) ⪰ +OP(φ, xk); moreover, there exists an LPB �m +i=1 a′ +ixi ≥ +d′ representing φ such that OP(φ, xi) = OP(φ, xk) im- +plies a′ +i = a′ +k. +In our algorithm, one notion used is that of symme- +try: two variables in a DNF are symmetric if exchanging +them yields the same DNF. For space reasons, we ne- +glect this aspect in the sequel and refer the reader to +(Smaus 2007a). +4.2 +Decomposing a DNF +We want to find an LPB representing φ if possible. +Using Lemma 4.1, we can establish the order of the +coefficients. Assume the numbering of the variables is +such that we have OP(φ, x1) ⪰ . . . ⪰ OP(φ, xm). +Consider now the maximal set X = {x1, . . . , xl} such +that OP(φ, x1) = . . . = OP(φ, xl) (=: OP(φ, X)). (Of + +course, it is very well possible that X = {x1}, i.e., +l = 1.) We want to divide φ into subproblems, and +for this we partition φ according to how many vari- +ables from X each clause contains. We then remove the +variables from X from each clause, which gives l + 1 +subproblems (DNFs). Theorem 4.6 below states under +which conditions solutions to these subproblems can be +combined to an LPB for φ. However, since the solutions +have to be similar in a certain sense, it turns out that +we cannot simply solve the subproblems independently +and then combine the solutions, but we must solve the +subproblems in parallel, as will be shown in Subsec. 4.3. +The following statements do not require X to be max- +imal, e.g. if {x1, . . . , x5} is the maximal set such that +OP(φ, x1) = . . . = OP(φ, x5), then the statements will +also hold for X = {x1, x2, x3}. From now on, the letter +X will always denote a set as just described, maximal +or not. +Definition 4.2 Let φ be a DNF and X a subset of its +variables with |X| = l. If φ contains a clause c ⊆ X, +then let kmax be the length of the longest such clause; +otherwise let kmax := ∞. For 0 ≤ k ≤ l, we define +S(φ, X, k) as the disjunction of clauses from φ contain- +ing exactly min{k, kmax} variables from X, with those +variables removed. +When constructing the S(φ, X, k) from φ, we say that +we split away the variables in X from φ. +Example 4.3 Let φ ≡ (x1) ∨ (x2) ∨ (x3 ∧ x4) and X = +{x1, x2}. We have kmax = 1. Then S(φ, X, 0) = (x3 ∧ +x4), S(φ, X, 1) = true (i.e. the disjunction of twice the +empty conjunction), and S(φ, X, 2) = true. +We must solve the l + 1 subproblems in such a way +that the resulting LPBs agree in all coefficients, and +that the degree difference of neighbouring LPBs is al- +ways the same. Before giving the theorem, we give two +examples for illustration. +Example 4.4 Consider φ ≡ (x1∧x2)∨(x1∧x3)∨(x1∧ +x4) ∨ (x2 ∧ x3 ∧ x4) and X = {x1}. Then S(φ, X, 0) = +x2 ∧x3 ∧x4, represented by x2 +x3 +x4 ≥ 3. Moreover, +S(φ, X, 1) = x2∨x3∨x4, represented by x2+x3+x4 ≥ 1. +Since the coefficients of the two LPBs agree, it turns +out that φ can be represented by 2x1 +x2 +x3 +x4 ≥ 3. +The coefficient of x1 is given by the difference of the +two degrees, i.e. 3 − 1. +Example 4.5 Consider φ ≡ (x1 ∧x2)∨(x1 ∧x3 ∧x4)∨ +(x2 ∧ x3 ∧ x4) and X = {x1, x2}. We have S(φ, X, 0) = +false, represented by x3 + x4 ≥ 4, S(φ, X, 1) = x3 ∧ +x4, represented by x3 + x4 ≥ 2, and S(φ, X, 2) = true, +represented by x3+x4 ≥ 0. The DNF φ is represented by +2x1+2x2+x3+x4 ≥ 4. The coefficient of x1, x2 is given +by 4 − 2 = 2 − 0 = 2 (the degrees are “equidistant”). +Theorem 4.6 Let φ be a DNF in variables x1, . . . , xm +and suppose X = {x1, . . . , xl} are symmetric variables +such that OP(φ, X) is maximal w.r.t. ⪯ in φ. Then φ +is represented by an LPB �m +i=1 aixi ≥ d, where a1 = +. . . = al, iff for all k ∈ [0..l], the DNF S(φ, X, k) is +represented by �m +i=l+1 aixi ≥ d − k · a1. +The remaining problem is that a DNF might be rep- +resented by various LPBs, and so even if the LPBs com- +puted recursively do not have agreeing coefficients and +equidistant degrees, one might find alternative LPBs +(such as the non-obvious LPB for false in Ex. 4.5) so +that Thm. 4.6 can be applied. +Before addressing this problem, we generalise LPBs +by recording to what extent degrees can be shifted with- +out changing the meaning. To formulate this, we tem- +porarily lift the restriction that coefficients and degrees +must be integers. How to obtain integers in the end is +explained at the end of Subsec. 4.3. +Definition 4.7 Given an LPB I ≡ �m +i=1 aixi ≥ d, we +call s the minimum degree of I if s is the smallest +number (possibly −∞) such that for any s′ ∈ (s, d], the +LPB �m +i=1 aixi ≥ s′ represents the same function as +I. We call b the maximum degree if b is the biggest +number (possibly ∞) such that �m +i=1 aixi ≥ b represents +the same function as I. +Note that the minimum degree of I is itself not a +possible degree of I. Since the minimum and maximum +degrees of an LPB are more informative than its actual +degree, we introduce the notation �m +i=1 aixi ≥ (s, b] for +denoting an LPB with minimum degree s and maximum +degree b. +The next lemma strengthens Thm. 4.6, stating that +information about minimum and maximum degrees can +be maintained with little overhead. +Lemma 4.8 Make +the +same +assumptions +as +in +Thm. 4.6, and assume that for all k ∈ [0..l], the DNF +S(φ, X, k) is represented by Ik ≡ �m +i=l+1 aixi ≥ d − +k · a1. Moreover, for all k ∈ [0..l], let sk, bk be min- +imum and maximum degrees of Ik, respectively. Then +s := maxk∈[0..l](sk+k·a1), b := mink∈[0..l](bk+k·ak) are +the minimum and maximum degrees of �m +i=1 aixi ≥ d. +4.3 +Composing LPBs +Theorem 4.6 suggests a recursive algorithm where, at +least conceptually, in the base case we have at most 2m +trivial problems of determining an LPB, trivial since +the formula for which we must find an LPB is either +true or false. +Example 4.9 Consider φ ≡ (x1∧x2)∨(x1∧x3)∨(x1∧ +x4)∨(x1 ∧x5)∨(x2 ∧x3)∨(x2 ∧x4)∨(x3 ∧x4 ∧x5). To +find an LPB for φ, we must find LPBs for S(φ, {x1}, 0) +and S(φ, {x1}, 1). To find an LPB for S(φ, {x1}, 0), +we must find LPBs for S(S(φ, {x1}, 0), {x2}, 0) and +S(S(φ, {x1}, 0), {x2}, 1), and so forth. Table 1 gives all +the formulae for which we must find LPBs. For a con- +cise notation we use some abbreviations which we ex- +plain using S(·, x3..5, 0) ≡ f in the top-right corner: +it stands for S((x3 ∧ x4 ∧ x5), {x3, x4, x5}, 0) ≡ false, +i.e. the ‘·’ stands for the nearest non-shaded formula to +the left, here (x3 ∧ x4 ∧ x5). Note how we arranged the +subproblem formulae in the table: e.g. (x3 ∧x4 ∧x5) has +three symmetric variables that are split away to obtain +the subproblems to be solved, so these subproblems are + +S(·, x3..4, 0) ≡ f +S(·, x3..5, 0) ≡ f +S(·, x1, 0) +S(·, x2, 0) ≡ +S(·, x3, 0) ≡ f +S(·, x3..4, 1) ≡ f +S(·, x3..5, 1) ≡ f +≡ (x2 ∧ x3)∨ +(x3 ∧ x4 ∧ x5) +S(·, x3, 1) +S(·, x3..4, 2) ≡ x5 +S(·, x3..5, 2) ≡ f +(x2 ∧ x4)∨ +≡ (x4 ∧ x5) +S(·, x3..5, 3) ≡ t +(x3 ∧ x4 ∧ x5) +S(·, x2, 1) ≡ +S(·, x3, 0) ≡ x4 +S(·, x3..4, 0) ≡ f +x3 ∨ x4 +S(·, x3, 1) ≡ t +S(·, x3..4, 1) ≡ t +φ +S(·, x3..4, 2) ≡ t +S(·, x2..3, 0) ≡ +S(·, x2..4, 0) ≡ x5 +S(·, x2..5, 0) ≡ f +S(·, x1, 1) +S(·, x2, 0) ≡ +x4 ∨ x5 +S(·, x2..4, 1) ≡ t +S(·, x2..5, 1) ≡ t +≡ x2 ∨ x3 +x3 ∨ x4 ∨ x5 +S(·, x2..3, 1) ≡ t +S(·, x2..4, 2) ≡ t +S(·, x2..5, 2) ≡ t +∨x4 ∨ x5 +S(·, x2, 1) ≡ t +S(·, x2..3, 2) ≡ t +S(·, x2..4, 3) ≡ t +S(·, x2..5, 3) ≡ t +S(·, x2..5, 4) ≡ t +Table 1: The recursive problems of Ex. 4.9 +located three columns to the right of (x3 ∧x4 ∧x5). The +two shaded boxes in between contain the subproblems ob- +tained by splitting away only {x3}, {x3, x4}, resp. Ob- +serve also the empty box in the last column, arising from +the fact that we do not attempt to split away x5 from +x3 ∨ x4. +The algorithm we propose is not a purely recursive +one, since the subproblems at each level must be solved +in parallel. Explained using the example, we first find +LPBs for the formulae in the rightmost column, which +have 0 variables and hence we must determine 0 coef- +ficients. Next to the left, we have formulae that con- +tain (at most) x5, and we determine LPBs representing +these, where we use the same a5 for all formulae! Then +we determine a4, and so forth. +Taking (x3 ∧ x4 ∧ x5) in Table 1 as an exam- +ple, Thm. 4.6 suggests that a3, a4, a5 should be equal +(x3, x4, x5 are symmetric) and determined in one go. +However, since a3, a4, a5 also have to represent other +subproblem formulae where x3, x4, x5 are not necessar- +ily symmetric, one cannot determine a3, a4, a5 in one +go, but rather first a5, then a4, then a3. Therefore, it +is necessary to define and interpret formulae obtained +by splitting away fewer variables than one could split +away, in the sense of Thm. 4.6. These are the shaded +formulae. +For each l ∈ {0, . . . m}, we call the formulae in column +l + 1 the l-successors. Shaded formulae are called aux- +iliary, the others are called main. Formulae that have +no further formulae to the right are called final. The +following definition formalises these notions. +Definition 4.10 Let φ be a DNF in m variables. Then +φ is the 0-successor of φ. Furthermore, φ is a main +successor of φ. Moreover, if φ′ is a main n-successor of +φ, and l is maximal so that xn+1, . . . , xn+l are symmet- +ric in φ′, then for all l′, k with 1 ≤ l′ ≤ l and 0 ≤ k ≤ l′, +we say that S(φ′, {xn+1, . . . , xn+l′}, k) is an (n + l′)- +successor of φ. The (n + l)-successors are called main, +and for l′ < l, the (n + l′)-successors are called auxil- +iary. A node that is a main node and true or false is +called final. +Note in particular x3 ∨ x4 in column 3 in Table 1. It +does not contain x5, and so we obtain final 4-successors +in the last-but-one column. Clearly, a final successor of +φ is either true or false. +Proposition 4.11 Assume φ, φ′, n, l as in Def. 4.10. +For 0 < l′ < l and 0 ≤ k ≤ l′, we have +S(S(φ′, {xn+1, . . . , xn+l′}, k), {xn+l′+1}, 0) ≡ +S(φ′, {xn+1, . . . , xn+l′+1}, k) +S(S(φ′, {xn+1, . . . , xn+l′}, k), {xn+l′+1}, 1) ≡ +S(φ′, {xn+1, . . . , xn+l′+1}, k + 1) +For example, consider S((x3 ∧ x4 ∧ x5), {x3}, 1) ≡ +(x4 ∧ x5) in Table 1. We have S((x4 ∧ x5), {x4}, 0) ≡ +S((x3 ∧x4 ∧x5), {x3, x4}, 1) and S((x4 ∧x5), {x4}, 1) ≡ +S((x3 ∧ x4 ∧ x5), {x3, x4}, 2). Generally, each non-final +successor is associated with two formulae in the column +right next to it, one slightly up and one slightly down, +obtained by splitting away the variable with the small- +est index. +This is not surprising per se and corresponds to a +na¨ıve approach where we always split away one variable +at a time (for applying Thm. 4.6), thereby construct- +ing 2m formulae in the rightmost column. The point of +Prop. 4.11 is that we can usually construct fewer formu- +lae since S(S(φ, {xn+1, . . . , xn+l′}, k), {xn+l′+1}, 1) and +S(S(φ, {xn+1, . . . , xn+l′}, k + 1), {xn+l′+1}, 0) coincide. +This means, φ′ triggers l + 1 main (n + l)-successors in- +stead of 2l. In Table 1, we have 12 final formulae rather +than 25 = 32. +It seems to be generally the case that the table has +much fewer final nodes that 2m. The many examples +we looked at strongly suggest that even if one tries to +construct an input DNF that has as few symmetries as +possible and hence would lead to a big table, the subfor- +mulae constructed by the splitting always exhibit many +symmetries. It would be interesting to have a theoreti- +cal statement about this observation. +The following theorem states if and how one can find +the next coefficient and degrees for representing all k- + +4x1 + 3x2+ +3x2+ +2x3 + 2x4+ +2x3 + 2x4+ +2x3 + 2x4+ +2x4+ +�5 +i=6 aixi +x5 ≥ . . . +x5 ≥ . . . +x5 ≥ . . . +x5 ≥ . . . +x5 ≥ . . . +≥ . . . +(1, ∞] +(0, ∞] +(3, ∞] +(1, ∞] +(0, ∞] +(4, 5] +(2, 3] +(0, 1] +(0, ∞] +(4, 5] +(−∞, 0] +(1, 2] +(1, ∞] +(1, 2] +(−∞, 0] +(−∞, 0] +(4, 5] +(−∞, 0] +(0, 1] +(0, ∞] +(0, 1] +(−∞, 0] +(−∞, 0] +(0, 1] +(0, 1] +(−∞, 0] +(−∞, 0] +(−∞, 0] +(−∞, 0] +(−∞, 0] +(−∞, 0] +(−∞, 0] +(−∞, 0] +Table 2: LPBs for Ex. 4.9 +successors of φ provided one has coefficients and degrees +for representing all (k + 1)-successors. +Theorem 4.12 Assume φ as in Thm. 4.6 and some +k with 0 +≤ +k +≤ +m − 1, and let Φk +be the +set of k-successors of φ. For every non-final φ′ ∈ +Φk, +suppose +we +have +two +LPBs +�m +i=k+2 aixi +≥ +(sφ′0, bφ′0] and �m +i=k+2 aixi ≥ (sφ′1, bφ′1], representing +S(φ′, {xk+1}, 0) and S(φ′, {xk+1}, 1), respectively. +If it is possible to choose ak+1 such that +max +φ′∈Φk(sφ′0 − bφ′1) < ak+1 < min +φ′∈Φk(bφ′0 − sφ′1), +(4) +then for all φ′ ∈ Φk, the LPB �m +i=k+1 aixi ≥ (sφ′, bφ′] +represents φ′, where +sφ′ = max{sφ′0, sφ′1 + ak+1}, +bφ′ = min{bφ′0, bφ′1 + ak+1} for non-final φ′; +(5) +sφ′ = −∞, bφ′ = 0 for φ′ ≡ true; +(6) +sφ′ = �m +i=k+1ai, bφ′ = ∞ for φ′ ≡ false. +(7) +If maxφ′∈Φk(sφ′0 − bφ′1) ≥ minφ′∈Φk(bφ′0 − sφ′1), then +no ak+1, sφ′, bφ′ exist such that �m +i=k+1 aixi ≥ (sφ′, bφ′] +represents φ′ for all φ′ ∈ Φk. +The m-successors of φ, i.e., the formulae in the +rightmost column, can only be false or true. They +are represented by LPBs with an empty sum as +l.h.s.: �m +i=m+1 aixi ≥ (0, ∞] for false, �m +i=m+1 aixi ≥ +(−∞, 0] for true. Then we proceed using Thm. 4.12, in +each step choosing an arbitrary ak+1 fulfilling (4). +Example 4.13 Consider again Ex. 4.9. Table 2 is ar- +ranged in strict correspondence to Table 1 and shows +LPBs for all successors of φ. In the top line we give +the l.h.s. of the LPBs, which is of course the same for +each LPB in a column. In the main table, we list the +minimum and maximum degree of each formula. +In the first step, applying (4), we have to choose a5 +so that +max{0 − ∞, 0 − ∞, 0 − 0, 0 − 0, −∞ − 0, −∞ − 0, +−∞ − 0} < a5 < min{∞ − 0, ∞ − 0, ∞ − −∞, +∞ − −∞, 0 − −∞, 0 − −∞, 0 − −∞}. +Choosing a5 = 1 will do. The minimum and maximum +degrees in column 5 are computed using (5); e.g. the +topmost (1, ∞] is (max{0, 0 + 1}, min{∞, ∞ + 1}]. +In the next step, we have to choose a4 so that +max{1 − ∞, 1 − 1, 1 − 0, −∞ − 0, 0 − 0, −∞ − 0, +−∞ − 0} < a4 < min{∞ − 1, ∞ − 0, ∞ − −∞, +0 − −∞, 1 − −∞, 0 − −∞, 0 − −∞}. +Choosing a4 = 2 will do. Note that the bound 1 − 0 < +a4 comes from the middle box of the fifth column and +thus ultimately from x3 ∨ x4. Our algorithm enforces +that a4 > a5, which must hold for an LPB representing +x3 ∨ x4. +In the next step, a3 can also be chosen to be any +number > 1 so we choose 2 again. In the next step, +2 < a2 < 4 must hold so we choose a2 = 3. Finally, +3 < a1 < 5 must hold so we choose a1 = 4. We obtain +the LPB 4x1 + 3x2 + 2x3 + 2x4 + x5 ≥ (4, 5]. +We have seen in the example how our algorithm +works. However, since the choice of ak+1 is not unique +in general, one might be worried that a bad choice of +ak+1 might later lead to non-applicability of Thm. 4.12. +Contrary to what was stated by Smaus (2007b), this is +indeed a problem. We have suggested to choose ak+1 +always as the smallest possible integer value to obtain +an LPB with small coefficients. But it turns out that +this strategy sometimes leads to a dead end. +Example 4.14 Consider the DNF +φ ≡ (x1 ∧ x2) ∨ (x1 ∧ x3) ∨ (x1 ∧ x4 ∧ x5) (x2 ∧ x3 ∧ x4) +∨ (x2 ∧ x3 ∧ x5) ∨ (x2 ∧ x4 ∧ x5) ∨ (x3 ∧ x4 ∧ x5 ∧ x6) + +(0, ∞] +(0, ∞] +(0, ∞] +(0, ∞] +(−∞, 0] +(1, ∞] +(1, ∞] +(1, ∞] +(0, 1] +(1, ∞] +(1, ∞] +(−∞, 0] +(−∞, 0] +(1, ∞] +(1, ∞] +(−∞, 0] +(3, ∞] +(3, ∞] +(2, 3] +(3, ∞] +(1, 2] +(−∞, 0] +(3, ∞] +(1, 2] +(5, ∞] +(4, 5] +(3, 4] +(1, 2] +(3, 4] +(−∞, 0] +(−∞, 0] +(?, ?] +(?, ?] +(?, ?] +(−∞, 0] +(?, ?] +(?, ?] +(?, ?] +0 ≥ . . . +x6 ≥ . . . +2x5 + x6 ≥ . . . +2x5 + x6 ≥ . . . +2x4 + +Table 3: LPBs for Ex. 4.14 +We apply the algorithm to create all successors of φ and +calculate LPBs for all recursive subproblems. The corre- +sponding LPBs can be found in Table 3. By applying the +strategy of choosing the coefficient as small as possible +we choose a6 = 1, a5 = 2, a4 = 2. We use the minimum +and maximum degrees in the fourth column to choose +the coefficient a3. We have to choose a3 s. t. +max {5 − 5, 3 − 2 , 3 − 0, −∞ − 0} < a3 < +min {∞ − 4, 4 − 1, 4 − −∞, 0 − −∞} +i. e. 3 < a3 < 3. This is, of course, not possible. But φ +can be represented by the LPB 9x1 + 7x2 + 6x3 + 4x4 + +4x5 + x6 ≥ 15. +The algorithm found solutions for all subproblems in +the fourth column. But we cannot combine the coeffi- +cients chosen so far to a solution representing all LPBs +in the third column. +Alternatively, we were allowed to choose a5 = a4 = 4, +and if we do so, we obtain an appropriate LPB. There- +fore the applicability of Thm. 4.12 depends on the choice +of the previous coefficients. +Another problem seems to be that ak+1 could be +forced to be between neighbouring integers, in which +case it cannot be an integer itself. However, in this case, +one can multiply all LPBs of the current system by 2 +(this obviously preserves the meaning of the LPBs) be- +fore proceeding so that ak+1 can be chosen to be an +integer. +From the construction of the successors (see Table 1) +it follows that all formulae in a column together have +size less than all formulae in the column to the left of it, +so that the entire table has size less than |φ|·(m+1). One +can thus show that the complexity of the algorithm is +polynomial in the size of φ, while the size of φ itself can +be exponential in m. In fact, this is the most interesting +case, because in this case an LPB representation may +yield an exponential saving. +13 14 15 16 17 18 19 20 21 22 23 24 25 +0 +20 +40 +60 +80 +variables +% conversion fails +Figure 1: Failure rate of the combinatorial algorithm +5 +Implementation +Both algorithms have been implemented in Java. They +share the same core classes representing the main com- +ponents such as DNFs and LPBs. The linear program +is solved by lp_solve. Both implementations can be +accessed and tested via a graphical user interface. +For testing the implementation we generated a full +enumeration of LPBs up to seven variables. For LPBs +with more variables we tested 180,000 randomly gen- +erated LPBs (with 8 to 25 variables). We transformed +the LPBs to DNFs (so we know that for these DNFs +there exists an LPB) to test the implementations. As +expected the linear programming algorithm solved all +tested input DNFs. +The combinatorial algorithm was able to solve all in- +put DNFs with up to five variables. But it fails on some +DNFs with six variables (with the strategy to choose + +ak+1 as small as possible). Our empirical analysis shows +that the more variables a DNF contains the more often +the conversion fails. Circa 8% of the tested DNFs with +seven variables cannot be converted, for DNFs with 25 +variables circa 86% cannot be converted. The failure +rate for 13 to 25 variables is illustrated in Figure 1. +The linear programming algorithm was faster in di- +rect runtime comparison, but we’re still working on im- +provements for the combinatorial algorithm. +As discussed in Subsec. 4.3, for the combinatorial al- +gorithm the number of final nodes is an important cri- +terion for its theoretical runtime. Figure 2 gives a first +impression. It is hard to judge whether the growth ex- +hibited is exponential, but in any case, the number of +final nodes is much smaller than 2m: around 50000 times +smaller for m = 25. +6 +Conclusion and future work +Linear pseudo-Boolean constraints have attracted inter- +est because they can often be used to represent Boolean +functions more compactly than CNFs or DNFs, and +because techniques applied in CNF-based propositional +satisfiability solving can be generalised to LPBs (Dixon +and Ginsberg 2000; Fr¨anzle and Herde 2007). +Some Boolean functions can be represented by a sin- +gle LPB. The problem of finding this LPB representa- +tion is called threshold recognition problem. In this work, +we have implemented two algorithms for this problem, a +classical one based on linear programming, and a more +recent one that we have previously presented. The most +important insight was that our algorithm is, unfortu- +nately, incomplete. +The most important topic for future work is, of +course, trying to reestablish completeness. +The obvious way to achieve this is to incorporate +some kind of backtracking into the algorithm: If a DNF +can be represented by an LPB and we cannot choose +ak+1, then this is because we must have chosen one +of the coefficients ak+2, . . . , am too small, because our +strategy so far was to chose the coefficients as small as +possible. In order to find a solution we increment the +coefficients ak+2, . . . , am and re-evaluate the LPBs. We +can use the minimum and maximum degrees to ensure +that we enumerate only legal candidates. We iteratively +increment the coefficients until we can choose the coef- +ficient ak+1. +One problem of this approach is that for a DNF that +cannot be represented by an LPB, termination is not +guaranteed, because frequently the choice of the next +coefficient is not bounded from above. However, we are +confident that this problem can be resolved because it +should be possible to derive some upper bound for each +variable in the sense that it is never necessary to choose +a coefficient bigger than this bound (something along +the lines: it is never necessary to choose a coefficient +more than m times bigger than the previous coefficient). +The other problem is of course that backtracking +worsens that runtime of the algorithm, and we very +much fear that it will destroy the polynomial runtime +of the algorithm. +The backtracking approach has been implemented +and was able to find a solution for each tested input +DNF. But the implementation has also shown that the +higher the dimension m, the more often we have to use +backtracking. +Alternatively, or more likely, additionally, one might +use the occurrence patterns for estimating the weight +ratio: In the example above we were able to represent +all LPBs in the fourth column but we were not able to +choose a3 such that we can represent all LPBs in the +third column with the configuration a6 = 1, a5 = a4 = +2. We need some global information that the distance +between a6 and a5, a4 will be too small in the sequel. +Maybe it is possible to use the occurrence patterns to +formulate such constraints, i.e., one might find a con- +straint of the form “in an LPB representing φ one has +to ensure that ai ≥ w · aj”. +It has to be said however that there have been previ- +ous attempts to somehow directly translate the occur- +rence patterns into numeric coefficients or better, co- +efficient ratios; the threshold recognition problem has +stubbornly resisted such attempts2. +However, even a rough estimate of the coefficient ra- +tion, based on the occurrence patterns, might be useful +for reducing if not eliminating the backtracking effort. +One other interesting topic is a more thorough anal- +ysis of the complexity of the combinatorial algorithm, +whether it is in its current state or after having achieved +completeness. In particular, as we have mentioned in +Subsec. 4.3, analysing the effect of exploiting the sym- +metries in the input DNF would be interesting. +Acknowledgements +We thank Yves Crama and +Utz-Uwe Haus for very fruitful discussions about this +work. +References +[Crama and Hammer 2011] Crama, Y., and Hammer, +P. L. 2011. Boolean Functions - Theory, Algorithms, +and Applications, volume 142 of Encyclopedia of math- +ematics and its applications. +Cambridge University +Press. +[Dixon and Ginsberg 2000] Dixon, H. E., and Ginsberg, +M. L. 2000. Combining satisfiability techniques from AI +and OR. The Knowledge Engineering Review 15(1):31– +45. +[Fr¨anzle and Herde 2007] Fr¨anzle, M., and Herde, C. +2007. HySAT: An efficient proof engine for bounded +model checking of hybrid systems. +Formal Methods +Syst. Des. 30(3):179–198. +[Hooker 1992] Hooker, J. N. 1992. Generalized resolu- +tion for 0-1 linear inequalities. Ann. Math. Artif. Intell. +6(1-3):271–286. +2Personal communication with Yves Crama + +13 +14 +15 +16 +17 +18 +19 +20 +21 +22 +23 +24 +25 +100 +200 +300 +400 +500 +600 +variables (m) +final node count +Average final node count for m variables +λ(m) +Figure 2: λ(m) is the average number of final nodes for DNFs with m variables. +[Peled and Simeone 1985] Peled, U. N., and Simeone, +B. 1985. Polynomial-time algorithms for regular set- +covering and threshold synthesis. Discret. Appl. Math. +12(1):57–69. +[Schilling 2011] Schilling, C. 2011. Solving the Thresh- +old Synthesis Problem of Boolean Functions by Trans- +lation to Linear Programming. Bachelor thesis, Albert- +Ludwigs-Universit¨at Freiburg. +[Smaus 2007a] Smaus, J. 2007a. On Boolean functions +encodable as a single linear pseudo-Boolean constraint. +In CPAIOR, volume 4510 of LNCS, 288–302. Springer. +[Smaus 2007b] Smaus, J. 2007b. On Boolean functions +encodable as a single linear pseudo-Boolean constraint. +Technical Report 230, Institut f¨ur Informatik, Univer- +sit¨at Freiburg. +Long version of (Smaus 2007a). Also +available as TR No. 13 on www.avacs.org. +[Wegener 1987] Wegener, I. +1987. +The complexity of +Boolean functions. Wiley-Teubner. +[Wenzelmann 2011] Wenzelmann, F. 2011. Solving the +Threshold Synthesis Problem of Boolean Functions by +a Combinatorial Algorithm. +Bachelor thesis, Albert- +Ludwigs-Universit¨at Freiburg. +[Winder 1962] Winder, +R. +O. +1962. +Threshold +Logic. Ph.D. Dissertation, Department of Mathemat- +ics, Princeton University, Princeton, U.S.A. + diff --git a/g9E2T4oBgHgl3EQfHQbG/content/tmp_files/load_file.txt b/g9E2T4oBgHgl3EQfHQbG/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0b3b8a60de3f2bd8f5335e93f75f40edb8614f48 --- /dev/null +++ b/g9E2T4oBgHgl3EQfHQbG/content/tmp_files/load_file.txt @@ -0,0 +1,603 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf,len=602 +page_content='Implementations of two Algorithms for the Threshold Synthesis Problem Jan-Georg Smaus IRIT Universit´e Paul Sabatier Toulouse France Christian Schilling Fabian Wenzelmann Institut f¨ur Informatik Albert-Ludwigs-Universit¨at Freiburg Germany Abstract A linear pseudo-Boolean constraint (LPB) is an expres- sion of the form a1 · ℓ1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' + am · ℓm ≥ d, where each ℓi is a literal (it assumes the value 1 or 0 depending on whether a propositional variable xi is true or false) and a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' , am, d are natural numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' An LPB represents a Boolean function, and those Boolean functions that can be represented by exactly one LPB are called threshold functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' The problem of finding an LPB representation of a Boolean function if possi- ble is called threshold recognition problem or threshold synthesis problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' The problem has an O(m7t5) algo- rithm using linear programming, where m is the di- mension and t the number of terms in the DNF input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' It has been an open question whether one can recog- nise threshold functions through an entirely combina- torial procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Smaus has developed such a proce- dure for doing this, which works by decomposing the DNF and “counting” the variable occurrences in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' We have implemented both algorithms as a thesis project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' We report here on this experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' The most important insight was that the algorithm by Smaus is, unfortu- nately, incomplete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' 1 Introduction A linear pseudo-Boolean constraint (LPB) (Dixon and Ginsberg 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Fr¨anzle and Herde 2007) is an expres- sion of the form a1ℓ1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' + amℓm ≥ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Here each ℓi is a literal of the form xi or ¯xi ≡ 1 − xi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' xi becomes 0 if xi is false and 1 if xi is true, and vice versa for ¯xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Moreover, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' , am, d are natural numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' An LPB can be used to represent a Boolean1 func- tion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' x1 + ¯x2 + x3 ≥ 3 represents the same func- tion as the propositional formula x1 ∧ ¬x2 ∧ x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' It has been observed that a function can be often represented more compactly as a set of LPBs than as a conjunc- tive or disjunctive normal form (CNF or DNF) (Dixon and Ginsberg 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Fr¨anzle and Herde 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' the LPB 2x1 + ¯x2 + x3 + x4 ≥ 2 corresponds to the DNF x1 ∨ (¬x2 ∧ x3) ∨ (¬x2 ∧ x4) ∨ (x3 ∧ x4), which has four clauses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' 1Whenever we say “function” we mean “Boolean func- tion”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' In this work we are concerned with functions that can be represented by a single LPB, the so-called threshold functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' The problem of recognising a Boolean func- tion given in DNF as threshold function and computing the LPB representation if possible, is called threshold recognition problem or threshold synthesis problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' The problem is known to have an O(m7t5) algorithm using linear programming, where m is the dimension and t the number of terms in the DNF (Crama and Hammer 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' It has been an open question for decades whether it is possible to recognise threshold functions through an entirely combinatorial procedure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=', without resorting to the equivalent linear program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Smaus has developed a procedure, which works by decomposing the DNF and “counting” its variable occurrences in an appropriate way (Smaus 2007a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Schilling and Wenzelmann, students of Freiburg Uni- versity, have implemented the classical linear pro- gramming algorithm and the more recent combinato- rial algorithm, respectively, as Bachelor thesis projects (Schilling 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Wenzelmann 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' We report here on this experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' The most important insight was that the algorithm by Smaus is, unfortunately, incomplete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' This paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' We continue with some preliminaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' 3 describes the linear program- ming algorithm, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' 4 the combinatorial procedure, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' 5 the implementation, and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' 6 concludes and discusses future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' 2 Preliminaries We assume the reader to be familiar with the basic no- tions of propositional logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' An m-dimensional Boolean function f is a func- tion Boolm → Bool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' A linear pseudo-Boolean con- straint (LPB) is an inequality of the form a1ℓ1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' + amℓm ≥ d ai ∈ N, d ∈ Z, ℓi ∈ {xi, ¯xi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' (1) We call the ai coefficients and d the degree (Hooker 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' An occurrence of a literal xi (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=', ¯xi) is called an occurrence of xi in positive (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=', negative) polar- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Note that if d ≤ 0, then the LPB is a tautology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' The reason for allowing for negative d will become apparent in Subsec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='03667v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='LO] 9 Jan 2023 A DNF is a formula of the form c1 ∨ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' ∨ cn where each clause cj is a conjunction of literals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Formally, a DNF is a set of sets of literals, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=', the order of clauses and the order of literals within a clause are insignificant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' For DNFs, we assume without loss of generality that no clause is a subset of another clause (the latter clause would be redundant since it is absorbed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' We call a DNF prime irredundant if every clause is a prime implicant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=', if for clause c1 there is no clause c2 ̸= c1 such that c1 ∨ c2 = c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Any Boolean function can be represented by a DNF (Wegener 1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' It is easy to see that an LPB can only represent mono- tone functions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=', functions represented by a DNF where each variable occurs in only one polarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Hence any DNF containing a variable in different polarities is immediately uninteresting for us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Without loss of gen- erality, we assume that this polarity is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' 3 The linear programming algorithm We shortly summarise the solution via linear program- ming, established by Peled & Simeone (Peled and Sime- one 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Crama and Hammer 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' For some DNFs, it is possible to establish a complete order ⪰ on the variables which, intuitively speaking, has the following meaning: xi ⪰ xj iff starting from any given input tuple X∗ ∈ Boolm, setting x∗ i to true is more likely to make the DNF true than setting x∗ j true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' The functions represented by such a DNF are called regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' The algorithm first tests the input DNF for the regu- larity property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' The property is weaker than the thresh- old property, and so if a DNF is not regular, then it is not convertible and we must give up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' The order is established by counting the variables in a special way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Intuitively, a variable is “important” if it occurs in many clauses and if it occurs in short clauses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' This is formalised as the so-called occurrence pattern of a variable x in φ, written OP(φ, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' For space reasons, we do not give the formal definition and refer the reader to (Smaus 2007a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Computing the set of occurrence patterns for all vari- ables in φ can be done in time linear in the size of φ as it can be done in a single pass over φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' In fact, the number of elements of all occurrence patterns is exactly the number of literals in φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Thus sorting the variables w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' the occurrence patterns can be done in time poly- nomial in |φ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' The notion of occurrence patterns is equivalent to the so-called Winder matrix (Winder 1962).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' We will need the concept again in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Provided the DNF is regular, we make use of the minimal true points of the DNF, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' the true tuples where we cannot set any 1-value to 0 without making the point false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' We also use the maximal false points de- fined analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Note that these together characterise the DNF uniquely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' In general, no polynomial algorithm is known to find these points (which is no surprise since the general task is NP-complete (Peled and Simeone 1985)), but for the special case that the input DNF is prime irredundant this is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' The reason is that the true points can be read directly from the clauses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' It is for this reason that we require the input DNF to be in prime irredundant form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Having these, there exists a polynomial time proce- dure to find the maximal false points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Then we can for- mulate the following linear program where the minimal true points are x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' , xk and the maximal false points are y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' , yl: m � i=1 aixj i ≥ d (1 ≤ j ≤ k) m � i=1 aiyj i < d (1 ≤ j ≤ l) ai ≥ 0 (1 ≤ i ≤ m) Note that the weights ai are the variables in the LP formulation and the threshold is d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Finally, the linear program is passed to an LP solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' The reason for the complexity blow-up (O(m7t5) where m is the dimension and t the number of terms in the DNF) is mainly due to the linear programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' The other parts run in O(m2t), so the whole procedure gains from future improvements of linear programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' It should be mentioned that for most inputs the well-known simplex method for solving linear programs runs in linear time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' 4 The combinatorial algorithm In this section we recall the results from our previous work (Smaus 2007a) and present an algorithm for the problem of converting a DNF to an equivalent LPB if possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='1 Determining the order of coefficients Given a DNF φ, if φ can be represented as an LPB at all, then the coefficients must respect the order ⪰ introduced in the previous section, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=', OP(φ, xi) ⪰ OP(φ, xk) implies that ai ≥ ak in the resulting LPB: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='1 Let φ be a DNF represented by the LPB �m i=1 aixi ≥ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Then ai ≥ ak implies OP(φ, xi) ⪰ OP(φ, xk);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' moreover, there exists an LPB �m i=1 a′ ixi ≥ d′ representing φ such that OP(φ, xi) = OP(φ, xk) im- plies a′ i = a′ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' In our algorithm, one notion used is that of symme- try: two variables in a DNF are symmetric if exchanging them yields the same DNF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' For space reasons, we ne- glect this aspect in the sequel and refer the reader to (Smaus 2007a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='2 Decomposing a DNF We want to find an LPB representing φ if possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='1, we can establish the order of the coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Assume the numbering of the variables is such that we have OP(φ, x1) ⪰ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' ⪰ OP(φ, xm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Consider now the maximal set X = {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' , xl} such that OP(φ, x1) = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' = OP(φ, xl) (=: OP(φ, X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' (Of course, it is very well possible that X = {x1}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=', l = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=') We want to divide φ into subproblems, and for this we partition φ according to how many vari- ables from X each clause contains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' We then remove the variables from X from each clause, which gives l + 1 subproblems (DNFs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='6 below states under which conditions solutions to these subproblems can be combined to an LPB for φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' However, since the solutions have to be similar in a certain sense, it turns out that we cannot simply solve the subproblems independently and then combine the solutions, but we must solve the subproblems in parallel, as will be shown in Subsec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' The following statements do not require X to be max- imal, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' if {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' , x5} is the maximal set such that OP(φ, x1) = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' = OP(φ, x5), then the statements will also hold for X = {x1, x2, x3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' From now on, the letter X will always denote a set as just described, maximal or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='2 Let φ be a DNF and X a subset of its variables with |X| = l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' If φ contains a clause c ⊆ X, then let kmax be the length of the longest such clause;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' otherwise let kmax := ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' For 0 ≤ k ≤ l, we define S(φ, X, k) as the disjunction of clauses from φ contain- ing exactly min{k, kmax} variables from X, with those variables removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' When constructing the S(φ, X, k) from φ, we say that we split away the variables in X from φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='3 Let φ ≡ (x1) ∨ (x2) ∨ (x3 ∧ x4) and X = {x1, x2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' We have kmax = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Then S(φ, X, 0) = (x3 ∧ x4), S(φ, X, 1) = true (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' the disjunction of twice the empty conjunction), and S(φ, X, 2) = true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' We must solve the l + 1 subproblems in such a way that the resulting LPBs agree in all coefficients, and that the degree difference of neighbouring LPBs is al- ways the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Before giving the theorem, we give two examples for illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='4 Consider φ ≡ (x1∧x2)∨(x1∧x3)∨(x1∧ x4) ∨ (x2 ∧ x3 ∧ x4) and X = {x1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Then S(φ, X, 0) = x2 ∧x3 ∧x4, represented by x2 +x3 +x4 ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Moreover, S(φ, X, 1) = x2∨x3∨x4, represented by x2+x3+x4 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Since the coefficients of the two LPBs agree, it turns out that φ can be represented by 2x1 +x2 +x3 +x4 ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' The coefficient of x1 is given by the difference of the two degrees, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' 3 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='5 Consider φ ≡ (x1 ∧x2)∨(x1 ∧x3 ∧x4)∨ (x2 ∧ x3 ∧ x4) and X = {x1, x2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' We have S(φ, X, 0) = false, represented by x3 + x4 ≥ 4, S(φ, X, 1) = x3 ∧ x4, represented by x3 + x4 ≥ 2, and S(φ, X, 2) = true, represented by x3+x4 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' The DNF φ is represented by 2x1+2x2+x3+x4 ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' The coefficient of x1, x2 is given by 4 − 2 = 2 − 0 = 2 (the degrees are “equidistant”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='6 Let φ be a DNF in variables x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' , xm and suppose X = {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' , xl} are symmetric variables such that OP(φ, X) is maximal w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' ⪯ in φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Then φ is represented by an LPB �m i=1 aixi ≥ d, where a1 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' = al, iff for all k ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='.l], the DNF S(φ, X, k) is represented by �m i=l+1 aixi ≥ d − k · a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' The remaining problem is that a DNF might be rep- resented by various LPBs, and so even if the LPBs com- puted recursively do not have agreeing coefficients and equidistant degrees, one might find alternative LPBs (such as the non-obvious LPB for false in Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='5) so that Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='6 can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Before addressing this problem, we generalise LPBs by recording to what extent degrees can be shifted with- out changing the meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' To formulate this, we tem- porarily lift the restriction that coefficients and degrees must be integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' How to obtain integers in the end is explained at the end of Subsec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='7 Given an LPB I ≡ �m i=1 aixi ≥ d, we call s the minimum degree of I if s is the smallest number (possibly −∞) such that for any s′ ∈ (s, d], the LPB �m i=1 aixi ≥ s′ represents the same function as I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' We call b the maximum degree if b is the biggest number (possibly ∞) such that �m i=1 aixi ≥ b represents the same function as I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Note that the minimum degree of I is itself not a possible degree of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Since the minimum and maximum degrees of an LPB are more informative than its actual degree, we introduce the notation �m i=1 aixi ≥ (s, b] for denoting an LPB with minimum degree s and maximum degree b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' The next lemma strengthens Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='6, stating that information about minimum and maximum degrees can be maintained with little overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='8 Make the same assumptions as in Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='6, and assume that for all k ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='.l], the DNF S(φ, X, k) is represented by Ik ≡ �m i=l+1 aixi ≥ d − k · a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Moreover, for all k ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='.l], let sk, bk be min- imum and maximum degrees of Ik, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Then s := maxk∈[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='.l](sk+k·a1), b := mink∈[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='.l](bk+k·ak) are the minimum and maximum degrees of �m i=1 aixi ≥ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='3 Composing LPBs Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='6 suggests a recursive algorithm where, at least conceptually, in the base case we have at most 2m trivial problems of determining an LPB, trivial since the formula for which we must find an LPB is either true or false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='9 Consider φ ≡ (x1∧x2)∨(x1∧x3)∨(x1∧ x4)∨(x1 ∧x5)∨(x2 ∧x3)∨(x2 ∧x4)∨(x3 ∧x4 ∧x5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' To find an LPB for φ, we must find LPBs for S(φ, {x1}, 0) and S(φ, {x1}, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' To find an LPB for S(φ, {x1}, 0), we must find LPBs for S(S(φ, {x1}, 0), {x2}, 0) and S(S(φ, {x1}, 0), {x2}, 1), and so forth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Table 1 gives all the formulae for which we must find LPBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' For a con- cise notation we use some abbreviations which we ex- plain using S(·, x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='.5, 0) ≡ f in the top-right corner: it stands for S((x3 ∧ x4 ∧ x5), {x3, x4, x5}, 0) ≡ false, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' the ‘·’ stands for the nearest non-shaded formula to the left, here (x3 ∧ x4 ∧ x5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Note how we arranged the subproblem formulae in the table: e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' (x3 ∧x4 ∧x5) has three symmetric variables that are split away to obtain the subproblems to be solved, so these subproblems are S(·, x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='.4, 0) ≡ f S(·, x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='.5, 0) ≡ f S(·, x1, 0) S(·, x2, 0) ≡ S(·, x3, 0) ≡ f S(·, x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='.4, 1) ≡ f S(·, x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='.5, 1) ≡ f ≡ (x2 ∧ x3)∨ (x3 ∧ x4 ∧ x5) S(·, x3, 1) S(·, x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='.4, 2) ≡ x5 S(·, x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='.5, 2) ≡ f (x2 ∧ x4)∨ ≡ (x4 ∧ x5) S(·, x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='.5, 3) ≡ t (x3 ∧ x4 ∧ x5) S(·, x2, 1) ≡ S(·, x3, 0) ≡ x4 S(·, x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='.4, 0) ≡ f x3 ∨ x4 S(·, x3, 1) ≡ t S(·, x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='.4, 1) ≡ t φ S(·, x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='.4, 2) ≡ t S(·, x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='.3, 0) ≡ S(·, x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='.4, 0) ≡ x5 S(·, x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='.5, 0) ≡ f S(·, x1, 1) S(·, x2, 0) ≡ x4 ∨ x5 S(·, x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='.4, 1) ≡ t S(·, x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='.5, 1) ≡ t ≡ x2 ∨ x3 x3 ∨ x4 ∨ x5 S(·, x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='.3, 1) ≡ t S(·, x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='.4, 2) ≡ t S(·, x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='.5, 2) ≡ t ∨x4 ∨ x5 S(·, x2, 1) ≡ t S(·, x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='.3, 2) ≡ t S(·, x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='.4, 3) ≡ t S(·, x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='.5, 3) ≡ t S(·, x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='.5, 4) ≡ t Table 1: The recursive problems of Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='9 located three columns to the right of (x3 ∧x4 ∧x5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' The two shaded boxes in between contain the subproblems ob- tained by splitting away only {x3}, {x3, x4}, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Ob- serve also the empty box in the last column, arising from the fact that we do not attempt to split away x5 from x3 ∨ x4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' The algorithm we propose is not a purely recursive one, since the subproblems at each level must be solved in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Explained using the example, we first find LPBs for the formulae in the rightmost column, which have 0 variables and hence we must determine 0 coef- ficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Next to the left, we have formulae that con- tain (at most) x5, and we determine LPBs representing these, where we use the same a5 for all formulae!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Then we determine a4, and so forth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Taking (x3 ∧ x4 ∧ x5) in Table 1 as an exam- ple, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='6 suggests that a3, a4, a5 should be equal (x3, x4, x5 are symmetric) and determined in one go.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' However, since a3, a4, a5 also have to represent other subproblem formulae where x3, x4, x5 are not necessar- ily symmetric, one cannot determine a3, a4, a5 in one go, but rather first a5, then a4, then a3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Therefore, it is necessary to define and interpret formulae obtained by splitting away fewer variables than one could split away, in the sense of Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' These are the shaded formulae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' For each l ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' m}, we call the formulae in column l + 1 the l-successors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Shaded formulae are called aux- iliary, the others are called main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Formulae that have no further formulae to the right are called final.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' The following definition formalises these notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='10 Let φ be a DNF in m variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Then φ is the 0-successor of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Furthermore, φ is a main successor of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Moreover, if φ′ is a main n-successor of φ, and l is maximal so that xn+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' , xn+l are symmet- ric in φ′, then for all l′, k with 1 ≤ l′ ≤ l and 0 ≤ k ≤ l′, we say that S(φ′, {xn+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' , xn+l′}, k) is an (n + l′)- successor of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' The (n + l)-successors are called main, and for l′ < l, the (n + l′)-successors are called auxil- iary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' A node that is a main node and true or false is called final.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Note in particular x3 ∨ x4 in column 3 in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' It does not contain x5, and so we obtain final 4-successors in the last-but-one column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Clearly, a final successor of φ is either true or false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='11 Assume φ, φ′, n, l as in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' For 0 < l′ < l and 0 ≤ k ≤ l′, we have S(S(φ′, {xn+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' , xn+l′}, k), {xn+l′+1}, 0) ≡ S(φ′, {xn+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' , xn+l′+1}, k) S(S(φ′, {xn+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' , xn+l′}, k), {xn+l′+1}, 1) ≡ S(φ′, {xn+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' , xn+l′+1}, k + 1) For example, consider S((x3 ∧ x4 ∧ x5), {x3}, 1) ≡ (x4 ∧ x5) in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' We have S((x4 ∧ x5), {x4}, 0) ≡ S((x3 ∧x4 ∧x5), {x3, x4}, 1) and S((x4 ∧x5), {x4}, 1) ≡ S((x3 ∧ x4 ∧ x5), {x3, x4}, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Generally, each non-final successor is associated with two formulae in the column right next to it, one slightly up and one slightly down, obtained by splitting away the variable with the small- est index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' This is not surprising per se and corresponds to a na¨ıve approach where we always split away one variable at a time (for applying Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='6), thereby construct- ing 2m formulae in the rightmost column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' The point of Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='11 is that we can usually construct fewer formu- lae since S(S(φ, {xn+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' , xn+l′}, k), {xn+l′+1}, 1) and S(S(φ, {xn+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' , xn+l′}, k + 1), {xn+l′+1}, 0) coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' This means, φ′ triggers l + 1 main (n + l)-successors in- stead of 2l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' In Table 1, we have 12 final formulae rather than 25 = 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' It seems to be generally the case that the table has much fewer final nodes that 2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' The many examples we looked at strongly suggest that even if one tries to construct an input DNF that has as few symmetries as possible and hence would lead to a big table, the subfor- mulae constructed by the splitting always exhibit many symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' It would be interesting to have a theoreti- cal statement about this observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' The following theorem states if and how one can find the next coefficient and degrees for representing all k- 4x1 + 3x2+ 3x2+ 2x3 + 2x4+ 2x3 + 2x4+ 2x3 + 2x4+ 2x4+ �5 i=6 aixi x5 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' x5 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' x5 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' x5 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' x5 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' (1, ∞] (0, ∞] (3, ∞] (1, ∞] (0, ∞] (4, 5] (2, 3] (0, 1] (0, ∞] (4, 5] (−∞, 0] (1, 2] (1, ∞] (1, 2] (−∞, 0] (−∞, 0] (4, 5] (−∞, 0] (0, 1] (0, ∞] (0, 1] (−∞, 0] (−∞, 0] (0, 1] (0, 1] (−∞, 0] (−∞, 0] (−∞, 0] (−∞, 0] (−∞, 0] (−∞, 0] (−∞, 0] (−∞, 0] Table 2: LPBs for Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='9 successors of φ provided one has coefficients and degrees for representing all (k + 1)-successors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='12 Assume φ as in Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='6 and some k with 0 ≤ k ≤ m − 1, and let Φk be the set of k-successors of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' For every non-final φ′ ∈ Φk, suppose we have two LPBs �m i=k+2 aixi ≥ (sφ′0, bφ′0] and �m i=k+2 aixi ≥ (sφ′1, bφ′1], representing S(φ′, {xk+1}, 0) and S(φ′, {xk+1}, 1), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' If it is possible to choose ak+1 such that max φ′∈Φk(sφ′0 − bφ′1) < ak+1 < min φ′∈Φk(bφ′0 − sφ′1), (4) then for all φ′ ∈ Φk, the LPB �m i=k+1 aixi ≥ (sφ′, bφ′] represents φ′, where sφ′ = max{sφ′0, sφ′1 + ak+1}, bφ′ = min{bφ′0, bφ′1 + ak+1} for non-final φ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' (5) sφ′ = −∞, bφ′ = 0 for φ′ ≡ true;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' (6) sφ′ = �m i=k+1ai, bφ′ = ∞ for φ′ ≡ false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' (7) If maxφ′∈Φk(sφ′0 − bφ′1) ≥ minφ′∈Φk(bφ′0 − sφ′1), then no ak+1, sφ′, bφ′ exist such that �m i=k+1 aixi ≥ (sφ′, bφ′] represents φ′ for all φ′ ∈ Φk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' The m-successors of φ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=', the formulae in the rightmost column, can only be false or true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' They are represented by LPBs with an empty sum as l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' : �m i=m+1 aixi ≥ (0, ∞] for false, �m i=m+1 aixi ≥ (−∞, 0] for true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Then we proceed using Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='12, in each step choosing an arbitrary ak+1 fulfilling (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='13 Consider again Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Table 2 is ar- ranged in strict correspondence to Table 1 and shows LPBs for all successors of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' In the top line we give the l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' of the LPBs, which is of course the same for each LPB in a column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' In the main table, we list the minimum and maximum degree of each formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' In the first step, applying (4), we have to choose a5 so that max{0 − ∞, 0 − ∞, 0 − 0, 0 − 0, −∞ − 0, −∞ − 0, −∞ − 0} < a5 < min{∞ − 0, ∞ − 0, ∞ − −∞, ∞ − −∞, 0 − −∞, 0 − −∞, 0 − −∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Choosing a5 = 1 will do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' The minimum and maximum degrees in column 5 are computed using (5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' the topmost (1, ∞] is (max{0, 0 + 1}, min{∞, ∞ + 1}].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' In the next step, we have to choose a4 so that max{1 − ∞, 1 − 1, 1 − 0, −∞ − 0, 0 − 0, −∞ − 0, −∞ − 0} < a4 < min{∞ − 1, ∞ − 0, ∞ − −∞, 0 − −∞, 1 − −∞, 0 − −∞, 0 − −∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Choosing a4 = 2 will do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Note that the bound 1 − 0 < a4 comes from the middle box of the fifth column and thus ultimately from x3 ∨ x4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Our algorithm enforces that a4 > a5, which must hold for an LPB representing x3 ∨ x4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' In the next step, a3 can also be chosen to be any number > 1 so we choose 2 again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' In the next step, 2 < a2 < 4 must hold so we choose a2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Finally, 3 < a1 < 5 must hold so we choose a1 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' We obtain the LPB 4x1 + 3x2 + 2x3 + 2x4 + x5 ≥ (4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' We have seen in the example how our algorithm works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' However, since the choice of ak+1 is not unique in general, one might be worried that a bad choice of ak+1 might later lead to non-applicability of Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Contrary to what was stated by Smaus (2007b), this is indeed a problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' We have suggested to choose ak+1 always as the smallest possible integer value to obtain an LPB with small coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' But it turns out that this strategy sometimes leads to a dead end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='14 Consider the DNF φ ≡ (x1 ∧ x2) ∨ (x1 ∧ x3) ∨ (x1 ∧ x4 ∧ x5) (x2 ∧ x3 ∧ x4) ∨ (x2 ∧ x3 ∧ x5) ∨ (x2 ∧ x4 ∧ x5) ∨ (x3 ∧ x4 ∧ x5 ∧ x6) (0, ∞] (0, ∞] (0, ∞] (0, ∞] (−∞, 0] (1, ∞] (1, ∞] (1, ∞] (0, 1] (1, ∞] (1, ∞] (−∞, 0] (−∞, 0] (1, ∞] (1, ∞] (−∞, 0] (3, ∞] (3, ∞] (2, 3] (3, ∞] (1, 2] (−∞, 0] (3, ∞] (1, 2] (5, ∞] (4, 5] (3, 4] (1, 2] (3, 4] (−∞, 0] (−∞, 0] (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=', ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='] (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=', ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='] (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=', ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='] (−∞, 0] (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=', ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='] (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=', ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='] (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=', ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='] 0 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' x6 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' 2x5 + x6 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' 2x5 + x6 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' 2x4 + Table 3: LPBs for Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='14 We apply the algorithm to create all successors of φ and calculate LPBs for all recursive subproblems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' The corre- sponding LPBs can be found in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' By applying the strategy of choosing the coefficient as small as possible we choose a6 = 1, a5 = 2, a4 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' We use the minimum and maximum degrees in the fourth column to choose the coefficient a3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' We have to choose a3 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' max {5 − 5, 3 − 2 , 3 − 0, −∞ − 0} < a3 < min {∞ − 4, 4 − 1, 4 − −∞, 0 − −∞} i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' 3 < a3 < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' This is, of course, not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' But φ can be represented by the LPB 9x1 + 7x2 + 6x3 + 4x4 + 4x5 + x6 ≥ 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' The algorithm found solutions for all subproblems in the fourth column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' But we cannot combine the coeffi- cients chosen so far to a solution representing all LPBs in the third column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Alternatively, we were allowed to choose a5 = a4 = 4, and if we do so, we obtain an appropriate LPB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' There- fore the applicability of Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='12 depends on the choice of the previous coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Another problem seems to be that ak+1 could be forced to be between neighbouring integers, in which case it cannot be an integer itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' However, in this case, one can multiply all LPBs of the current system by 2 (this obviously preserves the meaning of the LPBs) be- fore proceeding so that ak+1 can be chosen to be an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' From the construction of the successors (see Table 1) it follows that all formulae in a column together have size less than all formulae in the column to the left of it, so that the entire table has size less than |φ|·(m+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' One can thus show that the complexity of the algorithm is polynomial in the size of φ, while the size of φ itself can be exponential in m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' In fact, this is the most interesting case, because in this case an LPB representation may yield an exponential saving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' 13 14 15 16 17 18 19 20 21 22 23 24 25 0 20 40 60 80 variables % conversion fails Figure 1: Failure rate of the combinatorial algorithm 5 Implementation Both algorithms have been implemented in Java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' They share the same core classes representing the main com- ponents such as DNFs and LPBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' The linear program is solved by lp_solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Both implementations can be accessed and tested via a graphical user interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' For testing the implementation we generated a full enumeration of LPBs up to seven variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' For LPBs with more variables we tested 180,000 randomly gen- erated LPBs (with 8 to 25 variables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' We transformed the LPBs to DNFs (so we know that for these DNFs there exists an LPB) to test the implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' As expected the linear programming algorithm solved all tested input DNFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' The combinatorial algorithm was able to solve all in- put DNFs with up to five variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' But it fails on some DNFs with six variables (with the strategy to choose ak+1 as small as possible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Our empirical analysis shows that the more variables a DNF contains the more often the conversion fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Circa 8% of the tested DNFs with seven variables cannot be converted, for DNFs with 25 variables circa 86% cannot be converted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' The failure rate for 13 to 25 variables is illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' The linear programming algorithm was faster in di- rect runtime comparison, but we’re still working on im- provements for the combinatorial algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' As discussed in Subsec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='3, for the combinatorial al- gorithm the number of final nodes is an important cri- terion for its theoretical runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Figure 2 gives a first impression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' It is hard to judge whether the growth ex- hibited is exponential, but in any case, the number of final nodes is much smaller than 2m: around 50000 times smaller for m = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' 6 Conclusion and future work Linear pseudo-Boolean constraints have attracted inter- est because they can often be used to represent Boolean functions more compactly than CNFs or DNFs, and because techniques applied in CNF-based propositional satisfiability solving can be generalised to LPBs (Dixon and Ginsberg 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Fr¨anzle and Herde 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Some Boolean functions can be represented by a sin- gle LPB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' The problem of finding this LPB representa- tion is called threshold recognition problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' In this work, we have implemented two algorithms for this problem, a classical one based on linear programming, and a more recent one that we have previously presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' The most important insight was that our algorithm is, unfortu- nately, incomplete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' The most important topic for future work is, of course, trying to reestablish completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' The obvious way to achieve this is to incorporate some kind of backtracking into the algorithm: If a DNF can be represented by an LPB and we cannot choose ak+1, then this is because we must have chosen one of the coefficients ak+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' , am too small, because our strategy so far was to chose the coefficients as small as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' In order to find a solution we increment the coefficients ak+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' , am and re-evaluate the LPBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' We can use the minimum and maximum degrees to ensure that we enumerate only legal candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' We iteratively increment the coefficients until we can choose the coef- ficient ak+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' One problem of this approach is that for a DNF that cannot be represented by an LPB, termination is not guaranteed, because frequently the choice of the next coefficient is not bounded from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' However, we are confident that this problem can be resolved because it should be possible to derive some upper bound for each variable in the sense that it is never necessary to choose a coefficient bigger than this bound (something along the lines: it is never necessary to choose a coefficient more than m times bigger than the previous coefficient).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' The other problem is of course that backtracking worsens that runtime of the algorithm, and we very much fear that it will destroy the polynomial runtime of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' The backtracking approach has been implemented and was able to find a solution for each tested input DNF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' But the implementation has also shown that the higher the dimension m, the more often we have to use backtracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Alternatively, or more likely, additionally, one might use the occurrence patterns for estimating the weight ratio: In the example above we were able to represent all LPBs in the fourth column but we were not able to choose a3 such that we can represent all LPBs in the third column with the configuration a6 = 1, a5 = a4 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' We need some global information that the distance between a6 and a5, a4 will be too small in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Maybe it is possible to use the occurrence patterns to formulate such constraints, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=', one might find a con- straint of the form “in an LPB representing φ one has to ensure that ai ≥ w · aj”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' It has to be said however that there have been previ- ous attempts to somehow directly translate the occur- rence patterns into numeric coefficients or better, co- efficient ratios;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' the threshold recognition problem has stubbornly resisted such attempts2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' However, even a rough estimate of the coefficient ra- tion, based on the occurrence patterns, might be useful for reducing if not eliminating the backtracking effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' One other interesting topic is a more thorough anal- ysis of the complexity of the combinatorial algorithm, whether it is in its current state or after having achieved completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' In particular, as we have mentioned in Subsec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content='3, analysing the effect of exploiting the sym- metries in the input DNF would be interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Acknowledgements We thank Yves Crama and Utz-Uwe Haus for very fruitful discussions about this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' References [Crama and Hammer 2011] Crama, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=', and Hammer, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Boolean Functions - Theory, Algorithms, and Applications, volume 142 of Encyclopedia of math- ematics and its applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' Cambridge University Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E2T4oBgHgl3EQfHQbG/content/2301.03667v1.pdf'} +page_content=' [Dixon and Ginsberg 2000] Dixon, H.' 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Steinhardt +UC Berkeley +jsteinhardt@berkeley.edu +ABSTRACT +Neural networks often exhibit emergent behavior, where qualitatively new capa- +bilities arise from scaling up the amount of parameters, training data, or training +steps. One approach to understanding emergence is to find continuous progress +measures that underlie the seemingly discontinuous qualitative changes. We ar- +gue that progress measures can be found via mechanistic interpretability: reverse- +engineering learned behaviors into their individual components. As a case study, +we investigate the recently-discovered phenomenon of “grokking” exhibited by +small transformers trained on modular addition tasks. We fully reverse engineer +the algorithm learned by these networks, which uses discrete Fourier transforms +and trigonometric identities to convert addition to rotation about a circle. We +confirm the algorithm by analyzing the activations and weights and by perform- +ing ablations in Fourier space. Based on this understanding, we define progress +measures that allow us to study the dynamics of training and split training into +three continuous phases: memorization, circuit formation, and cleanup. Our re- +sults show that grokking, rather than being a sudden shift, arises from the gradual +amplification of structured mechanisms encoded in the weights, followed by the +later removal of memorizing components. +1 +INTRODUCTION +Neural networks often exhibit emergent behavior, in which qualitatively new capabilities arise from +scaling up the model size, training data, or number of training steps (Steinhardt, 2022; Wei et al., +2022a). This has led to a number of breakthroughs, via capabilities such as in-context learning (Rad- +ford et al., 2019; Brown et al., 2020) and chain-of-thought prompting (Wei et al., 2022b). However, +it also poses risks: Pan et al. (2022) show that scaling up the parameter count of models by as little +as 30% can lead to emergent reward hacking. +Emergence is most surprising when it is abrupt, as in the case of reward hacking, chain-of-thought +reasoning, or other phase transitions (Ganguli et al., 2022; Wei et al., 2022a). We could better +understand and predict these phase transitions by finding hidden progress measures (Barak et al., +2022): metrics that precede and are causally linked to the phase transition, and which vary more +smoothly. For example, Wei et al. (2022a) show that while large language models show abrupt +jumps in their performance on many benchmarks, their cross-entropy loss decreases smoothly with +model scale. However, cross-entropy does not explain why the phase changes happen. +In this work, we introduce a different approach to uncovering hidden progress measures: via mech- +anistic explanations.1 A mechanistic explanation aims to reverse engineer the mechanisms of the +network, generally by identifying the circuits (Cammarata et al., 2020; Elhage et al., 2021) within +a model that implement a behavior. Using such explanations, we study grokking, where models +1Interactive versions of figures, as well as the code to reproduce our results, are available at bit.ly/ +progress-measures-grokking-website. +1 +arXiv:2301.05217v1 [cs.LG] 12 Jan 2023 + +arXiv preprint +Figure 1: The algorithm implemented by the one-layer transformer for modular addition. Given two +numbers a and b, the model projects each point onto a corresponding rotation using its embedding +matrix. Using its attention and MLP layers, it then composes the rotations to get a representation of +a + b mod P. Finally, it “reads off” the logits for each c ∈ {0, 1, ..., P − 1}, by rotating by −c to +get cos(w(a + b − c)), which is maximized when a + b ≡ c mod P (since w is a multiple of 2π +P ). +abruptly transition to a generalizing solution after a large number of training steps, despite initially +overfitting (Power et al., 2022). Specifically, we study modular addition, where a model takes inputs +a, b ∈ {0, . . . , P −1} for some prime P and predicts their sum c mod P. Small transformers trained +with weight decay on this task consistently exhibit grokking (Figure 2, Appendix C.2). +We reverse engineer the weights of these transformers and find that they perform this task by map- +ping the inputs onto a circle and performing addition on the circle. Specifically, we show that the +embedding matrix maps the inputs a, b to sines and cosines at a sparse set of key frequencies wk. +The attention and MLP layers then combine these using trigonometric identities to compute the sine +and cosine of wk(a + b), and the output matrices shift and combine these frequencies. +We confirm this understanding with four lines of evidence (Section 4): (1) the network weights +and activations exhibit a consistent periodic structure; (2) the neuron-logit map WL is well approx- +imated by a sum of sinusoidal functions of the key frequencies, and projecting the MLP activations +onto these sinusoidal functions lets us “read off” trigonometric identities from the neurons; (3) the +attention heads and MLP neuron are well approximated by degree-2 polynomials of trigonometric +functions of a single frequency; and (4) ablating key frequencies used by the model reduces perfor- +mance to chance, while ablating the other 95% of frequencies slightly improves performance. +Using our understanding of the learned algorithm, we construct two progress measures for the mod- +ular addition task—restricted loss, where we ablate every non-key frequency, and excluded loss, +where we instead ablate all key frequencies. Both metrics improve continuously prior to when +grokking occurs. We use these metrics to understand the training dynamics underlying grokking +and find that training can be split into three phases: memorization of the training data; circuit for- +mation, where the network learns a mechanism that generalizes; and cleanup, where weight decay +removes the memorization components. Surprisingly, the sudden transition to perfect test accuracy +in grokking occurs during cleanup, after the generalizing mechanism is learned. These results show +that grokking, rather than being a sudden shift, arises from the gradual amplification of structured +mechanisms encoded in the weights, followed by the later removal of memorizing components. +2 +RELATED WORK +Phase Changes. Recent papers have observed that neural networks quickly develop novel quali- +tative behaviors as they are scaled up or trained longer (Ganguli et al., 2022; Wei et al., 2022a). +McGrath et al. (2021) find that AlphaZero quickly learns many human chess concepts between 10k +and 30k training steps and reinvents human opening theory between 25k and 60k training steps. +Grokking. Grokking was first reported in Power et al. (2022), which trained two-layer transformers +on several algorithmic tasks and found that test accuracy often increased sharply long after achieving +perfect train accuracy. Millidge (2022) suggests that this may be due to SGD being a random walk +on the optimal manifold. Our results echo Barak et al. (2022) in showing that the network instead +makes continuous progress toward the generalizing algorithm. Liu et al. (2022) construct small +2 + +Logits +cos w(a+ b- c) +Computes logits using further trig identities: +Unembed +Logit(c) α cos(w(a + b - c) += cos(w(a + b)) cos(wc) + sin(w(a + b)) sin(wc) +MLP m +Calculates sine and cosine of a + b using trig identities: +sin(w(a + b)) = sin(wa) cos(wb) + cos(wa) sin(wb) +cos(w(a + b)) = cos(wa) cos(wb) - sin(wa) sin(wb) +no +h2 +h3 +Translates one-hot a, b to Fourier basis: +Embed +a → sin(wa), cos(wa) +b → sin(wb), cos(wb) +TokensarXiv preprint +Figure 2: The train and test accuracy (left) and train and test loss (right) of one-layer transformers on +the modular addition task described in Section 3, over 5 random seeds. These models consistently +exhibit grokking: they quickly overfit early on in training, but then later learn to generalize. +examples of grokking, which they use to compute phase diagrams with four separate “phases” of +learning. Thilak et al. (2022) argue that grokking can arise without explicit regularization, from an +optimization anomaly they dub the slingshot mechanism, which may act as an implicit regularizer. +Circuits-style mechanistic interpretability. The style of post-hoc mechanistic interpretability in +Section 4 is heavily inspired by the Circuits approach of Cammarata et al. (2020), Elhage et al. +(2021), and Olsson et al. (2022). +Progress measures. Barak et al. (2022) introduce the notion of progress measures—metrics that +improve smoothly and that precede emergent behavior. They prove theoretically that training would +amplify a certain mechanism and heuristically define a progress measure. In contrast, we use mech- +anistic intepretability to discover progress measures empirically. +3 +SETUP AND BACKGROUND +We train transformers to perform addition mod P. The input to the model is of the form “a b =”, +where a and b are encoded as P-dimensional one-hot vectors, and = is a special token above which +we read the output c. In our mainline experiment, we take P = 113 and use a one-layer ReLU +transformer, token embeddings with d = 128, learned positional embeddings, 4 attention heads of +dimension d/4 = 32, and n = 512 hidden units in the MLP. In other experiments, we vary the depth +and dimension of the model. We did not use LayerNorm or tie our embed/unembed matrices. +Our mainline dataset consists of 30% of the entire set of possible inputs (that is, 30% of the 113 · +113 pairs of numbers mod P). We use full batch gradient descent using the AdamW optimizer +(Loshchilov & Hutter, 2017) with learning rate γ = 0.001 and weight decay parameter λ = 1. We +perform 40, 000 epochs of training. As there are only 113 · 113 possible pairs, we evaluate test loss +and accuracy on all pairs of inputs not used for training. +Networks trained on this task consistently exhibit grokking. As Figure 2 shows, our networks first +overfit the training set: train accuracy quickly converges to 100% and the train loss quickly declines, +while the test accuracy remains low and the test loss remains high. After around 10, 000 epochs, +the network generalizes and test accuracy increases to near 100%. In robustness experiments, we +confirm that grokking consistently occurs for other architectures and prime moduli (Appendix C.2). +In Section 5.3 we find that grokking does not occur without regularization. +To describe transformer components, we follow the conventions and notations laid out in Elhage +et al. (2021). We focus on the d×p embedding matrix WE, the d×n output matrix of the MLP layer +Wout, and the P × d unembedding matrix WU.2 Let Logits(a, b) denote the logit vector on inputs +a, b, and MLP(a, b) denote the MLP activations. Empirically, our networks do not significantly use +the skip connection around the MLP (Appendix A.1), so Logits(a, b) ≈ WUWoutMLP(a, b). We +therefore also study the P × n neuron-logit map WL = WUWout. +3.1 +THE FOURIER MULTIPLICATION ALGORITHM +We claim that the learned networks use the following algorithm (Figure 1): +2We ignore the embedding and unembedding of the ‘=’ token for simplicity. +3 + +0.8 +Accuracy +0.6 +0.4 +0.2 +-Average Train Accuracy +- Average Test Accuracy +0 +0 +5k +10k +15k +20k +EpochAverage Train Loss +Average Test Loss +Log Loss +0.01 +100μ +1μ +0 +5k +10k +15k +20k +EpocharXiv preprint +• Given two one-hot encoded tokens a, b map these to sin(wka), cos(wka), sin(wkb), and +cos(wkb) using the embedding matrix, for various frequencies wk = 2kπ +P , k ∈ N. +• Compute cos (wk(a + b)) and sin (wk(a + b)) using the trigonometric identities: +cos (wk(a + b)) = cos (wka) cos (wka) − sin (wka) sin (wkb) +sin (wk(a + b)) = sin(wka) cos (wkb) + cos (wka) sin (wkb) +In our networks, this is computed in the attention and MLP layers. +• For each output logit c, compute cos (wk(a + b − c)) using the trigonometric identity: +cos (wk(a + b − c)) = cos (wk(a + b)) cos (wkc) + sin (wk(a + b)) sin (wkc) . +(1) +This is a linear function of the already-computed values cos(wk(a + b)), sin(wk(a + b)) +and is implemented in the product of the output and unembedding matrices WL. +• The unembedding matrix also adds together cos (wk(a + b − c)) for the various ks. This +causes the cosine waves to constructively interfere at c∗ = a + b mod p (giving c∗ a large +logit), and destructively interfere everywhere else (thus giving small logits to other cs). +We refer to this algorithm as Fourier multiplication, and will justify our claim in detail in Section 4. +4 +REVERSE ENGINEERING A ONE-LAYER TRANSFORMER +In this section, we describe four lines of evidence that our transformers are using the Fourier mul- +tiplication algorithm described in Section 3.1. Here we apply our analysis to the mainline model +from Section 3; the results are broadly consistent for other models, including across different num- +ber of layers, different fractions of the training data, and different prime moduli (see Appendix C.2, +especially Table 5). +Our first line of evidence involves examining the network weights and activations and observing +consistent periodic structure that is unlikely to occur by chance (Section 4.1). Moreover, when we +take Fourier transforms, many components are either sparse or nearly sparse in the Fourier domain, +supported on a handful of key frequencies. +We next look into the actual mechanisms implemented in the model weights (Section 4.2). We show +that the unembedding matrix WL is (approximately) rank 10, where each direction corresponds to +the cosine or sine of one of 5 key frequencies. Projecting the MLP activations onto the components +of WL approximately produces multiples of the functions cos (wk(a + b)) and sin (wk(a + b)), +showing that the MLP layer does compute these sums. +To better understand the mechanism, we zoom in to individual neurons (Section 4.3). We find that +the attention heads and most neurons are well-approximated by degree-2 polynomials of sines and +cosines at a single frequency. Moreover, the corresponding direction in WL also contains only that +frequency. This suggests that the model’s computations are (1) localized across frequencies and (2) +mostly aligned with the neuron basis. +Finally, we use ablations to confirm that our interpretation is faithful (Section 4.4). We replace +various components of the model by the components of the Fourier multiplication algorithm and +find that doing so consistently does not harm and sometimes even improves model performance. +4.1 +SUGGESTIVE EVIDENCE: SURPRISING PERIODICITY +The first line of evidence that the network is using the algorithm described in Section 3.1 is the +surprising periodicity in the activations of the transformer. That is, the output of every part of the +network is periodic as a function of the input tokens. +Periodicity in the embeddings. We start by examining the embeddings. We apply a Fourier trans- +form along the input dimension of the embedding matrix WE then compute the ℓ2-norm along the +other dimension; results are shown in Figure 3. We plot only the components for the first 56 frequen- +cies, as the norm of the components for frequencies k and P − k are symmetric. The embedding +matrix WE is sparse in the Fourier basis–it only has significant nonnegligible norm at 6 frequencies. +Of these frequencies, only 5 appear to be used significantly in later parts of the model (corresponding +to k ∈ {14, 35, 41, 42, 52}). We dub these the key frequencies of the model. +4 + +arXiv preprint +0 +10 +20 +30 +40 +50 +0 +0.5 +1 +1.5 +2 +sin +cos +Fourier Components of Embedding Matrix +Frequency k +Norm of Fourier Component +0 +10 +20 +30 +40 +50 +0 +0.5 +1 +1.5 +2 +sin +cos +Fourier Components of Neuron-Logit Map +Frequency k +Norm of Fourier Component +Figure 3: (Left) The norms of the Fourier components in the embedding matrix WE. As discussed in +Section 4.1, the sparsity of WE in the Fourier basis is evidence that the network is operating in this +basis. Of the six non-zero frequencies, five “key frequencies” appear in later parts of the network, +corresponding to k ∈ {14, 35, 41, 42, 52}. (Right) Norm of Fourier components of the neuron-logit +map WL. A Fourier transform is taken over the logit axis, and then the norm is taken over the neuron +axis. As discussed in Section 4.2, WL is well-approximated by the 5 key frequencies wk. +0 +50 +100 +100 +80 +60 +40 +20 +0 +0 +0.2 +0.4 +0.6 +0.8 +1 +Attention Score for Head 0 +a +b +0 +50 +100 +100 +80 +60 +40 +20 +0 +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +Activations for Neuron 0 +a +b +Const +cos 5 +sin 9 +cos 14 +sin 18 +cos 23 +sin 27 +cos 32 +sin 36 +cos 41 +sin 45 +cos 50 +sin 54 +sin 54 +cos 50 +sin 45 +cos 41 +sin 36 +cos 32 +sin 27 +cos 23 +sin 18 +cos 14 +sin 9 +cos 5 +Const +−60 +−40 +−20 +0 +20 +40 +60 +Norms of Logits in 2D Fourier Basis +x Component +y Component +Figure 4: (Left) The attention score for head 0 from the token ‘=’ to ‘a’, as a function of inputs +a, b. (Center) The activations of MLP neuron 0 given inputs a, b. Both the attention scores and the +neuron activations are periodic (Section 4.1). (Right) The norm of the Fourier components of the +logits (2D Fourier transform is taken over the inputs a, b, and then norm is taken over the logit axis). +There are 20 significant components corresponding to the 5 key frequencies (Section 4.1). +Periodicity in attention heads and MLP neuron activations. +This periodic structure recurs +throughout the network. As an example, we plot the attention weight at position 0 for every combi- +nation of two inputs for head 0 in Figure 4. The attention exhibits a periodic structure with frequency +k = 35. In Figure 4, we also plot the activations of MLP neuron 0 for every combination of inputs. +The activations are periodic with frequency k = 42. We see similar patterns for other attention +heads and MLP neurons (Appendix C.1). +Periodicity in logits. Finally, the logits are also periodic. In Figure 4, we represent the logits in the +2D Fourier basis over the inputs, then take the ℓ2-norm over the output dimension. There are only +twenty components with significant norm, corresponding to the products of sines and cosines for the +five key frequencies wk. These show up as five 2 × 2 blocks in Figure 4. +4.2 +MECHANISTIC EVIDENCE: COMPOSING MODEL WEIGHTS +We now demonstrate that the model implements the trigonometric identity (1) as follows: the func- +tions cos (wk(a + b)), sin (wk(a + b)) are linearly represented in the MLP activations, and the un- +embed matrix reads these linear directions and multiplies them by cos (wkc), sin (wkc) respectively. +We will do this in two steps. First, we show that WL (the matrix mapping MLP activations to logits) +is (approximately) rank 10 and can be well approximated as: +WL = +� +k∈{14,35,41,42,52} cos (wk) uT +k + sin (wk) vT +k +(2) +for some uk, vk ∈ R512, where cos (wk) , sin (wk) ∈ R113 are vectors whose cth entry is cos (wkc) +and sin (wkc). Second, note that our model implements the logits for a, b as: +Logits(a, b) = WLMLP(a, b) ≈ +� +k cos (wk) uT +k MLP(a, b) + sin (wk) vT +k MLP(a, b) +(3) +5 + +arXiv preprint +WL Component +Fourier components of uT +k MLP(a, b) or vT +k MLP(a, b) +FVE +cos (w14c) +44.6 cos(w14a) cos(w14b) − 43.6 sin(w14a) sin(w14b) ≈ 44.1 cos (w14(a + b)) +93.2% +sin (w14c) +44.1 sin(w14a) cos(w14b) + 44.1 cos(w14a) sin(w14b) ≈ 44.1 sin (w14(a + b)) +93.5% +cos (w35c) +40.7 cos(w35a) cos(w35b) − 43.6 sin(w35a) sin(w35b) ≈ 42.2 cos (w35(a + b)) +96.8% +sin (w35c) +41.8 sin(w35a) cos(w35b) + 41.8 cos(w35a) sin(w35b) ≈ 41.8 sin (w35(a + b)) +96.5% +cos (w41c) +44.8 cos(w41a) cos(w41b) − 44.8 sin(w41a) sin(w41b) ≈ 44.8 cos (w41(a + b)) +97.0% +sin (w41c) +44.5 sin(w41a) cos(w41b) + 44.5 cos(w41a) sin(w41b) ≈ 44.5 sin (w41(a + b)) +97.0% +cos (w42c) +64.6 cos(w42a) cos(w42b) − 68.5 sin(w42a) sin(w42b) ≈ 66.6 cos (w42(a + b)) +96.4% +sin (w42c) +67.8 sin(w42a) cos(w42b) + 67.8 cos(w42a) sin(w42b) ≈ 67.8 sin (w42(a + b)) +96.4% +cos (w52c) +60.5 cos(w52a) cos(w52b) − 65.5 sin(w52a) sin(w52b) ≈ 63.0 cos (w52(a + b)) +97.4% +sin (w52c) +64.5 sin(w52a) cos(w52b) + 64.5 cos(w52a) sin(w52b) ≈ 64.5 sin (w52(a + b)) +98.2% +Table 1: For each of the directions uk or vk (corresponding to the cos(wk) and sin(wk) components +respectively) in the unembedding matrix, we take the dot product of the MLP activations with that +direction, then perform a Fourier transform (middle column; only two largest coefficients shown). +We then compute the fraction of variance explained (FVE) if we replace the projection with a single +term proportional to cos (wk(a + b)) or sin (wk(a + b)), and find that it is consistently close to 1. +We check empirically that the terms uT +k MLP(a, b) and vT +k MLP(a, b) are approximate multiples of +cos (wk(a + b)) and sin (wk(a + b)) (> 90% of variance explained). Thus the network computes +trigonometric functions in the MLP and reads them off as claimed. As a sanity check, we confirm +that the logits are indeed well-approximated by terms of the form cos (wk(a + b − c)) (95% of +variance explained). +WL is well approximated by cos (wkc) and sin (wkc). We perform a discrete Fourier transform +(DFT) on the logit axis of WL and look at the 10 directions uk, vk corresponding to sin (wk) and +cos (wk). When we approximate WL with � +k∈{14,35,41,42,52} cos (wk) uT +k +sin (wk) vT +k , the resid- +ual has Frobenius norm that is under 0.55% of the norm of WL. This shows that WL is well ap- +proximated by the 10 directions corresponding to cos (wk) and sin (wk) for each of the five key +frequencies. We also plot the norms of each direction in Figure 3, and find that no Fourier compo- +nent outside the 5 key frequencies has significant norm. +The unembedding matrix “reads off” terms of the form cos (wk(a + b)) and sin (wk(a + b)) +from the MLP neurons. Next, we take the dot product of the MLP activations with each of +the directions uk, vk for k ∈ {14, 35, 41, 42, 52}. Table 1 displays the results: the dot products +uT +k MLP(a, b) and vT +k MLP(a, b) are well approximated by a multiple of terms of the form +cos (wk(a + b)) = cos (wka) cos (wkb) − sin (wka) sin (wkb) , and +sin (wk(a + b)) = sin (wka) cos (wkb) + cos (wka) sin (wkb) . +That is, for each key frequency k, uk and vk are linear directions in the space of MLP neuron +activations that represent cos (wk(a + b)) and sin (wk(a + b)). +Taken together, these results confirm that the model computes sums of terms of the form +cos (wk(a + b − c)) = cos (wk(a + b)) cos (wkc) + sin (wk(a + b)) sin (wkc). +Logits are well approximated by a weighted sum of cos (wk(a + b − c))s. We approximate the +output logits as the sum � +k αk cos(wk(a+b−c)) for k ∈ {14, 35, 41, 42, 52} and fit the coefficients +αk via ordinary least squares. This approximation explains 95% of the variance in the original +logits. This is surprising—the output logits are a 113 · 113 · 113 dimensional vector, but are well- +approximated with just the 5 directions predicted by our interpretation. If we evaluate test loss using +this logit approximation, we actually see an improvement in loss, from 2.4 · 10−7 to 4.7 · 10−8. +4.3 +ZOOMING IN: APPROXIMATING NEURONS WITH SINES AND COSINES +In the previous section, we showed how the model computes its final logits by using WL to “read off” +trigonometric identities represented in the MLP neurons. In this section, we examine the attention +heads and MLP neurons to understand how the identities come to be represented at the MLP layer. +We show first that two of the attention heads approximately compute degree-2 polynomials of sines +and cosines of a particular frequency (and the other two are used to increase the magnitude of the +input embeddings in the residual stream), most neurons are also well-approximated by degree-2 +polynomials, and the map from neurons to logits is localized by frequency. +Attention heads approximately compute degree-2 polynomials of a single frequency or are +used to amplify WE. In order to compute terms like cos (wk(a + b)), the model needs to compute +6 + +arXiv preprint +0.4 +0.6 +0.8 +1 +0 +100 +200 +300 +400 +FVE by degree-2 polynomials +Fraction of variance explained +Number of neurons +Const +sin 3 +sin 6 +sin 9 +sin 12 +sin 15 +sin 18 +sin 21 +sin 24 +sin 27 +sin 30 +sin 33 +sin 36 +sin 39 +sin 42 +sin 45 +sin 48 +sin 51 +sin 54 +40 +30 +20 +10 +0 +−0.4 +−0.2 +0 +0.2 +0.4 +Components of W L corresponding to freq 14 neurons +Neuron +Figure 5: (Left) Most neurons are well-approximated by degree-2 polynomials of a single frequency. +(Right) A heatmap showing weights in WL corresponding to each of the 44 neurons of frequency +14. The non-trivial components correspond to sin (wk) and cos (wk) for k = 14. +the product of the sine and cosine embeddings output by WE. As the attention heads are approx- +imately bilinear (product of attention weights and OV circuit), they are a natural place to perform +this computation. Indeed, for each head, the attention scores’ Fourier transform is concentrated on +a single frequency wk. For two of the four heads, the corresponding OV circuit is concentrated on +that same frequency. Moreover, the softmax mapping the attention scores to attention weights is in +a regime where it behaves approximately linearly (and replacing it with a linear function actually +improves performance). Thus the attention weights multiply with the OV output to create degree-2 +polynomials of the frequency wk, as would be needed for the cosine/sine addition formulas. +For the remaining two heads, their attention scores approximately sum to one and the OV circuits +contain all five key frequencies, suggesting that they are used to increase the magnitude of key +frequencies in the residual stream. We confirm all of these claims in Appendix C.1.2. +Most MLP neurons approximately compute a degree-2 polynomial of a single frequency. We +next try to approximate the activations of each MLP neuron by a degree-2 polynomial of one of the +5 key frequencies. As shown in Figure 5, out of 512 total neurons, 433 (84.6%) have over 85% of +their variance explained with a single frequency. +Maps to the logits are localized by frequency. We partition these 433 neurons by the frequencies +with the highest variance explained. For each resulting subset, the map WL from neurons to logits +has only two non-trivial components, corresponding to sine and cosine at that frequency. For exam- +ple, in Figure 5 we plot the 44 columns of WL corresponding to the 44 neurons in the k = 14 cluster +and find that the only non-negligible components are sin +� 2kπ +P +� +and cos +� 2kπ +P +� +for k = 14. +4.4 +CORRECTNESS CHECKS: ABLATIONS +In previous sections, we showed that various components of the model were well-approximated by +sparse combinations of sines and cosines. We verify that these approximations are faithful to the +model’s functionality, by replacing each component with its approximation. This generally does not +hurt the performance of the model and in some cases improves it. +MLP neurons. In Section 4.3, we identified 433 neurons that were well-approximated by a degree-2 +polynomial. We replace each of these neurons’ activation value by the corresponding polynomial, +leaving the other neurons untouched. This increases loss by only 3% in relative terms (from 2.41 · +10−7 to 2.48 · 10−7) and has no effect on accuracy. +We can instead apply a stricter ablation to the MLP layer and restrict each neuron’s activation to +just the components of the polynomial corresponding to terms of the form cos(wk(a + b)) and +sin(wk(a + b)) in the key frequencies. This improves loss by 77% (to 5.54 · 10−8), validating that +the logits are calculated by trig identities of neurons as detailed in Section 4.2. +Logit frequencies. Next, we ablate various components of the final logits in the Fourier space. To +do so, we take a 2D DFT on the 113 · 113 · 113 logit matrix over all 113 · 113 pairs of inputs to get +the logits in the Fourier basis, then set various frequencies in this basis to 0. +We begin by ablating the components corresponding to each of the key frequencies. As reported in +Figure 6, ablating any key frequency causes a significant increase in loss. This confirms that the five +frequencies identified in previous sections are indeed necessary components of the transformer. In +contrast, ablating other frequencies does not hurt the model at all. +7 + +arXiv preprint +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +55 +10n +1μ +100μ +0.01 +1 +Restricted +Original +Other frequency +Key frequency +Loss after ablating frequencies +Log Loss +Figure 6: The loss of the transformer (lower=better) when ablating each frequency k ∈ {1, 2, ..., 56} +and everything except for the five key frequencies (restricted loss). We include the original unablated +loss for reference. Ablating key frequencies causes a performance drop, while the other ablations +do not harm performance. +We then ablate all 113 · 113 − 40 of the Fourier components besides key frequencies; this ablation +actually improves performance (loss drops 70% to 7.24 · 10−8). +Directions in WL. In Section 4.2, we found that WL is well approximated by the 10 directions +corresponding to the cosine and sine of key frequencies. If we project the MLP activations to these +10 directions, loss decreases 50% to 1.19 · 10−7. If we instead projected the MLP activations onto +the nullspace of these 10 directions, loss increases to 5.27—worse than uniform. This suggests that +the network achieves low loss using these and only these 10 directions. +5 +UNDERSTANDING GROKKING BEHAVIOR USING PROGRESS MEASURES +We now use our mechanistic understanding of the network to define two progress measures: metrics +that can be computed during training that track the progress of the model over the course of training, +including during phase transitions. This allows us to better understand how the network reaches its +final solution. +5.1 +PROGRESS MEASURES +We translate the ablations in Section 4.4 into two progress measures: restricted and excluded loss. +Restricted loss. Since the final network uses a sparse set of frequencies wk, it makes sense to check +how well intermediate versions of the model can do using only those frequencies. To measure this, +we perform a 2D DFT on the logits to write them as a linear combination of waves in a and b, +and set all terms besides the constant term and the 20 terms corresponding to cos(wk(a + b)) and +sin(wk(a + b)) for the five key frequencies to 0. We then measure the loss of the ablated network. +Excluded loss. Instead of keeping the important frequencies wk, we next remove only those key +frequencies from the logits but keep the rest. We measure this on the training data to track how +much of the performance comes from Fourier multiplication versus memorization. The idea is that +the memorizing solution should be spread out in the Fourier domain, so that ablating a few directions +will leave it mostly unaffected, while the generalizing solution will be hurt significantly. +Beyond these, we will also measure (1) the Gini coefficient (Hurley & Rickard, 2009) of the norms +of the Fourier components of WE and WL, which measures the sparsity of WE and WL in the +Fourier basis, and (2) the ℓ2-norm of the weights during training, since weight decay should push +these down once the train loss is near zero. +5.2 +PHASES OF GROKKING: MEMORIZATION, CIRCUIT FORMATION, AND CLEANUP +Using the mainline model from Section 4, we plot the excluded loss, restricted loss, Gini coefficient +of the matrices WU and WL, and sum of squared weights in Figure 7. We find that training splits +into three phases, which we call the memorization, circuit formation, and cleanup phases. (We show +similar results for other models in Appendix C.2.) +8 + +arXiv preprint +0 +5k +10k +15k +20k +25k +30k +1μ +100μ +0.01 +1 +Train Loss +Test Loss +Excluded Loss +Excluded Loss over All Frequencies +Epoch +Loss +0 +5k +10k +15k +20k +25k +30k +10n +1μ +100μ +0.01 +1 +Train loss +Test loss +Restricted loss +Restricted Loss +Epoch +Loss +0 +5k +10k +15k +20k +25k +30k +0 +0.2 +0.4 +0.6 +0.8 +Gini Coefficients of Embed Matrix and Neuron-logit Map +Epoch +Gini coefficient +0 +5k +10k +15k +20k +25k +30k +1000 +1500 +2000 +2500 +3000 +3500 +Total Sum of Squared Weights +Epoch +Sum of Squared Weights +Figure 7: How each of the progress measures in Section 5.1 changes over the course of training. +The lines delineate the 3 phases of training: memorization, circuit formation, and cleanup (and a +final stable phase). (Top Left) Excluded loss increases during circuit formation, while train and test +loss remain flat. (Top Right) The restricted loss begins declining before test loss declines, but has an +inflection point when grokking begins to occur. (Bottom Left) The Gini coefficient of the norms of +the Fourier components of WE and WL increase sharply during cleanup. (Bottom Right) The sums +of squared weights decreases smoothly during circuit formation and more sharply during cleanup, +indicating that both phases are linked to weight decay. +Memorization (Epochs 0k–1.4k). We first observe a decline of both excluded and train loss, with +test and restricted loss both remaining high and the Gini coefficient staying relatively flat. In other +words, the model memorizes the data, and the frequencies wk used by the final model are unused. +Circuit formation (Epochs 1.4k–9.4k). In this phase, excluded loss rises, sum of squared weights +falls, restricted loss starts to fall, and test and train loss stay flat. This suggests that the model’s +behavior on the train set transitions smoothly from the memorizing solution to the Fourier multi- +plication algorithm. The fall in the sum of squared weights suggests that circuit formation likely +happens due to weight decay. Notably, the circuit is formed well before grokking occurs. +Cleanup (Epochs 9.4k–14k). In this phase, excluded loss plateaus, restricted loss continues to drop, +test loss suddenly drops, and sum of squared weights sharply drops. As the completed Fourier +multiplication circuit both solves the task well and has lower weight than the memorization circuit, +weight decay encourages the network to shed the memorized solution in favor of focusing on the +Fourier multiplication circuit. This is most cleanly shown in the sharp increase in the Gini coefficient +for the matices WE and WL, which shows that the network is becoming sparser in the Fourier basis. +5.3 +GROKKING AND WEIGHT DECAY +In the previous section, we saw that each phase of grokking corresponded to an inflection point in the +ℓ2-norm of the weights. This suggests that weight decay is an important component of grokking and +drives progress towards the generalizing solution. In Appendix D.1, we provide additional evidence +that weight decay is necessary for grokking: smaller amounts of weight decay causes the network +to take significantly longer to grok (echoing the results on toy models from Liu et al. (2022)), and +our networks do not grok on the modular arithmetic task without weight decay or some other form +of regularization. In Appendix C.2, we also find that the amount of data affects grokking: when +networks are provided with enough data, there is no longer a gap between the train and test losses +(instead, both decline sharply some number of epochs into training). Finally, in Appendix D.3 we +replicate these results on several additional algorithmic tasks. +9 + +arXiv preprint +6 +CONCLUSION AND DISCUSSION +In this work, we use mechanistic interpretability to define progress measures for small transformers +trained on a modular addition task. We find that the transformers embed the input onto rotations +in R2 and compose the rotations using trigonometric identities to compute a + b mod 113. Using +our reverse-engineered algorithm, we define two progress measures, along which the network makes +continuous progress toward the final algorithm prior to the grokking phase change. We see this work +as a proof of concept for using mechanistic interpretability to understand emergent behavior. +Larger models and realistic tasks. In this work, we studied the behavior of small transformers +on a simple algorithmic task, solved with a single circuit. On the other hand, larger models use +larger, more numerous circuits to solve significantly harder tasks (Cammarata et al., 2020; Wang +et al., 2022). The analysis reported in this work required significant amounts of manual effort, and +our progress metrics are specific to small networks on one particular algorithmic task. Methods for +automating the analysis and finding task-independent progress measures seem necessary to scale to +other, larger models. We discuss possible scenarios for more realistic applications in Appendix F. +Discovering phase change thresholds. While the progress measures we defined in Section 5.1 in- +crease relatively smoothly before the phase transition (and suffice to allow us to understand grokking +for this task) we lack a general notion of criticality that would allow us to predict when the phase +transition will happen ex ante. Future work should develop theory and practice in order to apply +progress measures to predict the timing of emergent behavior. +REPRODUCIBILITY STATEMENT +An +annotated +Colab +notebook +containing +the +code +to +replicate +our +results, +includ- +ing download instructions for model checkpoints, +is available at https://bit.ly/ +grokking-progress-measures-website. +AUTHOR CONTRIBUTIONS +Neel Nanda was the primary research contributor. He reverse engineered the weights of the mainline +model to discover the Fourier multiplication algorithm and found the lines of evidence in Section 4. +He also discovered the restricted and excluded loss progress measures and that grokking in mainline +model could be divided into three discrete phases. Finally, he found the link between grokking, +limited data, and phase transitions by exhibiting grokking in other settings with phase transitions. +Lawrence Chan was invaluable to the framing and technical writing of this work. In addition, he +created the Gini coefficient progress measure and performed the analysis in the appendices exploring +to what extent the results on the mainline model applied to the other small transformer models, +including with other random seeds, architectures, prime moduli, and regularization methods. +Tom Lieberum contributed to the early stages of this work by creating a minimal setup of grokking +with a 1L Transformer on the modular addition task with no LayerNorm and finding the surprising +periodicity within the model’s internals. +Jess Smith performed experiments exploring grokking with different random seeds, architectures, +and other hyper-parameters. +Jacob Steinhardt helped clarify and distill the results, provided significant amounts of editing and +writing feedback, and suggested the progress measure frame. +ACKNOWLEDGMENTS +In writing this paper, our thinking and exposition was greatly clarified by correspondence with +and feedback from Oliver Balfour, David Bau, Sid Black, Nick Cammarata, Stephen Casper, Bilal +Chughtai, Arthur Conmy, Xander Davies, Ben Edelman, Nelson Elhage, Ryan Greenblatt, Jacob +Hilton, Evan Hubinger, Zac Kenton, Janos Kramar, Lauro Langosco, Tao Lin, David Lindner, Eric +Michaud, Vlad Mikulik, Noa Nabeshima, Chris Olah, Michela Paganini, Michela Paganini, Alex +10 + +arXiv preprint +Ray, Rohin Shah, Buck Shlegeris, Alex Silverstein, Ben Toner, Johannes Treutlein, Nicholas Turner, +Vikrant Varma, Vikrant Varma, Kevin Wang, Martin Wattenberg, John Wentworth, and Jeff Wu. +We’d also like to thank Adam Gleave and Chengcheng Tan for providing substantial editing help, +and Noa Nabeshima and Vlad Mikulik for pair programming with Neel. +This work draws heavily on the interpretability techniques and framework developed by Elhage et al. +(2021) and Olsson et al. (2022). +We trained our models using PyTorch (Paszke et al., 2019) and performed our data analysis using +NumPy (Harris et al., 2020), Pandas (Wes McKinney, 2010), and einops (Rogozhnikov, 2022). +Our figures were made using Plotly (Plotly Technologies Inc., 2015). +Neel would like to thank Jemima Jones for providing practical and emotional support as he navigated +personal challenges while contributing to this paper, and to the Schelling Residency for providing +an excellent research environment during the distillation stage. He would also like to thank the +Anthropic interpretability team, most notably Chris Olah, for an incredibly generous amount of +mentorship during his time there, without which this investigation would never have happened. +REFERENCES +Boaz Barak, Benjamin L Edelman, Surbhi Goel, Sham Kakade, Eran Malach, and Cyril Zhang. +Hidden progress in deep learning: Sgd learns parities near the computational limit. arXiv preprint +arXiv:2207.08799, 2022. +Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, Prafulla Dhariwal, +Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, et al. Language models are +few-shot learners. Advances in neural information processing systems, 33:1877–1901, 2020. +Nick Cammarata, Shan Carter, Gabriel Goh, Chris Olah, Michael Petrov, Ludwig Schubert, Chelsea +Voss, Ben Egan, and Swee Kiat Lim. Thread: Circuits. 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Inter- +pretability in the wild: a circuit for indirect object identification in gpt-2 small. arXiv preprint +arXiv:2211.00593, 2022. +Jason Wei, Yi Tay, Rishi Bommasani, Colin Raffel, Barret Zoph, Sebastian Borgeaud, Dani Yo- +gatama, Maarten Bosma, Denny Zhou, Donald Metzler, et al. Emergent abilities of large language +models. arXiv preprint arXiv:2206.07682, 2022a. +Jason Wei, Xuezhi Wang, Dale Schuurmans, Maarten Bosma, Ed Chi, Quoc Le, and Denny +Zhou. Chain of thought prompting elicits reasoning in large language models. arXiv preprint +arXiv:2201.11903, 2022b. +Wes McKinney. Data Structures for Statistical Computing in Python. In St´efan van der Walt and +Jarrod Millman (eds.), Proceedings of the 9th Python in Science Conference, pp. 56 – 61, 2010. +doi: 10.25080/Majora-92bf1922-00a. +12 + +arXiv preprint +A +MATHEMATICAL STRUCTURE OF THE TRANSFORMER +We follow the conventions and notation of Elhage et al. (2021) in describing our model. Here, we +briefly recap their notation and examine it in our specific case. +We denote our hyperparameters as follows: dvocab = 113 is the size of the input and output spaces +(treating ‘=’ separately), dmodel = 128 is the width of the residual stream (i.e. embedding size), +dhead = 32 is the size of query, key and value vectors for a single attention head, and dmlp = 512 +is the number of neurons. +We denote the parameters as follows: WE (embedding layer); Wpos (positional embedding); W j +Q +(queries), W j +K (keys), W j +V (values), W j +O (attention output) (the 4 weight matrices of head j in +the attention layer); Win and bin for the input linear map of the MLP layer; Wout and bout for +the output linear map of the MLP layer; and WU (unembedding layer). Note that we do not have +biases in our embedding, attention layer or unembedding, and we do not tie the matrices for the +embedding/unembedding layers. +We now describe the mathematical structure of our network. Note that loss is only calculated from +the logits on the final token, and information only moves between tokens during the attention layer, +so our variables from the end of the attention layer onwards only refer to the final token. We use +ti to denote the token in position i (as a one-hot encoded vector), pi to denote the ith positional +embedding, x(0) +i +to denote the initial residual stream on token with index i, A(i) to denote the +attention scores from = to all previous tokens from head i, x(1) to denote the residual stream after +the attention layer on the final token, MLP to denote the neuron activations in the MLP layer on the +final token, x(2) the final residual stream on the final token, Logits the logits on the final token. +The logits are calculated via the following equations: +x(0) +i += WEti + pi +Aj = softmax(x(0)T W jT +K W j +Qx(0) +2 ) +x(1) = [ +� +j +W j +OW j +V (x(0) · Aj)] + x(0) +2 +MLP = ReLU(Winx(1)) +x(2) = WoutN + x(1) = WoutReLU(Winx(1)) + x(1) +Logits = WUx(2) +As in Elhage et al. (2021), we refer to the term W j +OW j +V (x(0)) as the OV circuit for head j. +A.1 +EMPIRICAL MODEL SIMPLIFICATIONS +We make two empirical observations: +• The attention paid from ‘=’ to itself is trivial. In practice, the average attention paid is +0.1% to 0.4% for each head, and ablating this does not affect model performance at all. +• The skip connection around the MLP layer is not important for the model’s computation +and can be ignored. Concretely, if we set it to zero or to its average (zero or mean ablation) +then model accuracy is unchanged, and loss goes from 2.4 · 10−7 to 9.12 · 10−7 and 7.25 · +10−7 respectively. This is a significant increase in loss, but from such a small baseline that +we can still ignore it and reverse engineer the model’s computation. (That being said, both +the attention heads and the skip connection around them are crucial to the functioning of +the model: zero ablating attention heads increases loss to 24.3, while zero ablating the skip +connection around the attention heads increases loss to 19.1, both significantly worse than +chance.) +A consequence of the first observation is that the attention is now a softmax over 2 elements, i.e. a +sigmoid over the difference. And x(0) +2 +is constant, as it is independent of x and y, and the embedding +13 + +arXiv preprint +Figure 8: As discussed in Appendix B, while for every k ∈ [0, ...P − 1], cos +� 2kπ +P x +� +achieves its +maximum value (1) at x = 0 mod 113, it still has additional peaks at different values that are close +to the maximum value. However, by adding together cosine waves of the 5 keyfrequencies, the +model constructs a periodic function where the value at x = 0 mod 113 is significantly larger than +its value anywhere else. +and positional embedding of ‘=’ are fixed. So Aj +0 = σ +� +x(0) +2 +T W jT +Q WK(x(0) +0 +− x(0) +1 ) +� +(and Aj +1 = +1 − Aj +0) +A consequence of the second observation is that Logits ≈ WUWoutMLP, which we denote as +WL = WUWout. From the perspective of the network, WL is the meaningful matrix, not either of +its constituents, since they compose linearly. +B +WHY USE CONSTRUCTIVE INTEREFERENCE? +As demonstrated in Section 4 and Appendix C.2.1, small transformers trained on this task use several +different frequencies which they add together. The reason for this is to end up with a function whose +value at x = 0 mod 113 is significantly larger than any other x. +For example, consider the function f14(x) = cos +� 2π·14 +113 x +� +. This function has period 113 and is +maximized at x = 0 mod 113. However, other values of x cause this function to be close to 1: +f14(8) = f14(105) = 0.998, f14(16) = f14(89) = 0.994, etc. +Now consider f35(x) = cos +� 2π·35 +113 x +� +. While this function also has period 113 and is maximized +at x = 0 mod 113, it turns out that f35(8) = f35(105) = −0.990. This means that by adding +together f14 and f35, we end up with a function that is not close to 1 at x = 8 mod 113. Similarly, +while f35(16) = 0.961, f52(16) = −0.56, and so adding a third frequency reduces the peak at +x = 16 mod 113. +We show the constructive interference resulting from the cosine waves for the five frequencies used +by the mainline model in Figure 8. +C +SUPPORTING EVIDENCE FOR MECHANISTIC ANALYSIS OF MODULAR +ARITHMETIC NETWORKS +C.1 +FURTHER ANALYSIS OF THE SPECIFIC TRAINING RUN DISCUSSED IN THE PAPER +In this section, we provide additional evidence relating to the mainline model. +14 + +Constructive Interference of Cosine Waves of Different Freguencies +COS14 +cos 35 +cos 41 +31 +COS42 +2 +COS52 +Sum +-1 +-2 +-3 +0 +2 +6 +8arXiv preprint +0 +50 +100 +100 +80 +60 +40 +20 +0 +0 +50 +100 +0 +50 +100 +0 +50 +100 +0.2 +0.4 +0.6 +0.8 +Attention patterns by Head ('=' to a) +a +a +a +a +b +Figure 9: Attention patterns for each head, from the ‘=’ token at the third sequence position to the +a token at the first sequence position, as a heatmap over the inputs. All four attention heads exhibit +striking periodicity. +Head +k +αj +βj +FVE +0 +35 +−0.26 +−0.14 +99.03% +1 +42 +0.27 +−0.04 +98.49% +2 +52 +0.29 +−0.05 +99.07% +3 +42 +−0.26 +0.04 +97.91% +Table 2: For each attention head, we show the pattern from ‘=’ to a is well approximated by 0.5 + +α(cos(wka)−cos(wkb))+β(sin(wka)−sin(wkb)) and give the coefficients and fraction of variance +explained for this approximation. +C.1.1 +PERIODICITY IN THE ACTIVATIONS OF OTHER ATTENTION HEADS +In Figure 9 we plot the attention patterns from the final token ‘=’ to the first token a for all 4 +attention heads, as a heatmap over the inputs a and b, as this is a scalar for each head. We observe a +striking periodicity and further that heads 1 and 3 represent the same frequency while heads 0 and 2 +are different. +As shown in Appendix A.1, the attention paid from ‘=’ to itself is negligible, so Aj +0 = 1 − Aj +1 and +it suffices to plot attention to a. +C.1.2 +THE ATTENTION PATTERN WEIGHTS ARE WELL APPROXIMATED BY DIFFERENCES OF +SINES AND COSINES OF A SINGLE FREQUENCY. +The periodicity of the attention heads has a striking form—Aj +0 is well approximated by 0.5 + +αj(cos(wka) − cos(wkb)) + βj(sin(wka) − sin(wkb)), for some frequency wk and constants αj +and βj (which may differ for each head). Note further that this simplifies to 0.5 + γ(cos(wk(a + +θ))−cos(wk(b+θ))) for some constants γ and θ. We show the coefficients and fraction of variance +explained in Table 1 +Mechanistic Analysis of Attention Patterns. +We can further mechanistically analyse how the +model achieves this form. The following is a high-level sketch of what is going on: +First, note that the attention score on position 0 and head j is just a lookup table on the input token +a (of size P). To see why, note that Aj +0 = mx(0) +0 +T W jT +K W j +Qx(0) +2 . x(0) +2 +is constant since the token +is always ‘=’ and x(0) +0 += WEt0 + p0. So this reduces to t0 · Cj + D for some constant vector +Cj = W T +E W jT +K W j +Qx(0) +2 +∈ Rp and some scalar D = pT +0 W j +K +T W j +Qx(0) +2 . As t0 is one-hot encoded, +this is just a lookup table, which we may instead denote as Cj[a] +Next, note that the attention pattern from =→ 0 is σ(Cj[a] − Cj[b]). As argued in Appendix A.1, +the attention paid =→= is negligible and can be ignored. So the softmax reduces to a softmax over +two elements, which is a sigmoid on their difference. As form of Cj does not mention the token +index or value, it is the same for position 0 and 1. +We now show that Cj is well-approximated by a wave of frequency wkj for some integer kj. That is, +Cj[a] ≈ Fj cos(wkja)+Gj sin(wkja). We do this by simply computing Cj and fitting the constants +15 + +arXiv preprint +0 +10 +20 +30 +40 +50 +0 +2 +4 +6 +8 +sin +cos +Fourier components for C 0 +Frequency k +Coefficient of Fourier Component +0 +10 +20 +30 +40 +50 +−8 +−6 +−4 +−2 +0 +sin +cos +Fourier components for C 1 +Frequency k +Coefficient of Fourier Component +0 +10 +20 +30 +40 +50 +−10 +−8 +−6 +−4 +−2 +0 +2 +sin +cos +Fourier components for C 2 +Frequency k +Coefficient of Fourier Component +0 +10 +20 +30 +40 +50 +0 +2 +4 +6 +8 +sin +cos +Fourier components for C 3 +Frequency k +Coefficient of Fourier Component +Figure 10: We plot the attention pattern weights Cj in the Fourier basis for each of the four heads +j ∈ {0, 1, 2, 3}. We observe significant sparsity, with almost all of each term being associated with +a single frequency. +Fj and Gj to minimize ℓ2 loss, and display the resulting coefficients for each head in Figure 10. This +fit explain 99.02%, 95.21%, 99.10%, 92.42% of the variance of Cj respectively. Interestingly, the +coefficients of heads 1 and 3 are almost exactly the opposite of each other. +For each head j, σ(Cj[a] − Cj[b]) ≈ 0.5 + Ej(Cj[a] − Cj[b]) for some constant Ej—that is, the +sigmoid has some linear approximation. (The intercept will be 0.5 by symmetry.) The striking +thing is that, because the inputs to the sigmoid for the attention heads are over a fairly wide range +([−5, 5] roughly), the linear approximation to the sigmoid is a fairly good fit, explaining 97.5% of +the variance. +We validate that this is all that is going on, by replacing the sigmoid with the best linear fit. This +improves performance, decreasing test loss from 2.41 · 10−7 to 2.12 · 10−7. +By properties of sinusoidal functions, the attention patterns of each head will be well approximated +by 0.5 ± Cj(cos(wkj(a + θj)) − cos(wkj(b + θj))) - the softmax is linear, with an intercept of 0.5, +and the weights Cj map each token to a score that is a wave in a single frequency. This exactly gives +us the periodic form shown in Figure 9. +Finally, for each head j, we plot the output of the OV circuit W j +OW j +V x(0) in the Fourier basis and +display the results in Figure 11). The largest component of each head corresponding to the frequency +of the attention pattern Cj, with heads 0 and 2 being almost entirely composed of a sines and cosines +of a single frequency. On the other hand, the norms for the components of heads 1 and 3 are almost +exactly the same, and contain all five key frequencies. As the coefficients of the attention pattern +weights have the opposite non-constant components (Table 2, Figure 10), their attention scores sum +almost exactly to 1 across all inputs. This implies that heads 1 and 3 are used to output the first +order terms sin (wk) , cos (wk) in the five key frequencies. We speculate that this is because of +weight decay encouraging the embeddings WE to be small, causing the network to allocate two of +its attention heads to effectively increasing the size of WE. +Bringing it all together, this implies that attention heads 0 and 2 are approximately computing a +degree 2 polynomial of cosines and sines of a single frequency each, while heads 1 and 3 amplify +the key frequencies in the residual stream. +16 + +arXiv preprint +0 +10 +20 +30 +40 +50 +0 +0.5 +1 +1.5 +2 +2.5 +3 +cos +sin +Fourier components of OV circuit for head 0 +Frequency k +Norm of Fourier Component +0 +10 +20 +30 +40 +50 +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +cos +sin +Fourier components of OV circuit for head 1 +Frequency k +Norm of Fourier Component +0 +10 +20 +30 +40 +50 +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +cos +sin +Fourier components of OV circuit for head 2 +Frequency k +Norm of Fourier Component +0 +10 +20 +30 +40 +50 +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +cos +sin +Fourier components of OV circuit for head 3 +Frequency k +Norm of Fourier Component +Figure 11: We plot the output of the OV circuit W j +OW j +V x(0) in the Fourier basis for each of the +four heads j ∈ {0, 1, 2, 3}. As with the attention pattern weights Cj in Figure 10, we observe that +the only components with significant norm are those corresponding to key frequencies, and that the +largest component corresponds to the frequencies of the attention patterns of the attention heads. +As attention pattern of heads 1 and 3 are sum to one, but their OV circuits are almost exactly the +same and consist of all five key frequencies, this implies that heads 1 and 3 are used to increase the +magnitude of key frequencies in the residual stream (Section C.1.2). +0 +50 +100 +100 +80 +60 +40 +20 +0 +0 +50 +100 +0 +50 +100 +0 +50 +100 +0 +1 +2 +3 +4 +Neuron Activations for Additional Neurons +a +a +a +a +b +Figure 12: Plots of neuron activations for MLP neurons 1, 2, 3 and 4, for inputs a, b ∈ {0, 1, ..., 112}. +As with Neuron 0, all of the activation patterns are periodic in both inputs. +17 + +海arXiv preprint +0 +10k +20k +30k +0 +0.2 +0.4 +0.6 +0.8 +1 +Train Accuracy +Test Accuracy +Restricted Accuracy (Train) +Restricted Accuracy (Test) +Pure Restricted Accuracy +Epoch +Figure 13: Accuracy when restricting Fourier Components to the five key frequencies. As with +restricted loss, this shows that the model figures out how to generalize modulo deleting noise before +it removes the noise. +0 +5k +10k +15k +20k +25k +30k +0 +5 +10 +15 +20 +25 +30 +Freq +14 +35 +41 +42 +52 +Coefficients of cos(w k (a+b-c)) in the Logits +Epoch +Figure 14: The coefficients of cos(w(a + b − c)) in the logits over the model’s training. As with the +metrics in the paper, this shows a nice interpolation and growth of each cosine term. +C.1.3 +PERIODICITY IN THE ACTIVATIONS OF ADDITIONAL NEURONS +In Figure 12, we display the activations of four more MLP neurons, as a function of the inputs. As +with neuron 0, the activations of these neurons are also periodic in the inputs. +C.1.4 +ADDITIONAL GROKKING FIGURES FOR MAINLINE RUN +In Figure 13, we display the accuracy of the model when restricting the model to use only the five +key frequencies. As with restricted loss, this improves model performance during training. +In Figure 14, we show the coefficients of the five key frequencies in the logits, calculated by regress- +ing the logits against the five cos (wk(a + b − c)) terms. +In Figure 15, we plot the excluded loss if we exclude each of the five key frequencies (as opposed to +all five key frequencies). +All three of these figures have inflection points corresponding to the relevant phases of grokking, +discussed in Section 5.1. +18 + +arXiv preprint +0 +5k +10k +15k +20k +25k +30k +0 +0.2 +0.4 +0.6 +0.8 +1 +Train accuracy +Test Accuracy +k=14 +k=35 +k=41 +k=42 +k=52 +Accuracy when Excluding Key Frequencies +Epoch +Accuracy +0 +5k +10k +15k +20k +25k +30k +1μ +100μ +0.01 +1 +Train loss +Test Loss +k=14 +k=35 +k=41 +k=42 +k=52 +Loss when Excluding Key Frequencies +Epoch +Loss +Figure 15: The excluded accuracy (left) and loss (right) if we exclude each of the five key frequencies +for our mainline model. As with the excluded loss results in Section 5.1, this shows that the model +interpolates between memorising and generalising. +C.2 +ADDITIONAL RESULTS FROM DIFFERENT RUNS +In this section, we plot relevant figures from other runs, either with the same architecture (Appendix +C.2.1) or with different architectures or experimental setups (Appendix C.2.2). Note that in general, +while all models learn to use variants of the modular arithmetic algorithm, they use a varying number +of different key frequencies. In order to find the key frequencies to calculate the excluded and +restricted loss, we perform a DFT on the neuron-logit map WL, then take the frequencies with +nontrivial coefficients.3 +C.2.1 +ADDITIONAL RESULTS FOR DIFFERENT RUNS WITH THE SAME ARCHITECTURE +In this section, we provide evidence that all 4 other runs (i.e., random seeds) using the experimental +setup of our mainline model also use the Fourier multiplication algorithm, and then confirm that the +same phases of grokking also occur on these runs. +Confirming that the other seeds use the Fourier Multiplication Algorithm. In Figure 16, we +show the norms of the Fourier components of the embedding matrix WE for each of the 4 other +random seeds. As with the mainline model, the matrices are sparse in the Fourier basis. In Figure +17, we show the norms of the Fourier components of the neuron-logit map WL for the 4 other random +seeds. The matrices are sparse in the Fourier basis, enabling us to identify 3 or 4 key frequencies for +each of the seeds. Again, note that these are different frequencies per seed. +Using the key frequencies identified in the neuron-logit map, we repeat the experiment in Sec- +tion 4.2, where we “read off” the MLP activations in the 6 or 8 directions corresponding to the +key frequencies. As with our mainline model, this lets us identify the trigonometric identities for +cos (wk(a + b)) and sin (wk(a + b)) being computed at the MLP layer. We confirm that the trigono- +metric identities are a good approximation by approximating the activations with a single term of +the form cos (wk(a + b)) or sin (wk(a + b))—as with the mainline model, the fraction of variance +explained is consistently close to 100%. +Next, we ablate the key frequencies from the logits as in Section 4.4 and report the results in Table +4. As with the mainline model, ablating all of the key frequencies reduces performance to worse +than chance, while ablating everything but the key frequencies improves test performance. +Progress measures and grokking. Finally, we confirm the progress measure and grokking results +from the mainline model on other runs with the same architecture. In Figure 18, we display the +train, test, and restricted loss for each of the four other random seeds. In Figure 19, we display +the Gini coefficients of the Fourier components of the embedding matrix WE and the neuron-logit +map WL for each of the four other random seeds. The shape of the curves are very similar to those +of the mainline model, allowing us to divide grokking on these models into the same three phases +identified in the main text. Interestingly, while all of the models complete memorization by around +1400 epochs, circuit formation and cleanup occur at different times. +3One method for getting a general (model-independent) progress measure for this task is to compute the +excluded loss for each of the 56 unique frequencies and then take the max. We omit the plots for this variant of +the excluded loss as they are broadly similar. +19 + +arXiv preprint +0 +10 +20 +30 +40 +50 +0 +0.5 +1 +1.5 +2 +2.5 +cos +sin +Embedding Matrix (Seed 1) +Frequency k +Norm of Fourier Component +0 +10 +20 +30 +40 +50 +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +cos +sin +Embedding Matrix (Seed 2) +Frequency k +Norm of Fourier Component +0 +10 +20 +30 +40 +50 +0 +0.5 +1 +1.5 +2 +2.5 +cos +sin +Embedding Matrix (Seed 3) +Frequency k +Norm of Fourier Component +0 +10 +20 +30 +40 +50 +0 +0.5 +1 +1.5 +2 +2.5 +cos +sin +Embedding Matrix (Seed 4) +Frequency k +Norm of Fourier Component +Figure 16: The norms of the Fourier components in the embedding matrix WE for each of four +other random seeds for the original (1 layer) architecture. As discussed in Section 4.1 and Appendix +C.2.1, the sparsity of WE in the Fourier basis is evidence that the network is operating in a Fourier +basis. +0 +10 +20 +30 +40 +50 +0 +0.5 +1 +1.5 +2 +2.5 +cos +sin +Neuron-Logit Map (Seed 1) +Frequency k +Norm of Fourier Component +0 +10 +20 +30 +40 +50 +0 +0.5 +1 +1.5 +2 +2.5 +3 +cos +sin +Neuron-Logit Map (Seed 2) +Frequency k +Norm of Fourier Component +0 +10 +20 +30 +40 +50 +0 +0.5 +1 +1.5 +2 +2.5 +3 +cos +sin +Neuron-Logit Map (Seed 3) +Frequency k +Norm of Fourier Component +0 +10 +20 +30 +40 +50 +0 +0.5 +1 +1.5 +2 +2.5 +cos +sin +Neuron-Logit Map (Seed 4) +Frequency k +Norm of Fourier Component +Figure 17: The norms of the direction corresponding to sine and cosine waves in the neuron-logit +map weights WL. As with the mainline model discussed in the main body and discussed in Appendix +C.2.1, WL is consistently sparse, providing is evidence that all four are operating in a Fourier basis. +20 + +arXiv preprint +WL Component +Fourier components of uT +k MLP(a, b) or vT +k MLP(a, b) +FVE +cos (w2c) +147.4 cos (w2a) cos (w2b) − 145.8 sin (w2a) sin (w2b) ≈ 146.6 cos (w2(a + b)) +99.2% +sin (w2c) +145.5 cos (w2a) sin (w2b) + 145.6 sin (w2a) cos (w2b) ≈ 145.5 sin (w2(a + b)) +99.1% +cos (w9c) +49.3 cos (w9a) cos (w9b) − 48.0 sin (w9a) sin (w9b) ≈ 48.6 cos (w9(a + b)) +96.4% +sin (w9c) +48.6 cos (w9a) sin (w9b) + 48.5 sin (w9a) cos (w9b) ≈ 48.5 sin (w9(a + b)) +96.7% +cos (w19c) +58.0 cos (w19a) cos (w19b) − 58.3 sin (w19a) sin (w19b) ≈ 58.2 cos (w19(a + b)) +95.4% +sin (w19c) +59.3 cos (w19a) sin (w19b) + 59.4 sin (w19a) cos (w19b) ≈ 59.4 sin (w19(a + b)) +93.9% +cos (w31c) +94.4 cos (w31a) cos (w31b) − 96.4 sin (w31a) sin (w31b) ≈ 95.4 cos (w31(a + b)) +98.4% +sin (w31c) +97.2 cos (w31a) sin (w31b) + 97.1 sin (w31a) cos (w31b) ≈ 97.2 sin (w31(a + b)) +98.7% +(a) Seed 1 +WL Component +Fourier components of uT +k MLP(a, b) or vT +k MLP(a, b) +FVE +cos (w40c) +97.0 cos (w40a) cos (w40b) − 99.4 sin (w40a) sin (w40b) ≈ 98.2 cos (w40(a + b)) +97.3% +sin (w40c) +81.3 cos (w40a) sin (w40b) + 81.3 sin (w40a) cos (w40b) ≈ 81.3 sin (w40(a + b)) +92.7% +cos (w44c) +309.1 cos (w44a) cos (w44b) − 338.7 sin (w44a) sin (w44b) ≈ 323.9 cos (w44(a + b)) +98.5% +sin (w44c) +327.3 cos (w44a) sin (w44b) + 327.2 sin (w44a) cos (w44b) ≈ 327.3 sin (w44(a + b)) +98.9% +cos (w53c) +192.1 cos (w53a) cos (w53b) − 192.2 sin (w53a) sin (w53b) ≈ 192.1 cos (w53(a + b)) +97.3% +sin (w53c) +166.7 cos (w53a) sin (w53b) + 166.8 sin (w53a) cos (w53b) ≈ 166.8 sin (w53(a + b)) +95.7% +(b) Seed 2 +WL Component +Fourier components of uT +k MLP(a, b) or vT +k MLP(a, b) +FVE +cos (w31c) +156.1 cos (w31a) cos (w31b) − 156.5 sin (w31a) sin (w31b) ≈ 156.3 cos (w31(a + b)) +99.3% +sin (w31c) +150.7 cos (w31a) sin (w31b) + 150.7 sin (w31a) cos (w31b) ≈ 150.7 sin (w31(a + b)) +98.9% +cos (w45c) +72.5 cos (w45a) cos (w45b) − 76.8 sin (w45a) sin (w45b) ≈ 74.6 cos (w45(a + b)) +95.9% +sin (w45c) +74.7 cos (w45a) sin (w45b) + 74.6 sin (w45a) cos (w45b) ≈ 74.6 sin (w45(a + b)) +96.6% +cos (w49c) +45.9 cos (w49a) cos (w49b) − 45.5 sin (w49a) sin (w49b) ≈ 45.7 cos (w49(a + b)) +97.0% +sin (w49c) +45.8 cos (w49a) sin (w49b) + 45.8 sin (w49a) cos (w49b) ≈ 45.8 sin (w49(a + b)) +96.9% +cos (w52c) +71.6 cos (w52a) cos (w52b) − 72.1 sin (w52a) sin (w52b) ≈ 71.9 cos (w52(a + b)) +98.5% +sin (w52c) +68.7 cos (w52a) sin (w52b) + 68.7 sin (w52a) cos (w52b) ≈ 68.7 sin (w52(a + b)) +97.9% +(c) Seed 3 +WL Component +Fourier components of uT +k MLP(a, b) or vT +k MLP(a, b) +FVE +cos (w17c) +66.0 cos (w17a) cos (w17b) − 63.5 sin (w17a) sin (w17b) ≈ 64.8 cos (w17(a + b)) +96.4% +sin (w17c) +66.4 cos (w17a) sin (w17b) + 66.4 sin (w17a) cos (w17b) ≈ 66.4 sin (w17(a + b)) +94.9% +cos (w32c) +68.7 cos (w32a) cos (w32b) − 68.4 sin (w32a) sin (w32b) ≈ 68.5 cos (w32(a + b)) +96.2% +sin (w32c) +68.0 cos (w32a) sin (w32b) + 68.0 sin (w32a) cos (w32b) ≈ 68.0 sin (w32(a + b)) +96.3% +cos (w42c) +100.4 cos (w42a) cos (w42b) − 96.0 sin (w42a) sin (w42b) ≈ 98.2 cos (w42(a + b)) +97.9% +sin (w42c) +100.2 cos (w42a) sin (w42b) + 100.1 sin (w42a) cos (w42b) ≈ 100.1 sin (w42(a + b)) +98.6% +cos (w51c) +118.0 cos (w51a) cos (w51b) − 116.2 sin (w51a) sin (w51b) ≈ 117.1 cos (w51(a + b)) +99.0% +sin (w51c) +114.3 cos (w51a) sin (w51b) + 114.2 sin (w51a) cos (w51b) ≈ 114.2 sin (w51(a + b)) +98.5% +(d) Seed 4 +Table 3: For each of the directions in the neuron-logit map WL of the final models from 4 other ran- +dom seeds (Appendix C.2.1), we project the MLP activations in that direction then perform a Fourier +transform. For brevity, we omit terms with coefficients less than 15% of the largest coefficient. We +then compute the fraction of variance explained (FVE) if we replace the projection with a multiple +of a single term of the form cos (wk(a + b)) or sin (wk(a + b)), and find that this is consistently +close to 1. +Seed +Test Loss +Loss (Key frequencies removed) +Loss (All other frequencies removed) +1 +2.07 · 10−7 +6.5 · 100 +5.7 · 10−8 +2 +2.1 · 10−7 +1.1 · 101 +6.2 · 10−8 +3 +2.05 · 10−7 +6.7 · 100 +5.5 · 10−8 +4 +2.33 · 10−7 +6.8 · 100 +6.0 · 10−8 +Table 4: As discussed in Appendix C.2.1, ablating the key frequencies for each of the networks re- +duces performance to worse than chance, while ablating all other frequencies improves performance. +21 + +arXiv preprint +0 +5k +10k +15k +20k +25k +10n +1μ +100μ +0.01 +1 +Train loss +Test loss +Restricted loss +Restricted Loss for Seed 1 +Epoch +Loss +0 +5k +10k +15k +20k +25k +1μ +100μ +0.01 +1 +Train loss +Test loss +Restricted loss +Restricted Loss for Seed 2 +Epoch +Loss +0 +5k +10k +15k +20k +25k +10n +1μ +100μ +0.01 +1 +Train loss +Test loss +Restricted loss +Restricted Loss for Seed 3 +Epoch +Loss +0 +5k +10k +15k +20k +25k +30k +1μ +100μ +0.01 +1 +Train loss +Test loss +Restricted loss +Restricted Loss for Seed 4 +Epoch +Loss +Figure 18: The train, test, and restricted loss for each of the four other random seeds described in +Appendix C.2.1. The lines delineate the 3 phases of training: memorization, circuit formation, and +cleanup (and a final stable phase). As with the mainline model, restricted loss consistently declines +prior to train loss. Note that while the shapes of the loss curves are similar to each other and those of +the mainline model, the exact time that grokking occurs (and thus the dividers between the phases +of grokking) differ by random seed. Interestingly, memorization is complete by around 1400 steps +for all five runs. +22 + +arXiv preprint +0 +5k +10k +15k +20k +25k +0 +0.2 +0.4 +0.6 +0.8 +Gini Coefficients, Seed 1 +Epoch +Gini coefficient +0 +5k +10k +15k +20k +25k +0 +0.2 +0.4 +0.6 +0.8 +Gini Coefficients, Seed 2 +Epoch +Gini coefficient +0 +5k +10k +15k +20k +25k +0 +0.2 +0.4 +0.6 +0.8 +Gini Coefficients, Seed 3 +Epoch +Gini coefficient +0 +5k +10k +15k +20k +25k +0 +0.2 +0.4 +0.6 +0.8 +Gini Coefficients, Seed 4 +Epoch +Gini coefficient +Figure 19: The Gini coefficients (a measure of sparsity) of the Fourier components of the embedding +matrix WE and the neuron-logit map WL for each of the four other random seeds. The lines delineate +the 3 phases of training: memorization, circuit formation, and cleanup (and a final stable phase). As +with the mainline model, sparsity increases slowly during memorization and circuit formation, and +then quickly during cleanup. +C.2.2 +RESULTS FOR OTHER EXPERIMENTAL SETUPS +In this section, we provide further evidence that small transformers grok on the modular addition +task, by varying the size of the network, the amount of training data, and the size of the prime P. +1-Layer Transformers with Varying Fractions of Training Data. +We find that grokking occurs +for the modular addition task with P = 113 for many data fractions (that is, the fraction of the +113 · 113 pairs of inputs that the model sees during training), as shown in Figure 20. Smaller +amount lead to slower grokking, but sufficiently large fractions of data (≥ 60%) lead to immediate +generalization, as shown in Figures 20 and 21. +As with the results in Appendix C.2.1, all of the 1-layer transformers in this section also converge +to using the Fourier multiplication algorithm. +2-Layer Transformers. +As shown in Figure 22, 2-layer transformers also exhibit some degree +of grokking. However, this is complicated by the slingshot mechanism (Thilak et al., 2022). We +display the excluded loss of a 2-layer transformer in Figure 23 and find it shows a similar pattern to +the mainline 1-layer transformer, in that it improves relatively smoothly before grokking occurs. +Smaller and larger primes. +We also examined smaller and larger prime moduli. For P = 53 +(Figure 24), we explored a variety of weight decays to observe grokking in the small prime case. +With the original weight decay setting of λ = 1, we found that the models never generalized. +However, increasing the weight decay to λ = 5 does allow the model to grok. We speculate that +this is because the memorization solution is significantly smaller (since there are only 53 · 53 total +pairs), thereby requiring more aggressive weight decay for the generalizing solution to be favored. +For P = 109, we saw exactly the same behavior as with the mainline model. +For P = 401 (Figure 25), we could not get grokking, even by varying the weight decay parame- +ter λ ∈ {0.3, 0.5, 1, 3, 5, 8}. Instead, the model immediately learns the generalizing solution. We +23 + +arXiv preprint +0 +5k +10k +15k +20k +1μ +100μ +0.01 +1 +100 +Train loss +Test loss +Data Fraction 0.1 +Epoch +Loss +0 +5k +10k +15k +20k +1μ +100μ +0.01 +1 +Train loss +Test loss +Data Fraction 0.2 +Epoch +Loss +0 +5k +10k +15k +20k +1μ +100μ +0.01 +1 +Train loss +Test loss +Data Fraction 0.3 +Epoch +Loss +0 +1000 +2000 +3000 +4000 +1μ +100μ +0.01 +1 +Train loss +Test loss +Data Fraction 0.4 +Epoch +Loss +0 +1000 +2000 +3000 +4000 +1μ +100μ +0.01 +1 +Train loss +Test loss +Data Fraction 0.5 +Epoch +Loss +0 +1000 +2000 +3000 +4000 +1μ +100μ +0.01 +1 +Train loss +Test loss +Data Fraction 0.6 +Epoch +Loss +0 +1000 +2000 +3000 +4000 +1μ +100μ +0.01 +1 +Train loss +Test loss +Data Fraction 0.7 +Epoch +Loss +0 +1000 +2000 +3000 +4000 +1μ +100μ +0.01 +1 +Train loss +Test loss +Data Fraction 0.8 +Epoch +Loss +0 +1000 +2000 +3000 +4000 +1μ +100μ +0.01 +1 +Train loss +Test loss +Data Fraction 0.9 +Epoch +Loss +Figure 20: Training and test losses for a 1-layer transformer on the modular addition task with +P = 113, with varying fractions of the 113 · 113 pairs of possible inputs used in training. Grokking +occurs when between 30 − 50% of the dataset is used during training and lower fractions of data +lead to slower grokking. Using ≥ 60% data leads to immediate generalization, while using 10% or +20% of the data doesn’t lead to grokking even after 40k epochs. Note the different x-axes: we only +show 5k epochs for the runs with data fraction ≥ 40% for more detail. +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +2k +4k +6k +8k +10k +12k +14k +16k +18k +Train loss +Test loss +Training Epochs until <1e-06 Loss +Fraction of train data +Number of steps +Figure 21: Number of steps for train/test loss to be < 10−6, as a function of the amount of training +data. While train loss immediately converges to below 10−6 for all data fractions, generalization +takes significantly longer with lower fractions of data. Note that the plots for other thresholds are +also qualitatively similar. +24 + +arXiv preprint +Figure 22: Training and test loss for a 2-layer version of the original architecture. Average across 5 +random seeds is in bold. +Figure 23: Training, test, and full excluded loss for a 2-layer version of the original architecture. +One random seed chosen for readability. +Figure 24: The training and test losses for P = 53 and all other hyperparameters except weight +decay (γ = 5) the same as the main training run discussed in the paper. The averages are bold, and +all contributing runs are partially transparent. Note that grokking occurs. +25 + +AverageTrain Loss +Average TestLoss +Loss +0.01 +Log +100u +1μ +10n +0 +5k +10k +15k +20k +EpochExcluded Loss Over All Frequencies +100 +Excluded Loss +Train Loss +TestLoss +0.01 +LOSS +100μ +1μ +10n +0 +5k +10k +15k +20k +25k +30k +35k +Epoch-Average Train Loss +Average Test Loss +Log Loss +0.01 +100μ +1μ +0 +5k +10k +15k +20k +EpocharXiv preprint +Figure 25: The training and test losses for P = 401 and all other hyperparameters the same as the +main training run discussed in the paper. Grokking doesn’t occur (the model generalizes immedi- +ately), even across a variety of weight decays. +believe this is because the amount of data seen by the model is greatly increased compared to the +P = 113 case (from 30% of 113 · 113 pairs to 30% of 401 · 401 pairs), thereby favoring the gen- +eralizing solution from the start. We then trained 3 models each using 5%, 10%, 20% of the pairs +of training data with λ = 1, and found that the models trained on 5% and 10% of the data imme- +diately overfit and never generalized, while the models trained on 20% of the data also generalized +immediately. +C.2.3 +GENERALIZING MODELS CONSISTENTLY USE THE FOURIER MULTIPLICATION +ALGORITHM +For each of the models in Appendix C.2.2 that achieve low test loss, we repeated the analysis per- +formed in the mainline model, and summarize the results in Table 5. We list their key frequencies, +Gini coefficients, and relevant FVEs. We find that every model trained with weight decay and that +generalizes correctly implements some variation of the Fourier multiplication algorithm. +Interestingly, the embedding and unembedding matrices of the models trained with dropout are not +sparse in the Fourier basis, and the logits for the p = 0.2 models are not as well explained by a sum +of cosines as the other models (likely because the p = 0.2 models are simply worse at the task). We +speculate that this is likely due to a combination of insufficient training epochs (as dropout models +seem to take much longer to grok) and the inherent need for redundancy for networks trained via +dropout. +As with the mainline model, we ignore the final skip connection (around the final MLP), as all of the +generalizing models studied do not suffer significant performance penalties if the skip connection is +zero or mean ablated (Table 6). +D +ADDITIONAL RESULTS ON GROKKING +D.1 +BOTH REGULARIZATION AND LIMITED DATA ARE NECESSARY FOR GROKKING +As discussed in Section 7 and Appendix C.2, the weight decay and the amount of data seem to have +a strong effect on whether grokking occurs. To confirm this, we experiment with removing weight +decay and varying the amount of data on 1-layer transformers. In Figure 26, we give the training, +test, and full excluded loss for a typical training run with λ = 0 (no weight decay). As the figure +shows, no grokking occurs, and excluded loss does not increase, suggesting that the model does not +form the circuit for generalizing algorithm at all. +26 + +Train Loss +- Tost Loss +Log Loss +0.01 +100μ +1μ +0 +5k +10k +15k +20k +EpocharXiv preprint +Model +Test Loss +Gini(WE) +Gini(WL) +Key Frequencies +Logit FVE +MLP FVE +40% Training Data +1.98 · 10−7 +0.76 +0.79 +[17, 43, 49, 55] +94.9% +83.3% [26.1%] +50% Training Data +1.68 · 10−7 +0.75 +0.77 +[2, 17, 31, 41, 44] +91.2% +85.2% [28.2%] +60% Training Data +1.23 · 10−7 +0.79 +0.84 +[2, 23, 34, 51] +96.4% +95.7% [1.4%] +70% Training Data +9.85 · 10−8 +0.80 +0.91 +[14, 15, 26] +99.0% +98.9% [0.4%] +80% Training Data +5.83 · 10−7 +0.62 +0.80 +[38, 41] +63.9% +94.1% [2.5%] +90% Training Data +1.11 · 10−7 +0.79 +0.88 +[3, 26, 34, 43] +98.6% +98.7% [0.3%] +2 Layer Transformer +9.54 · 10−7 +0.59 +0.80 +[14, 18, 29] +91.8% +95.2% [1.9%] +2 Layer Transformer +4.41 · 10−5 +0.55 +0.73 +[7, 12, 35, 49] +86.1% +86.2% [6.4%] +2 Layer Transformer +6.50 · 10−2 +0.66 +0.80 +[4, 9, 28] +88.5% +85.4% [5.9%] +2 Layer Transformer +4.18 · 10−2 +0.56 +0.76 +[4, 5, 15, 54] +91.4% +81.2% [17.8%] +2 Layer Transformer +1.75 · 10−2 +0.68 +0.71 +[3, 4, 13, 30, 38] +84.0% +71.9% [19.5%] +P = 53 +3.00 · 10−4 +0.61 +0.68 +[6, 9, 16, 21] +91.2% +90.2% [5.8%] +P = 53 +1.03 · 10−4 +0.56 +0.72 +[4, 13, 16] +94.8% +93.1% [6.4%] +P = 53 +1.21 · 10−5 +0.66 +0.79 +[13, 22, 23] +98.2% +97.6% [0.9%] +P = 53 +3.95 · 10−6 +0.66 +0.74 +[3, 14, 15] +88.5% +91.8% [4.6%] +P = 53 +5.56 · 10−6 +0.67 +0.80 +[10, 14, 22] +98.1% +98.3% [0.6%] +P = 109 +2.02 · 10−7 +0.76 +0.83 +[6, 7, 22, 25] +98.0% +97.3% [1.9%] +P = 109 +2.95 · 10−7 +0.69 +0.82 +[8, 14, 29, 32, 41] +95.2% +94.7% [2.3%] +P = 109 +1.66 · 10−7 +0.78 +0.86 +[13, 23, 39, 45] +98.5% +97.6% [0.9%] +P = 109 +2.50 · 10−7 +0.68 +0.82 +[8, 13, 32, 41] +96.8% +95.5% [2.3%] +P = 109 +2.77 · 10−7 +0.76 +0.85 +[29, 37, 38, 49] +97.9% +98.1% [0.8%] +Dropout p = 0.2 +2.65 · 10−1 +0.19 +0.46 +[1, 4, 7, 17, 22, 33, 40, 49, 55] +71.3% +65.0% [17.5%] +Dropout p = 0.2 +4.52 · 10−1 +0.19 +0.46 +[3, 8, 19, 28, 32, 34, 40, 44] +73.3% +71.4% [10.7%] +Dropout p = 0.2 +2.03 · 10−1 +0.20 +0.45 +[4, 5, 32, 38, 41, 44, 49, 50] +74.2% +71.1% [10.6%] +Dropout p = 0.5 +< 10−8 +0.26 +0.56 +[1, 4, 26, 46, 47, 55] +89.4% +88.9% [3.5%] +Dropout p = 0.5 +2.01 · 10−2 +0.20 +0.49 +[16, 21, 35, 47, 53] +88.4% +88.4% [3.0%] +Dropout p = 0.5 +< 10−8 +0.25 +0.54 +[1, 4, 7, 19, 29, 31, 42] +86.1% +85.6% [4.0%] +Table 5: For each of the models in Appendices C.2.3 and D.1 that generalizes to test data, we +report the test loss, the Gini coefficients of the norms of the Fourier components of WE and WL +(Section 5.1), the key frequencies of the network, and the fraction of variance in logits explained by +a weighted sum of cos (wk(a + b − c))s over the key frequencies (Section 4.2). +In addition, we find the components uk, vk of WL that correspond to cosines and sines of +the key frequencies, and then report the average fraction of variance of uT +k MLP(a, b) and +vT +k MLP(a, b) explained by a single term of form cos (wk(a + b)) or sin (wk(a + b)) respectively +(Section 4.2). Numbers in square brackets represent the standard deviation. For 2 Layer models, we +use the final layer MLP activations for MLP(a, b). +We omit test accuracy because every model on this list except for the dropout p = 0.2 mod- +els achieves > 99.95% test accuracy, while the dropout p = 0.2 models achieve around 99.6% test +accuracy. +Model Type +Loss +Accuracy +Ablated Loss +Ablated Acuracy +Varying Data Fraction +1.83 · 10−7 (1.65 · 10−7) +100% +7.74 · 10−7 (6.74 · 10−7) +100% +2 Layer Transformer +1.97 · 10−2 (2.41 · 10−2 +99.6% +4.63 · 10−2 (6.72 · 10−2 +98.7% +P = 53 +5.96 · 10−5 (8.91 · 10−5) +100% +1.5 · 10−4 (2.70 · 10−4) +100% +P = 109 +1.94 · 10−7 (3.74 · 10−8) +100% +6.53 · 10−7 (1.41 · 10−7) +100% +Dropout p = 0.2 +0.215 (0.091) +99.7% +0.205 (0.075) +99.7% +Dropout p = 0.5 +4.68 · 10−3 (8.11 · 10−3) +100% +3.6 · 10−3 (5.82 · 10−3) +100% +Table 6: We confirm that the skip connection around the final MLP layer is not important for perfor- +mance by mean ablating the skip connection and computing loss and accuracy over the entire dataset +for each problem, averaged over all runs. (We report the standard deviation of loss over the runs in +parentheses.) While loss does increase a small amount, accuracy remains consistently high and the +loss of the ablated model remains low. Results with zero ablations are also similar. +27 + +arXiv preprint +Figure 26: Training, test, and full excluded loss for a 1-layer version of the original architecture +without weight decay. One random seed chosen for readability. Note that not having weight decay +prevents grokking. +0 +5k +10k +15k +20k +25k +10n +1μ +100μ +0.01 +1 +Weight Decay 0.3 +Epoch +Log Loss +0 +5k +10k +15k +20k +25k +1μ +10μ +100μ +0.001 +0.01 +0.1 +1 +10 +Average Train Loss +Average Test Loss +Weight Decay 3.0 +Epoch +Log Loss +Loading [MathJax]/extensions/MathMenu.js +Figure 27: The train and test loss over the course of training with weight decay λ = 0.3 (left) and +λ = 3.0 (right). Less aggressive weight decay leads to slower grokking. +In Figure 20, we show the test loss curves for models trained with weight decay λ = 1 and on +various fractions of the data. Though all the train losses are approximately the same—that is, they +memorize at the same rate, models trained on smaller fractions of data take longer to grok. +In Figure 27, we display the test and train loss of models trained with λ = 0.3 and λ = 3.0. Smaller +amounts of weight decay lead to slower grokking, while larger amounts of weight decay lead to faster +grokking—on average, it takes around 3k epochs for models to grok with weight decay λ = 0.3, +5-10k epochs for the models to grok with weight decay λ = 1.0, and 20k epochs for the models to +grok with weight decay λ = 3.0. +Finally, we test whether other forms of regularization can also induce grokking. We replaced weight +decay with the following types of regularization while keeping all other hyperpameters the same: +1. Dropout We add dropout Srivastava et al. (2014) to the MLP neurons, with p +∈ +{0.2, 0.5, 0.8}. That is, for each individual neuron, we set it to 0 with probability p during +training, and also multiply the outputs of the other neurons by +1 +1−p. +2. ℓ1 Regularization We add an ℓ1 penalty to the loss term. We use λ ∈ {1, 10, 100}. Note +that we do not decouple the updates with respect to the ℓ1 penalty from optimization steps +done with respect to the log loss (as is done for ℓ2 regularization via AdamW Loshchilov +& Hutter (2017)). +In each case, we ran three random seeds. We show the results in Figure 28. While grokking did +not occur with ℓ1 regularization, we found that it does occur for all three seeds using dropout with +p = 0.2 or p = 0.5. We speculate that this is because both dropout and weight decay encourage +the network to spread out computation (which is required for the Fourier multiplication algorithm), +while ℓ1 regularization encourages the network to become more sparse in the neuron basis and thus +28 + +Excluded Loss Over All Frequencies +100 +Train Loss + Test Loss +ExcludedLoss +0.01 +Loss +100μ +1μ +10n +100p +0 +5k +10k +15k +20k +25k +30k +35k +EpocharXiv preprint +0 +10k +20k +30k +40k +1μ +10μ +100μ +0.001 +0.01 +0.1 +1 +10 +100 +Average Train Loss +Average Test Loss +Dropout p=0.2 +Epoch +Log Loss +0 +10k +20k +30k +40k +100p +10n +1μ +100μ +0.01 +1 +100 +Average Train Loss +Average Test Loss +L1 penalty, 1.0 +Epoch +Log Loss +0 +10k +20k +30k +40k +1μ +10μ +100μ +0.001 +0.01 +0.1 +1 +10 +100 +Average Train Loss +Average Test Loss +Dropout p=0.5 +Epoch +Log Loss +0 +10k +20k +30k +40k +100p +10n +1μ +100μ +0.01 +1 +100 +Average Train Loss +Average Test Loss +L1 penalty, 10 +Epoch +Log Loss +0 +10k +20k +30k +40k +1μ +10μ +100μ +0.001 +0.01 +0.1 +1 +10 +100 +Average Train Loss +Average Test Loss +Dropout p=0.8 +Epoch +Log Loss +0 +10k +20k +30k +40k +100p +10n +1μ +100μ +0.01 +1 +100 +Average Train Loss +Average Test Loss +L1 penalty, 100 +Epoch +Log Loss +Figure 28: The train and test loss over the course of training with two types of regularization, dropout +and ℓ1 regularization. Grokking occurs with some runs for dropout but never for ℓ1 regularization. +less sparse in the Fourier basis, preventing the network from learning the Fourier Multiplication +Algorithm. +D.2 +THE SLINGSHOT MECHANISM OFTEN OCCURS, BUT IS UNNECESSARY FOR GROKKING +As noted in Section C.2, our 2-layer transformers exhibit significant slingshots (Thilak et al., 2022) +during training. We speculate that this is due to how gradients of different scale interact with adaptive +optimizers. We were even able to induce slingshots on a 1-layer by reducing the precision of the loss +calculations (as this causes many gradients to round to 0 and thus greatly increases the differences +in scale of gradients). +However, as many of our 1-layer models do not exhibit slingshots but nonetheless grok, the slingshot +mechanism is unnecessary for grokking to occur, in the presence of weight decay or other regular- +ization. We speculate that the slingshots of Thilak et al. (2022) (which co-occur with grokking for +training runs without weight decay) serve as an implicit regularization mechanism that favors the +simpler, generalizing solution over the more complicated +29 + +arXiv preprint +0 +500 +1000 +1500 +2000 +2500 +0 +0.5 +1 +1.5 +2 +2.5 +Train/Test Loss for 5 Digit Addition w/ Infinite Data +Steps +Loss +0 +500 +1000 +1500 +2000 +2500 +0 +0.5 +1 +1.5 +2 +2.5 +Token 0 loss +Token 1 loss +Token 2 loss +Token 3 loss +Token 4 loss +Token 5 loss +Per Token Train/Test Loss for 5 Digit Addition w/ Infinite Data +Steps +Loss +Figure 29: (Top) The training/test loss for 5 Digit Addition trained on randomly generated data. Note +that training and test loss coincide, as the model does not see repeated pairs.(Bottom) The train/test +loss per token for 5 Digit Addition, trained with randomly generated data at each step. Note that +phase changes in the average loss correspond to phase changes in individual tokens, though one +phase change (token 1, around step 270) is not visible on the averaged loss as it overlaps with the +end of the first phase change (token 0, starting around step 150). +30 + +arXiv preprint +0 +5k +10k +15k +0 +2 +4 +6 +8 +Train Loss +Test Loss +Train/Test Loss for 5 Digit Addition with 700 Data Points +Epochs +Loss +Figure 30: The train and test loss for 5 Digit Addition trained on 700 data points. Unlike the infinite, +randomly generated data case, this shows both a sharp phase change and clear train test divergence. +D.3 +ADDITIONAL EVIDENCE FROM OTHER ALGORITHMIC TASKS +We now provide addition analysis of grokking phenomena on 3 additional algorithmic tasks and +confirm that limited data is an important part of grokking: +1. 5 digit addition. We sample pairs of random 5 digit numbers and have the model predict +their sum +2. Predicting repeated subsequences. We take a uniform random sequence of tokens, ran- +domly choose a subsequence to repeat, and train the model to predict the repeated tokens. +3. Skip trigram. We feed in a sequence of tokens from 0 to 19, of which exactly one is +greater than or equal to 10, and the model needs to output the token that is ≥ 10. This can +be solved with learning 10 skip trigrams. +We use a 1-layer full transformer for 5-digit addition, a 2-layer attention only transformer for pre- +dicting repeated subsequences, and a 1-layer attention only transformer for the skip trigram task. +Otherwise, we use the same hyperparameters as in the mainline model. +5 Digit Addition +We first consider the case where we train on the approximately infinite data +regime. For each minibatch, we randomly new sample 5 digit numbers. We report the results in +Figure 29. Train loss coincides with test loss, so grokking does not occur, as the model almost never +sees the same pair of 5 digit numbers twice, with 1010 such pairs. Interestingly, the various small +bumps in Figure 29 correspond to the model learning how to calculate each of the 6 tokens in the +output. However, grokking does occur when we restrict the model to only see 700 data points, as +shown in Figure 30. +Repeated subsequence +As with the 5-digit addition task, we find that restricting the amount of +data is necessary and sufficient for grokking on the repeated subsequence task. In Figure 31, the +model sees new data at every step exhibits no grokking. In contrast, clear grokking occurs when we +restrict the model to only see 512 data points in Figure 32. +Skip trigram +As with the previous tasks, we find that restricting the amount of data is necessary +and sufficient for grokking on the skip trigram task. The model that sees new data at every step +exhibits no grokking in Figure 33. Meanwhile, the model restricted to only see 512 data points +exhibits clear grokking in Figure 34. +Taken together, these results echo the importance of limited data for grokking. +31 + +arXiv preprint +0 +1000 +2000 +3000 +4000 +5000 +0 +1 +2 +3 +4 +5 +Repeated Subsequence Prediction w/ Infinite data +Step +Loss +Figure 31: The training/test loss for repeated subsequences trained on randomly generated data. +Note that training and test loss coincide, as the model does not see repeated pairs. There sharp phase +change corresponds to the model forming induction heads. (Olsson et al., 2022) +0 +50k +100k +150k +200k +0 +2 +4 +6 +8 +10 +12 +14 +16 +Train Loss +Test Loss +Repeated Subsequence Prediction w/ 512 Data Points +Epoch +Loss +Figure 32: The train and test loss for the repeated subsequence task, trained on 512 data points. +Unlike the infinite, randomly generated data case, this shows both a sharp phase change and clear +train test divergence. +0 +500 +1000 +1500 +2000 +2500 +0 +0.5 +1 +1.5 +2 +2.5 +Train/Test Loss for Skip Trigram Task w/ Infinite Data +Epoch +Loss +Figure 33: The training/test loss for the skip trigram task, trained on randomly generated data. Note +that training and test loss coincide, as the model does not see repeated pairs. The sharp phase change +corresponds to the network learning all of the skip trigrams. +32 + +arXiv preprint +0 +1000 +2000 +3000 +4000 +0 +0.5 +1 +1.5 +2 +2.5 +Train Loss +Test Loss +Train/Test Loss for Skip Trigram Task w/ 50 Data Points +Epoch +Loss +Figure 34: The train and test loss for the skip trigram task, trained on 512 data points. Unlike the +infinite, randomly generated data case, this shows both a sharp phase change and clear train test +divergence. +E +FURTHER SPECULATIONS ON GROKKING +E.1 +AN INTUITIVE EXPLANATION OF GROKKING +In this section, we speculate on what might be happening “under the hood” when a model groks and +explore why this phenomena happens. The evidence is only suggestive, so this a promising direction +for future research. +Grokking occurs when models, trained on algorithmic tasks with certain hyperparameters, initially +overfit the training data where train loss significantly improves while test loss worsens and the two +diverge. But later in training, there is a sudden improvement in test loss, so test and train loss +converge. In contrast to Power et al. (2022) but in line with Liu et al. (2022), grokking does not +occur when both train and test loss improve together without the initial divergence, as shown in +many of the figures in this paper, for example Figures 2 and 18. +The core issue is that the model has two possible solutions: memorization (with low train loss +and high test loss) and a generalization (with low train loss and low test loss). In our case, the +Fourier Multiplication Algorithm is the generalization solution. Intuitively, with very little training +data, the model will overfit and memorize. With more training data, the model must generalize +or suffer poor performance on both train and test loss. Since neural networks have an inductive +bias favoring “simpler” solutions, memorization complexity scales with the size of the training set, +whereas generalization complexity is constant. The two must cross at some point! Yet, the surprising +aspect of grokking is the abrupt shift during training, when the model switches from memorization +to generalization. +The other component of grokking is phase transitions - the phenomena where models trained on a +certain task develop a specific capability fairly rapidly during a brief period of training, as shown +for the case of induction heads forming in transformer language models in Olsson et al. (2022) and +our results in Appendix D.3. That is, rather than slowly forming that capability over training, the +model rapidly goes from being bad at it to being good at it. One interpretation of a phase transition +is that there’s some feature of the loss landscape that makes the generalising solution harder to reach +- rather than a smooth gradient for the model to follow, it instead initially finds it difficult to make +progress, but then crosses some threshold where it can rapidly make progress. +Therefore, grokking occurs with phase transitions, limited data, and regularization. Models exhibit +phase transitions despite having enough training data to avoid overfitting. Regularization (weight +decay in our case) favors simpler solutions over complex ones. The model has enough data to +marginally prefer generalization over memorization. The phase transition indicates that generaliza- +tion is “hard to reach” while the model has no problems with memorization. But as it memorizes, the +network becomes more complex until the weight decay prevents further memorization then moves +towards equilibrium. The gradient to memorize balances the gradient towards smaller weights. With +33 + +arXiv preprint +generalization, the model is incentivized to both memorize and simplify. Strikingly, it is capable of +both while maintaining a somewhat constant training performance in this circuit formation phase. +Next, as the model approaches generalization, the memorization weights are removed in the cleanup +phase. The cost from complexity outweighs the benefit from lower loss. Due to the phase transition +during this training period, as model’s progress towards generalization accelerates, the cleanup rate +sharpens as well. +A model that learns a perfect solution and is trained with weight decay has competing incentives: +larger weights (for more extreme logits and thus lower loss) and smaller weights (from weight +decay). So for any solution and any level of weight decay, there will always be a level of train loss +where these two forces equilibrate. Thus, memorization is not necessarily a “simpler” solution than +generalization. The key is that generalization will have smaller weights holding train loss fixed. In +fact, weight decay should be expected to equilibrate at a slightly lower train loss in generalization, +since the base solution is simpler. This matches what we observe in practice. 4 +E.2 +HYPOTHESIS: PHASE TRANSITIONS ARE INHERENT TO COMPOSITION +A promising line of work in the growing field of mechanistic interpretability suggests that models +form circuits (Cammarata et al., 2020) – clean interpretable algorithms formed by subnetworks of +the model, such as curve detectors (Cammarata et al., 2020) in image classification networks and +induction heads (Elhage et al., 2021; Olsson et al., 2022) in LLMs. This is surprisingly true! A +circuit represents the model learning an algorithm, a fundamentally discrete thing; each step in the +algorithm only makes sense if the other steps are present. But neural networks are fundamentally +continuous, trained to follow gradients towards lower loss and struggle to jump to new optima with- +out following a smooth gradient. So how can a model learn a discrete algorithm? +As a concrete example, let’s consider the case of induction heads in LLMs. There is a subnetwork +of a next-token prediction autoregressive language model that learns to continue repeated subse- +quences. It detects whether the current token occurred earlier in the context. If so, it predicts the +same token after that previous occurrence will also come next. The circuit consists of a previous +token head, which attends to each previous token and copies the context of the previous token to +the current token, and an induction head which attends to the token after a previous occurrence of +the current token. The induction head composes with the previous token head by forming a query +vector representing the current token and a key vector representing the previous token head’s output +using K-Composition, the context of the previous token. It attends to a token where this query and +key match. +This circuit significantly improves loss but only in the context of the other heads present. Before +either head is present, no gradient encourages the formation of either head. At initialization, we +have neither head, so gradient descent should never discover this circuit. Naively, we might predict +that neural networks will only produce circuits analogous to linear regression, where each weight +will marginally improve performance as it continuously trains. And yet in practice, neural networks +indeed form such sophisticated circuits, involving several parts interacting in non-trivial, algorithmic +ways. So how can this be? +A few possible explanations: +• Lottery tickets (Frankle & Carbin, 2018): Initially, each layer of the network is the +superposition of many partial circuit components, and the output of each layer is the average +of the output of each component. The full output of the network is the average of many +different circuits, with significant interference from non-linear interaction. Some of these +circuits are systematically useful to reducing loss, but most aren’t. Gradients for useless +circuits will have zero mean, while gradients for useful circuits will have non-zero mean, +with a lot of noise. SGD reinforces relevant circuits and suppresses useless ones, so circuits +will gradually form. +4One subtlety: the grokking phenomena is often incorrectly summarized as “the model learned to generalize +even after achieving zero loss.” Zero loss does not exist with cross-entropy loss. Although the model achieves +perfect accuracy, it is trained to optimize loss not accuracy. This means the model is always incentivized to +further improve. In particular, the easiest way to improve performance with perfect accuracy is by scaling up +the logits. This lowers the temperature and pushes the softmax closer to an argmax. +34 + +arXiv preprint +• Random walk: The network wanders randomly around the loss landscape until it encoun- +ters a half-formed previous token head and induction head that somewhat compose. This +half-formed circuit becomes useful for reducing loss, so gradient descent completes the +circuit. +• Evolution: A similar mystery arises from how organisms develop sophisticated machinery, +like the human eye. Each part is only useful in the context of other parts. A compelling +explanation is a component first developed that was somewhat useful in its own right, like +a light-detecting membrane. It was reinforced as a useful component. Then, later compo- +nents developed depending on the first, like the lens of the eye. +Evolution is a natural explanation, However, based on our toy tasks, it cannot be the whole story. +In the repeated subsequence task, we have a sequence of uniform randomly generated tokens, apart +from a repeated subsequence at an arbitrary location, e.g. 7 2 8 3 1 9 3 8 3 1 9 9 2 5 END. This means +all pairs of tokens are independent, apart from pairs of equal tokens in the repeated subsequence. In +particular, this means that a previous token head can never reduce loss for the current token. The +previous token will always be independent of the next token. So a previous token head is only useful +in the context of an induction-like head that completes the circuit. Likewise, an induction head relies +on K-composition with a previous token head and so cannot be useful on its own. Yet the model +eventually forms an induction circuit! +A priori, the random walk seems insufficient on its own. An induction circuit is relatively compli- +cated, representing a small region in model space. So a random walk is unlikely to stumble upon +it. Concretely, in our modular addition case, progress measures show significant hidden progress +pre-grokking, indicating the model did not stumble upon the solution by chance. +Thus, the lottery ticket hypothesis seems the most explanatory. An induction head is useless without +a previous token head but might be slightly useful when composing with a head that uniformly +attends to prior tokens, since part of its output will include the previous token! Nevertheless, we +suspect that all explanations contribute to the entire picture. This seems most plausible if the uniform +head just so happens to attend a bit more to the previous token via a random walk. +Returning to phase transitions, the lottery ticket-style explanation suggests that we might expect +phase transitions as circuits form. Early in circuit formation, each part of the circuit is rough, so the +effect on the loss of improving any individual component is weak, meaning gradients will be small. +As each component develops, other components will become more useful, meaning all gradients will +increase together non-linearly. As the circuit nears completion, we should expect an acceleration in +the loss curve for this circuit, resulting in a phase transition. +F +FURTHER DISCUSSION ON USING MECHANISTIC INTERPRETABILITY AND +PROGRESS MEASURES FOR STUDYING EMERGENT PHENOMENA +While we find approach of using mechanistic interpretability to define progress measures rela- +tively promising, there remains significant uncertainty as to how scalable existing mechanistic inter- +pretability approaches really are. Broadly speaking, depending on the success of future mechanistic +interpretability work, we think there are three methods through which mechanistic interpretability +and progress measures can help with understanding and predicting emergent phenomena: +1. If mechanistic interpretability can be scaled to large models to the level where we can un- +derstand the mechanisms behind significant portions of their behavior, we could perform +the same style of analysis as was done in this work. We believe it’s currently unclear as to +whether or not mechanistic interpretability will successfully scale to large models to this +extent (or even if there exist human-understandable explanations for all of their sophisti- +cated behavior). That being said, in cases where mechanistic interpretability does recover +human-understandable mechanisms, we could simply use the parts of the mechanism as +progress measures. +2. If future mechanistic interpretability can only recover parts of the mechanism of larger +models (as in Wang et al. (2022)) and can only generate comprehensive understanding +of the mechanisms of smaller models, we might still be able to use our understanding +from smaller models to guide the development measures that track parts of the behavior of +35 + +arXiv preprint +the larger model. We find this scenario relatively plausible, as existing mechanistic inter- +pretability work already allows us to recover fragments of large model behavior and un- +derstand these fragments by analogy to smaller models. For example, Olsson et al. (2022) +use this approach to understand the emergence of in-context learning in medium-sized lan- +guage transformers. +3. Even if mechanistic interpretability fails to recover understandable mechanisms at all on +large models, we might still be able to derive progress measures that don’t require human +understanding. For example, if we end up with automated mechanistic interpretability (that +nonetheless still fails to recover human-understandable mechanisms), we might be able to +use the outputs of those opaque processes. +Another approach is task-independent progress measures: if we can discover progress mea- +sures that don’t depend on the task, perhaps using many small, interpretable models as +testbeds, we might be able to apply these progress measures to large models. +That being said, we think the future work outlined in Section 6 is necessary to successfully apply +our approach to predict and understand emergent behavior in existing large language models, and so +remain cautiously optimistic. +36 + diff --git a/jtE4T4oBgHgl3EQfsg2s/content/tmp_files/load_file.txt b/jtE4T4oBgHgl3EQfsg2s/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..35f43e5cfd4c2acca410f1e359693db434149325 --- /dev/null +++ b/jtE4T4oBgHgl3EQfsg2s/content/tmp_files/load_file.txt @@ -0,0 +1,1667 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf,len=1666 +page_content='arXiv preprint PROGRESS MEASURES FOR GROKKING VIA MECHANISTIC INTERPRETABILITY Neel Nanda Independent neelnanda27@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='com Lawrence Chan UC Berkeley chanlaw@berkeley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='edu Tom Lieberum Independent tlieberum3141@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='com Jess Smith Independent smith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='jessk@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='com Jacob Steinhardt UC Berkeley jsteinhardt@berkeley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='edu ABSTRACT Neural networks often exhibit emergent behavior, where qualitatively new capa- bilities arise from scaling up the amount of parameters, training data, or training steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' One approach to understanding emergence is to find continuous progress measures that underlie the seemingly discontinuous qualitative changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We ar- gue that progress measures can be found via mechanistic interpretability: reverse- engineering learned behaviors into their individual components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' As a case study, we investigate the recently-discovered phenomenon of “grokking” exhibited by small transformers trained on modular addition tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We fully reverse engineer the algorithm learned by these networks, which uses discrete Fourier transforms and trigonometric identities to convert addition to rotation about a circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We confirm the algorithm by analyzing the activations and weights and by perform- ing ablations in Fourier space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Based on this understanding, we define progress measures that allow us to study the dynamics of training and split training into three continuous phases: memorization, circuit formation, and cleanup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Our re- sults show that grokking, rather than being a sudden shift, arises from the gradual amplification of structured mechanisms encoded in the weights, followed by the later removal of memorizing components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 1 INTRODUCTION Neural networks often exhibit emergent behavior, in which qualitatively new capabilities arise from scaling up the model size, training data, or number of training steps (Steinhardt, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=', 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' This has led to a number of breakthroughs, via capabilities such as in-context learning (Rad- ford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=', 2020) and chain-of-thought prompting (Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=', 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' However, it also poses risks: Pan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' (2022) show that scaling up the parameter count of models by as little as 30% can lead to emergent reward hacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Emergence is most surprising when it is abrupt, as in the case of reward hacking, chain-of-thought reasoning, or other phase transitions (Ganguli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=', 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We could better understand and predict these phase transitions by finding hidden progress measures (Barak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=', 2022): metrics that precede and are causally linked to the phase transition, and which vary more smoothly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' For example, Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' (2022a) show that while large language models show abrupt jumps in their performance on many benchmarks, their cross-entropy loss decreases smoothly with model scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' However, cross-entropy does not explain why the phase changes happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' In this work, we introduce a different approach to uncovering hidden progress measures: via mech- anistic explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1 A mechanistic explanation aims to reverse engineer the mechanisms of the network, generally by identifying the circuits (Cammarata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Elhage et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=', 2021) within a model that implement a behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Using such explanations, we study grokking, where models 1Interactive versions of figures, as well as the code to reproduce our results, are available at bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='ly/ progress-measures-grokking-website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='05217v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='LG] 12 Jan 2023 arXiv preprint Figure 1: The algorithm implemented by the one-layer transformer for modular addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Given two numbers a and b, the model projects each point onto a corresponding rotation using its embedding matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Using its attention and MLP layers, it then composes the rotations to get a representation of a + b mod P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Finally, it “reads off” the logits for each c ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=', P − 1}, by rotating by −c to get cos(w(a + b − c)), which is maximized when a + b ≡ c mod P (since w is a multiple of 2π P ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' abruptly transition to a generalizing solution after a large number of training steps, despite initially overfitting (Power et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Specifically, we study modular addition, where a model takes inputs a, b ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' , P −1} for some prime P and predicts their sum c mod P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Small transformers trained with weight decay on this task consistently exhibit grokking (Figure 2, Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We reverse engineer the weights of these transformers and find that they perform this task by map- ping the inputs onto a circle and performing addition on the circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Specifically, we show that the embedding matrix maps the inputs a, b to sines and cosines at a sparse set of key frequencies wk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' The attention and MLP layers then combine these using trigonometric identities to compute the sine and cosine of wk(a + b), and the output matrices shift and combine these frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We confirm this understanding with four lines of evidence (Section 4): (1) the network weights and activations exhibit a consistent periodic structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' (2) the neuron-logit map WL is well approx- imated by a sum of sinusoidal functions of the key frequencies, and projecting the MLP activations onto these sinusoidal functions lets us “read off” trigonometric identities from the neurons;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' (3) the attention heads and MLP neuron are well approximated by degree-2 polynomials of trigonometric functions of a single frequency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' and (4) ablating key frequencies used by the model reduces perfor- mance to chance, while ablating the other 95% of frequencies slightly improves performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Using our understanding of the learned algorithm, we construct two progress measures for the mod- ular addition task—restricted loss, where we ablate every non-key frequency, and excluded loss, where we instead ablate all key frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Both metrics improve continuously prior to when grokking occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We use these metrics to understand the training dynamics underlying grokking and find that training can be split into three phases: memorization of the training data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' circuit for- mation, where the network learns a mechanism that generalizes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' and cleanup, where weight decay removes the memorization components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Surprisingly, the sudden transition to perfect test accuracy in grokking occurs during cleanup, after the generalizing mechanism is learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' These results show that grokking, rather than being a sudden shift, arises from the gradual amplification of structured mechanisms encoded in the weights, followed by the later removal of memorizing components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 2 RELATED WORK Phase Changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Recent papers have observed that neural networks quickly develop novel quali- tative behaviors as they are scaled up or trained longer (Ganguli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=', 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' McGrath et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' (2021) find that AlphaZero quickly learns many human chess concepts between 10k and 30k training steps and reinvents human opening theory between 25k and 60k training steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Grokking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Grokking was first reported in Power et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' (2022), which trained two-layer transformers on several algorithmic tasks and found that test accuracy often increased sharply long after achieving perfect train accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Millidge (2022) suggests that this may be due to SGD being a random walk on the optimal manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Our results echo Barak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' (2022) in showing that the network instead makes continuous progress toward the generalizing algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' (2022) construct small 2 Logits cos w(a+ b- c) Computes logits using further trig identities: Unembed Logit(c) α cos(w(a + b - c) = cos(w(a + b)) cos(wc) + sin(w(a + b)) sin(wc) MLP m Calculates sine and cosine of a + b using trig identities: sin(w(a + b)) = sin(wa) cos(wb) + cos(wa) sin(wb) cos(w(a + b)) = cos(wa) cos(wb) - sin(wa) sin(wb) no h2 h3 Translates one-hot a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' b to Fourier basis: Embed a → sin(wa),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' cos(wa) b → sin(wb),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' cos(wb) TokensarXiv preprint Figure 2: The train and test accuracy (left) and train and test loss (right) of one-layer transformers on the modular addition task described in Section 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' over 5 random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' These models consistently exhibit grokking: they quickly overfit early on in training, but then later learn to generalize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' examples of grokking, which they use to compute phase diagrams with four separate “phases” of learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Thilak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' (2022) argue that grokking can arise without explicit regularization, from an optimization anomaly they dub the slingshot mechanism, which may act as an implicit regularizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Circuits-style mechanistic interpretability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' The style of post-hoc mechanistic interpretability in Section 4 is heavily inspired by the Circuits approach of Cammarata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' (2020), Elhage et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' (2021), and Olsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Progress measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Barak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' (2022) introduce the notion of progress measures—metrics that improve smoothly and that precede emergent behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' They prove theoretically that training would amplify a certain mechanism and heuristically define a progress measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' In contrast, we use mech- anistic intepretability to discover progress measures empirically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 3 SETUP AND BACKGROUND We train transformers to perform addition mod P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' The input to the model is of the form “a b =”, where a and b are encoded as P-dimensional one-hot vectors, and = is a special token above which we read the output c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' In our mainline experiment, we take P = 113 and use a one-layer ReLU transformer, token embeddings with d = 128, learned positional embeddings, 4 attention heads of dimension d/4 = 32, and n = 512 hidden units in the MLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' In other experiments, we vary the depth and dimension of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We did not use LayerNorm or tie our embed/unembed matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Our mainline dataset consists of 30% of the entire set of possible inputs (that is, 30% of the 113 · 113 pairs of numbers mod P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We use full batch gradient descent using the AdamW optimizer (Loshchilov & Hutter, 2017) with learning rate γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='001 and weight decay parameter λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We perform 40, 000 epochs of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' As there are only 113 · 113 possible pairs, we evaluate test loss and accuracy on all pairs of inputs not used for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Networks trained on this task consistently exhibit grokking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' As Figure 2 shows, our networks first overfit the training set: train accuracy quickly converges to 100% and the train loss quickly declines, while the test accuracy remains low and the test loss remains high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' After around 10, 000 epochs, the network generalizes and test accuracy increases to near 100%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' In robustness experiments, we confirm that grokking consistently occurs for other architectures and prime moduli (Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='3 we find that grokking does not occur without regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' To describe transformer components, we follow the conventions and notations laid out in Elhage et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We focus on the d×p embedding matrix WE, the d×n output matrix of the MLP layer Wout, and the P × d unembedding matrix WU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2 Let Logits(a, b) denote the logit vector on inputs a, b, and MLP(a, b) denote the MLP activations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Empirically, our networks do not significantly use the skip connection around the MLP (Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1), so Logits(a, b) ≈ WUWoutMLP(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We therefore also study the P × n neuron-logit map WL = WUWout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1 THE FOURIER MULTIPLICATION ALGORITHM We claim that the learned networks use the following algorithm (Figure 1): 2We ignore the embedding and unembedding of the ‘=’ token for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='8 Accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2 Average Train Accuracy Average Test Accuracy 0 0 5k 10k 15k 20k EpochAverage Train Loss Average Test Loss Log Loss 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='01 100μ 1μ 0 5k 10k 15k 20k EpocharXiv preprint Given two one-hot encoded tokens a, b map these to sin(wka), cos(wka), sin(wkb), and cos(wkb) using the embedding matrix, for various frequencies wk = 2kπ P , k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Compute cos (wk(a + b)) and sin (wk(a + b)) using the trigonometric identities: cos (wk(a + b)) = cos (wka) cos (wka) − sin (wka) sin (wkb) sin (wk(a + b)) = sin(wka) cos (wkb) + cos (wka) sin (wkb) In our networks, this is computed in the attention and MLP layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' For each output logit c, compute cos (wk(a + b − c)) using the trigonometric identity: cos (wk(a + b − c)) = cos (wk(a + b)) cos (wkc) + sin (wk(a + b)) sin (wkc) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' (1) This is a linear function of the already-computed values cos(wk(a + b)), sin(wk(a + b)) and is implemented in the product of the output and unembedding matrices WL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' The unembedding matrix also adds together cos (wk(a + b − c)) for the various ks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' This causes the cosine waves to constructively interfere at c∗ = a + b mod p (giving c∗ a large logit), and destructively interfere everywhere else (thus giving small logits to other cs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We refer to this algorithm as Fourier multiplication, and will justify our claim in detail in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 4 REVERSE ENGINEERING A ONE-LAYER TRANSFORMER In this section, we describe four lines of evidence that our transformers are using the Fourier mul- tiplication algorithm described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Here we apply our analysis to the mainline model from Section 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' the results are broadly consistent for other models, including across different num- ber of layers, different fractions of the training data, and different prime moduli (see Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2, especially Table 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Our first line of evidence involves examining the network weights and activations and observing consistent periodic structure that is unlikely to occur by chance (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Moreover, when we take Fourier transforms, many components are either sparse or nearly sparse in the Fourier domain, supported on a handful of key frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We next look into the actual mechanisms implemented in the model weights (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We show that the unembedding matrix WL is (approximately) rank 10, where each direction corresponds to the cosine or sine of one of 5 key frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Projecting the MLP activations onto the components of WL approximately produces multiples of the functions cos (wk(a + b)) and sin (wk(a + b)), showing that the MLP layer does compute these sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' To better understand the mechanism, we zoom in to individual neurons (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We find that the attention heads and most neurons are well-approximated by degree-2 polynomials of sines and cosines at a single frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Moreover, the corresponding direction in WL also contains only that frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' This suggests that the model’s computations are (1) localized across frequencies and (2) mostly aligned with the neuron basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Finally, we use ablations to confirm that our interpretation is faithful (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We replace various components of the model by the components of the Fourier multiplication algorithm and find that doing so consistently does not harm and sometimes even improves model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1 SUGGESTIVE EVIDENCE: SURPRISING PERIODICITY The first line of evidence that the network is using the algorithm described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1 is the surprising periodicity in the activations of the transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' That is, the output of every part of the network is periodic as a function of the input tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Periodicity in the embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We start by examining the embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We apply a Fourier trans- form along the input dimension of the embedding matrix WE then compute the ℓ2-norm along the other dimension;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' results are shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We plot only the components for the first 56 frequen- cies, as the norm of the components for frequencies k and P − k are symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' The embedding matrix WE is sparse in the Fourier basis–it only has significant nonnegligible norm at 6 frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Of these frequencies, only 5 appear to be used significantly in later parts of the model (corresponding to k ∈ {14, 35, 41, 42, 52}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We dub these the key frequencies of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 4 arXiv preprint 0 10 20 30 40 50 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 2 sin cos Fourier Components of Embedding Matrix Frequency k Norm of Fourier Component 0 10 20 30 40 50 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 2 sin cos Fourier Components of Neuron-Logit Map Frequency k Norm of Fourier Component Figure 3: (Left) The norms of the Fourier components in the embedding matrix WE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' As discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1, the sparsity of WE in the Fourier basis is evidence that the network is operating in this basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Of the six non-zero frequencies, five “key frequencies” appear in later parts of the network, corresponding to k ∈ {14, 35, 41, 42, 52}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' (Right) Norm of Fourier components of the neuron-logit map WL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' A Fourier transform is taken over the logit axis, and then the norm is taken over the neuron axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' As discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2, WL is well-approximated by the 5 key frequencies wk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 0 50 100 100 80 60 40 20 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='8 1 Attention Score for Head 0 a b 0 50 100 100 80 60 40 20 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 4 Activations for Neuron 0 a b Const cos 5 sin 9 cos 14 sin 18 cos 23 sin 27 cos 32 sin 36 cos 41 sin 45 cos 50 sin 54 sin 54 cos 50 sin 45 cos 41 sin 36 cos 32 sin 27 cos 23 sin 18 cos 14 sin 9 cos 5 Const −60 −40 −20 0 20 40 60 Norms of Logits in 2D Fourier Basis x Component y Component Figure 4: (Left) The attention score for head 0 from the token ‘=’ to ‘a’, as a function of inputs a, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' (Center) The activations of MLP neuron 0 given inputs a, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Both the attention scores and the neuron activations are periodic (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' (Right) The norm of the Fourier components of the logits (2D Fourier transform is taken over the inputs a, b, and then norm is taken over the logit axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' There are 20 significant components corresponding to the 5 key frequencies (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Periodicity in attention heads and MLP neuron activations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' This periodic structure recurs throughout the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' As an example, we plot the attention weight at position 0 for every combi- nation of two inputs for head 0 in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' The attention exhibits a periodic structure with frequency k = 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' In Figure 4, we also plot the activations of MLP neuron 0 for every combination of inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' The activations are periodic with frequency k = 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We see similar patterns for other attention heads and MLP neurons (Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Periodicity in logits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Finally, the logits are also periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' In Figure 4, we represent the logits in the 2D Fourier basis over the inputs, then take the ℓ2-norm over the output dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' There are only twenty components with significant norm, corresponding to the products of sines and cosines for the five key frequencies wk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' These show up as five 2 × 2 blocks in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2 MECHANISTIC EVIDENCE: COMPOSING MODEL WEIGHTS We now demonstrate that the model implements the trigonometric identity (1) as follows: the func- tions cos (wk(a + b)), sin (wk(a + b)) are linearly represented in the MLP activations, and the un- embed matrix reads these linear directions and multiplies them by cos (wkc), sin (wkc) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We will do this in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' First, we show that WL (the matrix mapping MLP activations to logits) is (approximately) rank 10 and can be well approximated as: WL = � k∈{14,35,41,42,52} cos (wk) uT k + sin (wk) vT k (2) for some uk, vk ∈ R512, where cos (wk) , sin (wk) ∈ R113 are vectors whose cth entry is cos (wkc) and sin (wkc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Second, note that our model implements the logits for a, b as: Logits(a, b) = WLMLP(a, b) ≈ � k cos (wk) uT k MLP(a, b) + sin (wk) vT k MLP(a, b) (3) 5 arXiv preprint WL Component Fourier components of uT k MLP(a, b) or vT k MLP(a, b) FVE cos (w14c) 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='6 cos(w14a) cos(w14b) − 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='6 sin(w14a) sin(w14b) ≈ 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1 cos (w14(a + b)) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2% sin (w14c) 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1 sin(w14a) cos(w14b) + 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1 cos(w14a) sin(w14b) ≈ 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1 sin (w14(a + b)) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5% cos (w35c) 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='7 cos(w35a) cos(w35b) − 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='6 sin(w35a) sin(w35b) ≈ 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2 cos (w35(a + b)) 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='8% sin (w35c) 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='8 sin(w35a) cos(w35b) + 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='8 cos(w35a) sin(w35b) ≈ 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='8 sin (w35(a + b)) 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5% cos (w41c) 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='8 cos(w41a) cos(w41b) − 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='8 sin(w41a) sin(w41b) ≈ 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='8 cos (w41(a + b)) 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='0% sin (w41c) 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 sin(w41a) cos(w41b) + 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 cos(w41a) sin(w41b) ≈ 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 sin (w41(a + b)) 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='0% cos (w42c) 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='6 cos(w42a) cos(w42b) − 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 sin(w42a) sin(w42b) ≈ 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='6 cos (w42(a + b)) 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4% sin (w42c) 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='8 sin(w42a) cos(w42b) + 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='8 cos(w42a) sin(w42b) ≈ 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='8 sin (w42(a + b)) 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4% cos (w52c) 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 cos(w52a) cos(w52b) − 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 sin(w52a) sin(w52b) ≈ 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='0 cos (w52(a + b)) 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4% sin (w52c) 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 sin(w52a) cos(w52b) + 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 cos(w52a) sin(w52b) ≈ 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 sin (w52(a + b)) 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2% Table 1: For each of the directions uk or vk (corresponding to the cos(wk) and sin(wk) components respectively) in the unembedding matrix, we take the dot product of the MLP activations with that direction, then perform a Fourier transform (middle column;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' only two largest coefficients shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We then compute the fraction of variance explained (FVE) if we replace the projection with a single term proportional to cos (wk(a + b)) or sin (wk(a + b)), and find that it is consistently close to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We check empirically that the terms uT k MLP(a, b) and vT k MLP(a, b) are approximate multiples of cos (wk(a + b)) and sin (wk(a + b)) (> 90% of variance explained).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Thus the network computes trigonometric functions in the MLP and reads them off as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' As a sanity check, we confirm that the logits are indeed well-approximated by terms of the form cos (wk(a + b − c)) (95% of variance explained).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' WL is well approximated by cos (wkc) and sin (wkc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We perform a discrete Fourier transform (DFT) on the logit axis of WL and look at the 10 directions uk, vk corresponding to sin (wk) and cos (wk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' When we approximate WL with � k∈{14,35,41,42,52} cos (wk) uT k +sin (wk) vT k , the resid- ual has Frobenius norm that is under 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='55% of the norm of WL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' This shows that WL is well ap- proximated by the 10 directions corresponding to cos (wk) and sin (wk) for each of the five key frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We also plot the norms of each direction in Figure 3, and find that no Fourier compo- nent outside the 5 key frequencies has significant norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' The unembedding matrix “reads off” terms of the form cos (wk(a + b)) and sin (wk(a + b)) from the MLP neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Next, we take the dot product of the MLP activations with each of the directions uk, vk for k ∈ {14, 35, 41, 42, 52}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Table 1 displays the results: the dot products uT k MLP(a, b) and vT k MLP(a, b) are well approximated by a multiple of terms of the form cos (wk(a + b)) = cos (wka) cos (wkb) − sin (wka) sin (wkb) , and sin (wk(a + b)) = sin (wka) cos (wkb) + cos (wka) sin (wkb) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' That is, for each key frequency k, uk and vk are linear directions in the space of MLP neuron activations that represent cos (wk(a + b)) and sin (wk(a + b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Taken together, these results confirm that the model computes sums of terms of the form cos (wk(a + b − c)) = cos (wk(a + b)) cos (wkc) + sin (wk(a + b)) sin (wkc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Logits are well approximated by a weighted sum of cos (wk(a + b − c))s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We approximate the output logits as the sum � k αk cos(wk(a+b−c)) for k ∈ {14, 35, 41, 42, 52} and fit the coefficients αk via ordinary least squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' This approximation explains 95% of the variance in the original logits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' This is surprising—the output logits are a 113 · 113 · 113 dimensional vector, but are well- approximated with just the 5 directions predicted by our interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' If we evaluate test loss using this logit approximation, we actually see an improvement in loss, from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4 · 10−7 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='7 · 10−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='3 ZOOMING IN: APPROXIMATING NEURONS WITH SINES AND COSINES In the previous section, we showed how the model computes its final logits by using WL to “read off” trigonometric identities represented in the MLP neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' In this section, we examine the attention heads and MLP neurons to understand how the identities come to be represented at the MLP layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We show first that two of the attention heads approximately compute degree-2 polynomials of sines and cosines of a particular frequency (and the other two are used to increase the magnitude of the input embeddings in the residual stream), most neurons are also well-approximated by degree-2 polynomials, and the map from neurons to logits is localized by frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Attention heads approximately compute degree-2 polynomials of a single frequency or are used to amplify WE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' In order to compute terms like cos (wk(a + b)), the model needs to compute 6 arXiv preprint 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='8 1 0 100 200 300 400 FVE by degree-2 polynomials Fraction of variance explained Number of neurons Const sin 3 sin 6 sin 9 sin 12 sin 15 sin 18 sin 21 sin 24 sin 27 sin 30 sin 33 sin 36 sin 39 sin 42 sin 45 sin 48 sin 51 sin 54 40 30 20 10 0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4 Components of W L corresponding to freq 14 neurons Neuron Figure 5: (Left) Most neurons are well-approximated by degree-2 polynomials of a single frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' (Right) A heatmap showing weights in WL corresponding to each of the 44 neurons of frequency 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' The non-trivial components correspond to sin (wk) and cos (wk) for k = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' the product of the sine and cosine embeddings output by WE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' As the attention heads are approx- imately bilinear (product of attention weights and OV circuit), they are a natural place to perform this computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Indeed, for each head, the attention scores’ Fourier transform is concentrated on a single frequency wk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' For two of the four heads, the corresponding OV circuit is concentrated on that same frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Moreover, the softmax mapping the attention scores to attention weights is in a regime where it behaves approximately linearly (and replacing it with a linear function actually improves performance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Thus the attention weights multiply with the OV output to create degree-2 polynomials of the frequency wk, as would be needed for the cosine/sine addition formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' For the remaining two heads, their attention scores approximately sum to one and the OV circuits contain all five key frequencies, suggesting that they are used to increase the magnitude of key frequencies in the residual stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We confirm all of these claims in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Most MLP neurons approximately compute a degree-2 polynomial of a single frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We next try to approximate the activations of each MLP neuron by a degree-2 polynomial of one of the 5 key frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' As shown in Figure 5, out of 512 total neurons, 433 (84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='6%) have over 85% of their variance explained with a single frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Maps to the logits are localized by frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We partition these 433 neurons by the frequencies with the highest variance explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' For each resulting subset, the map WL from neurons to logits has only two non-trivial components, corresponding to sine and cosine at that frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' For exam- ple, in Figure 5 we plot the 44 columns of WL corresponding to the 44 neurons in the k = 14 cluster and find that the only non-negligible components are sin � 2kπ P � and cos � 2kπ P � for k = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4 CORRECTNESS CHECKS: ABLATIONS In previous sections, we showed that various components of the model were well-approximated by sparse combinations of sines and cosines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We verify that these approximations are faithful to the model’s functionality, by replacing each component with its approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' This generally does not hurt the performance of the model and in some cases improves it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' MLP neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='3, we identified 433 neurons that were well-approximated by a degree-2 polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We replace each of these neurons’ activation value by the corresponding polynomial, leaving the other neurons untouched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' This increases loss by only 3% in relative terms (from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='41 · 10−7 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='48 · 10−7) and has no effect on accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We can instead apply a stricter ablation to the MLP layer and restrict each neuron’s activation to just the components of the polynomial corresponding to terms of the form cos(wk(a + b)) and sin(wk(a + b)) in the key frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' This improves loss by 77% (to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='54 · 10−8), validating that the logits are calculated by trig identities of neurons as detailed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Logit frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Next, we ablate various components of the final logits in the Fourier space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' To do so, we take a 2D DFT on the 113 · 113 · 113 logit matrix over all 113 · 113 pairs of inputs to get the logits in the Fourier basis, then set various frequencies in this basis to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We begin by ablating the components corresponding to each of the key frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' As reported in Figure 6, ablating any key frequency causes a significant increase in loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' This confirms that the five frequencies identified in previous sections are indeed necessary components of the transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' In contrast, ablating other frequencies does not hurt the model at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 7 arXiv preprint 5 10 15 20 25 30 35 40 45 50 55 10n 1μ 100μ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='01 1 Restricted Original Other frequency Key frequency Loss after ablating frequencies Log Loss Figure 6: The loss of the transformer (lower=better) when ablating each frequency k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=', 56} and everything except for the five key frequencies (restricted loss).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We include the original unablated loss for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Ablating key frequencies causes a performance drop, while the other ablations do not harm performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We then ablate all 113 · 113 − 40 of the Fourier components besides key frequencies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' this ablation actually improves performance (loss drops 70% to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='24 · 10−8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Directions in WL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2, we found that WL is well approximated by the 10 directions corresponding to the cosine and sine of key frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' If we project the MLP activations to these 10 directions, loss decreases 50% to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='19 · 10−7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' If we instead projected the MLP activations onto the nullspace of these 10 directions, loss increases to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='27—worse than uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' This suggests that the network achieves low loss using these and only these 10 directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 5 UNDERSTANDING GROKKING BEHAVIOR USING PROGRESS MEASURES We now use our mechanistic understanding of the network to define two progress measures: metrics that can be computed during training that track the progress of the model over the course of training, including during phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' This allows us to better understand how the network reaches its final solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1 PROGRESS MEASURES We translate the ablations in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4 into two progress measures: restricted and excluded loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Restricted loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Since the final network uses a sparse set of frequencies wk, it makes sense to check how well intermediate versions of the model can do using only those frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' To measure this, we perform a 2D DFT on the logits to write them as a linear combination of waves in a and b, and set all terms besides the constant term and the 20 terms corresponding to cos(wk(a + b)) and sin(wk(a + b)) for the five key frequencies to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We then measure the loss of the ablated network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Excluded loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Instead of keeping the important frequencies wk, we next remove only those key frequencies from the logits but keep the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We measure this on the training data to track how much of the performance comes from Fourier multiplication versus memorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' The idea is that the memorizing solution should be spread out in the Fourier domain, so that ablating a few directions will leave it mostly unaffected, while the generalizing solution will be hurt significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Beyond these, we will also measure (1) the Gini coefficient (Hurley & Rickard, 2009) of the norms of the Fourier components of WE and WL, which measures the sparsity of WE and WL in the Fourier basis, and (2) the ℓ2-norm of the weights during training, since weight decay should push these down once the train loss is near zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2 PHASES OF GROKKING: MEMORIZATION, CIRCUIT FORMATION, AND CLEANUP Using the mainline model from Section 4, we plot the excluded loss, restricted loss, Gini coefficient of the matrices WU and WL, and sum of squared weights in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We find that training splits into three phases, which we call the memorization, circuit formation, and cleanup phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' (We show similar results for other models in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=') 8 arXiv preprint 0 5k 10k 15k 20k 25k 30k 1μ 100μ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='01 1 Train Loss Test Loss Excluded Loss Excluded Loss over All Frequencies Epoch Loss 0 5k 10k 15k 20k 25k 30k 10n 1μ 100μ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='01 1 Train loss Test loss Restricted loss Restricted Loss Epoch Loss 0 5k 10k 15k 20k 25k 30k 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='8 Gini Coefficients of Embed Matrix and Neuron-logit Map Epoch Gini coefficient 0 5k 10k 15k 20k 25k 30k 1000 1500 2000 2500 3000 3500 Total Sum of Squared Weights Epoch Sum of Squared Weights Figure 7: How each of the progress measures in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1 changes over the course of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' The lines delineate the 3 phases of training: memorization, circuit formation, and cleanup (and a final stable phase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' (Top Left) Excluded loss increases during circuit formation, while train and test loss remain flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' (Top Right) The restricted loss begins declining before test loss declines, but has an inflection point when grokking begins to occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' (Bottom Left) The Gini coefficient of the norms of the Fourier components of WE and WL increase sharply during cleanup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' (Bottom Right) The sums of squared weights decreases smoothly during circuit formation and more sharply during cleanup, indicating that both phases are linked to weight decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Memorization (Epochs 0k–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We first observe a decline of both excluded and train loss, with test and restricted loss both remaining high and the Gini coefficient staying relatively flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' In other words, the model memorizes the data, and the frequencies wk used by the final model are unused.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Circuit formation (Epochs 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4k–9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' In this phase, excluded loss rises, sum of squared weights falls, restricted loss starts to fall, and test and train loss stay flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' This suggests that the model’s behavior on the train set transitions smoothly from the memorizing solution to the Fourier multi- plication algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' The fall in the sum of squared weights suggests that circuit formation likely happens due to weight decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Notably, the circuit is formed well before grokking occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Cleanup (Epochs 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4k–14k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' In this phase, excluded loss plateaus, restricted loss continues to drop, test loss suddenly drops, and sum of squared weights sharply drops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' As the completed Fourier multiplication circuit both solves the task well and has lower weight than the memorization circuit, weight decay encourages the network to shed the memorized solution in favor of focusing on the Fourier multiplication circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' This is most cleanly shown in the sharp increase in the Gini coefficient for the matices WE and WL, which shows that the network is becoming sparser in the Fourier basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='3 GROKKING AND WEIGHT DECAY In the previous section, we saw that each phase of grokking corresponded to an inflection point in the ℓ2-norm of the weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' This suggests that weight decay is an important component of grokking and drives progress towards the generalizing solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' In Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1, we provide additional evidence that weight decay is necessary for grokking: smaller amounts of weight decay causes the network to take significantly longer to grok (echoing the results on toy models from Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' (2022)), and our networks do not grok on the modular arithmetic task without weight decay or some other form of regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' In Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2, we also find that the amount of data affects grokking: when networks are provided with enough data, there is no longer a gap between the train and test losses (instead, both decline sharply some number of epochs into training).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Finally, in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='3 we replicate these results on several additional algorithmic tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 9 arXiv preprint 6 CONCLUSION AND DISCUSSION In this work, we use mechanistic interpretability to define progress measures for small transformers trained on a modular addition task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We find that the transformers embed the input onto rotations in R2 and compose the rotations using trigonometric identities to compute a + b mod 113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Using our reverse-engineered algorithm, we define two progress measures, along which the network makes continuous progress toward the final algorithm prior to the grokking phase change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We see this work as a proof of concept for using mechanistic interpretability to understand emergent behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Larger models and realistic tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' In this work, we studied the behavior of small transformers on a simple algorithmic task, solved with a single circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' On the other hand, larger models use larger, more numerous circuits to solve significantly harder tasks (Cammarata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' The analysis reported in this work required significant amounts of manual effort, and our progress metrics are specific to small networks on one particular algorithmic task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Methods for automating the analysis and finding task-independent progress measures seem necessary to scale to other, larger models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We discuss possible scenarios for more realistic applications in Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Discovering phase change thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' While the progress measures we defined in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1 in- crease relatively smoothly before the phase transition (and suffice to allow us to understand grokking for this task) we lack a general notion of criticality that would allow us to predict when the phase transition will happen ex ante.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Future work should develop theory and practice in order to apply progress measures to predict the timing of emergent behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' REPRODUCIBILITY STATEMENT An annotated Colab notebook containing the code to replicate our results, includ- ing download instructions for model checkpoints, is available at https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='ly/ grokking-progress-measures-website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' AUTHOR CONTRIBUTIONS Neel Nanda was the primary research contributor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' He reverse engineered the weights of the mainline model to discover the Fourier multiplication algorithm and found the lines of evidence in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' He also discovered the restricted and excluded loss progress measures and that grokking in mainline model could be divided into three discrete phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Finally, he found the link between grokking, limited data, and phase transitions by exhibiting grokking in other settings with phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Lawrence Chan was invaluable to the framing and technical writing of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' In addition, he created the Gini coefficient progress measure and performed the analysis in the appendices exploring to what extent the results on the mainline model applied to the other small transformer models, including with other random seeds, architectures, prime moduli, and regularization methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Tom Lieberum contributed to the early stages of this work by creating a minimal setup of grokking with a 1L Transformer on the modular addition task with no LayerNorm and finding the surprising periodicity within the model’s internals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Jess Smith performed experiments exploring grokking with different random seeds, architectures, and other hyper-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Jacob Steinhardt helped clarify and distill the results, provided significant amounts of editing and writing feedback, and suggested the progress measure frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' ACKNOWLEDGMENTS In writing this paper,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' our thinking and exposition was greatly clarified by correspondence with and feedback from Oliver Balfour,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' David Bau,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Sid Black,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Nick Cammarata,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Stephen Casper,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Bilal Chughtai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Arthur Conmy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Xander Davies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Ben Edelman,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Nelson Elhage,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Ryan Greenblatt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Jacob Hilton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Evan Hubinger,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Zac Kenton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Janos Kramar,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Lauro Langosco,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Tao Lin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' David Lindner,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Eric Michaud,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Vlad Mikulik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Noa Nabeshima,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Chris Olah,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Michela Paganini,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Michela Paganini,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Alex 10 arXiv preprint Ray,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Rohin Shah,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Buck Shlegeris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Alex Silverstein,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Ben Toner,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Johannes Treutlein,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Nicholas Turner,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Vikrant Varma,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Vikrant Varma,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Kevin Wang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Martin Wattenberg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' John Wentworth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' and Jeff Wu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We’d also like to thank Adam Gleave and Chengcheng Tan for providing substantial editing help, and Noa Nabeshima and Vlad Mikulik for pair programming with Neel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' This work draws heavily on the interpretability techniques and framework developed by Elhage et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' (2021) and Olsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We trained our models using PyTorch (Paszke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=', 2019) and performed our data analysis using NumPy (Harris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=', 2020), Pandas (Wes McKinney, 2010), and einops (Rogozhnikov, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Our figures were made using Plotly (Plotly Technologies Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Neel would like to thank Jemima Jones for providing practical and emotional support as he navigated personal challenges while contributing to this paper, and to the Schelling Residency for providing an excellent research environment during the distillation stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' He would also like to thank the Anthropic interpretability team, most notably Chris Olah, for an incredibly generous amount of mentorship during his time there, without which this investigation would never have happened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} 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+page_content=' Vimal Thilak, Etai Littwin, Shuangfei Zhai, Omid Saremi, Roni Paiss, and Joshua Susskind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' The slingshot mechanism: An empirical study of adaptive optimizers and the grokking phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' arXiv preprint arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='04817, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Kevin Wang, Alexandre Variengien, Arthur Conmy, Buck Shlegeris, and Jacob Steinhardt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Inter- pretability in the wild: a circuit for indirect object identification in gpt-2 small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' arXiv preprint arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='00593, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Jason Wei, Yi Tay, Rishi Bommasani, Colin Raffel, Barret Zoph, Sebastian Borgeaud, Dani Yo- gatama, Maarten Bosma, Denny Zhou, Donald Metzler, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Emergent abilities of large language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' arXiv preprint arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='07682, 2022a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Jason Wei, Xuezhi Wang, Dale Schuurmans, Maarten Bosma, Ed Chi, Quoc Le, and Denny Zhou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Chain of thought prompting elicits reasoning in large language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' arXiv preprint arXiv:2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='11903, 2022b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Wes McKinney.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Data Structures for Statistical Computing in Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' In St´efan van der Walt and Jarrod Millman (eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' ), Proceedings of the 9th Python in Science Conference, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 56 – 61, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='25080/Majora-92bf1922-00a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 12 arXiv preprint A MATHEMATICAL STRUCTURE OF THE TRANSFORMER We follow the conventions and notation of Elhage et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' (2021) in describing our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Here, we briefly recap their notation and examine it in our specific case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We denote our hyperparameters as follows: dvocab = 113 is the size of the input and output spaces (treating ‘=’ separately), dmodel = 128 is the width of the residual stream (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' embedding size), dhead = 32 is the size of query, key and value vectors for a single attention head, and dmlp = 512 is the number of neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We denote the parameters as follows: WE (embedding layer);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Wpos (positional embedding);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' W j Q (queries), W j K (keys), W j V (values), W j O (attention output) (the 4 weight matrices of head j in the attention layer);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Win and bin for the input linear map of the MLP layer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Wout and bout for the output linear map of the MLP layer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' and WU (unembedding layer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Note that we do not have biases in our embedding, attention layer or unembedding, and we do not tie the matrices for the embedding/unembedding layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We now describe the mathematical structure of our network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Note that loss is only calculated from the logits on the final token, and information only moves between tokens during the attention layer, so our variables from the end of the attention layer onwards only refer to the final token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We use ti to denote the token in position i (as a one-hot encoded vector),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' pi to denote the ith positional embedding,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' x(0) i to denote the initial residual stream on token with index i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' A(i) to denote the attention scores from = to all previous tokens from head i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' x(1) to denote the residual stream after the attention layer on the final token,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' MLP to denote the neuron activations in the MLP layer on the final token,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' x(2) the final residual stream on the final token,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Logits the logits on the final token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' The logits are calculated via the following equations: x(0) i = WEti + pi Aj = softmax(x(0)T W jT K W j Qx(0) 2 ) x(1) = [ � j W j OW j V (x(0) · Aj)] + x(0) 2 MLP = ReLU(Winx(1)) x(2) = WoutN + x(1) = WoutReLU(Winx(1)) + x(1) Logits = WUx(2) As in Elhage et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' (2021), we refer to the term W j OW j V (x(0)) as the OV circuit for head j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1 EMPIRICAL MODEL SIMPLIFICATIONS We make two empirical observations: The attention paid from ‘=’ to itself is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' In practice, the average attention paid is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1% to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4% for each head, and ablating this does not affect model performance at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' The skip connection around the MLP layer is not important for the model’s computation and can be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Concretely, if we set it to zero or to its average (zero or mean ablation) then model accuracy is unchanged, and loss goes from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4 · 10−7 to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='12 · 10−7 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='25 · 10−7 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' This is a significant increase in loss, but from such a small baseline that we can still ignore it and reverse engineer the model’s computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' (That being said, both the attention heads and the skip connection around them are crucial to the functioning of the model: zero ablating attention heads increases loss to 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='3, while zero ablating the skip connection around the attention heads increases loss to 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1, both significantly worse than chance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=') A consequence of the first observation is that the attention is now a softmax over 2 elements, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' a sigmoid over the difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' And x(0) 2 is constant, as it is independent of x and y, and the embedding 13 arXiv preprint Figure 8: As discussed in Appendix B, while for every k ∈ [0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='P − 1], cos � 2kπ P x � achieves its maximum value (1) at x = 0 mod 113, it still has additional peaks at different values that are close to the maximum value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' However, by adding together cosine waves of the 5 keyfrequencies, the model constructs a periodic function where the value at x = 0 mod 113 is significantly larger than its value anywhere else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' and positional embedding of ‘=’ are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' So Aj 0 = σ � x(0) 2 T W jT Q WK(x(0) 0 − x(0) 1 ) � (and Aj 1 = 1 − Aj 0) A consequence of the second observation is that Logits ≈ WUWoutMLP, which we denote as WL = WUWout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' From the perspective of the network, WL is the meaningful matrix, not either of its constituents, since they compose linearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' B WHY USE CONSTRUCTIVE INTEREFERENCE?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' As demonstrated in Section 4 and Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1, small transformers trained on this task use several different frequencies which they add together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' The reason for this is to end up with a function whose value at x = 0 mod 113 is significantly larger than any other x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' For example, consider the function f14(x) = cos � 2π·14 113 x � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' This function has period 113 and is maximized at x = 0 mod 113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' However, other values of x cause this function to be close to 1: f14(8) = f14(105) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='998, f14(16) = f14(89) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='994, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Now consider f35(x) = cos � 2π·35 113 x � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' While this function also has period 113 and is maximized at x = 0 mod 113, it turns out that f35(8) = f35(105) = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' This means that by adding together f14 and f35, we end up with a function that is not close to 1 at x = 8 mod 113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Similarly, while f35(16) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='961, f52(16) = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='56, and so adding a third frequency reduces the peak at x = 16 mod 113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We show the constructive interference resulting from the cosine waves for the five frequencies used by the mainline model in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' C SUPPORTING EVIDENCE FOR MECHANISTIC ANALYSIS OF MODULAR ARITHMETIC NETWORKS C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1 FURTHER ANALYSIS OF THE SPECIFIC TRAINING RUN DISCUSSED IN THE PAPER In this section, we provide additional evidence relating to the mainline model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 14 Constructive Interference of Cosine Waves of Different Freguencies COS14 cos 35 cos 41 31 COS42 2 COS52 Sum 1 2 3 0 2 6 8arXiv preprint 0 50 100 100 80 60 40 20 0 0 50 100 0 50 100 0 50 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content="8 Attention patterns by Head ('=' to a) a a a a b Figure 9: Attention patterns for each head, from the ‘=’ token at the third sequence position to the a token at the first sequence position, as a heatmap over the inputs." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' All four attention heads exhibit striking periodicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Head k αj βj FVE 0 35 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='26 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='14 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='03% 1 42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='27 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='04 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='49% 2 52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='29 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='05 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='07% 3 42 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='04 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='91% Table 2: For each attention head, we show the pattern from ‘=’ to a is well approximated by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 + α(cos(wka)−cos(wkb))+β(sin(wka)−sin(wkb)) and give the coefficients and fraction of variance explained for this approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1 PERIODICITY IN THE ACTIVATIONS OF OTHER ATTENTION HEADS In Figure 9 we plot the attention patterns from the final token ‘=’ to the first token a for all 4 attention heads, as a heatmap over the inputs a and b, as this is a scalar for each head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We observe a striking periodicity and further that heads 1 and 3 represent the same frequency while heads 0 and 2 are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' As shown in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1, the attention paid from ‘=’ to itself is negligible, so Aj 0 = 1 − Aj 1 and it suffices to plot attention to a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2 THE ATTENTION PATTERN WEIGHTS ARE WELL APPROXIMATED BY DIFFERENCES OF SINES AND COSINES OF A SINGLE FREQUENCY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' The periodicity of the attention heads has a striking form—Aj 0 is well approximated by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 + αj(cos(wka) − cos(wkb)) + βj(sin(wka) − sin(wkb)), for some frequency wk and constants αj and βj (which may differ for each head).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Note further that this simplifies to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 + γ(cos(wk(a + θ))−cos(wk(b+θ))) for some constants γ and θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We show the coefficients and fraction of variance explained in Table 1 Mechanistic Analysis of Attention Patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We can further mechanistically analyse how the model achieves this form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' The following is a high-level sketch of what is going on: First, note that the attention score on position 0 and head j is just a lookup table on the input token a (of size P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' To see why, note that Aj 0 = mx(0) 0 T W jT K W j Qx(0) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' x(0) 2 is constant since the token is always ‘=’ and x(0) 0 = WEt0 + p0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' So this reduces to t0 · Cj + D for some constant vector Cj = W T E W jT K W j Qx(0) 2 ∈ Rp and some scalar D = pT 0 W j K T W j Qx(0) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' As t0 is one-hot encoded, this is just a lookup table, which we may instead denote as Cj[a] Next, note that the attention pattern from =→ 0 is σ(Cj[a] − Cj[b]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' As argued in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1, the attention paid =→= is negligible and can be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' So the softmax reduces to a softmax over two elements, which is a sigmoid on their difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' As form of Cj does not mention the token index or value, it is the same for position 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We now show that Cj is well-approximated by a wave of frequency wkj for some integer kj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' That is, Cj[a] ≈ Fj cos(wkja)+Gj sin(wkja).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We do this by simply computing Cj and fitting the constants ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='arXiv preprint ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='sin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='cos ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='Fourier components for C 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='Frequency k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='Coefficient of Fourier Component ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='−8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='−6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='−4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='sin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='cos ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='Fourier components for C 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='Frequency k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='Coefficient of Fourier Component ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='−10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='−8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='−6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='−4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='sin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='cos ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='Fourier components for C 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='Frequency k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='Coefficient of Fourier Component ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='sin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='cos ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='Fourier components for C 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='Frequency k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='Coefficient of Fourier Component ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='Figure 10: We plot the attention pattern weights Cj in the Fourier basis for each of the four heads ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='j ∈ {0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We observe significant sparsity, with almost all of each term being associated with a single frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Fj and Gj to minimize ℓ2 loss, and display the resulting coefficients for each head in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' This fit explain 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='02%, 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='21%, 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='10%, 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='42% of the variance of Cj respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Interestingly, the coefficients of heads 1 and 3 are almost exactly the opposite of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' For each head j, σ(Cj[a] − Cj[b]) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 + Ej(Cj[a] − Cj[b]) for some constant Ej—that is, the sigmoid has some linear approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' (The intercept will be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 by symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=') The striking thing is that, because the inputs to the sigmoid for the attention heads are over a fairly wide range ([−5, 5] roughly), the linear approximation to the sigmoid is a fairly good fit, explaining 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5% of the variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We validate that this is all that is going on, by replacing the sigmoid with the best linear fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' This improves performance, decreasing test loss from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='41 · 10−7 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='12 · 10−7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' By properties of sinusoidal functions, the attention patterns of each head will be well approximated by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 ± Cj(cos(wkj(a + θj)) − cos(wkj(b + θj))) - the softmax is linear, with an intercept of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5, and the weights Cj map each token to a score that is a wave in a single frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' This exactly gives us the periodic form shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Finally, for each head j, we plot the output of the OV circuit W j OW j V x(0) in the Fourier basis and display the results in Figure 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' The largest component of each head corresponding to the frequency of the attention pattern Cj, with heads 0 and 2 being almost entirely composed of a sines and cosines of a single frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' On the other hand, the norms for the components of heads 1 and 3 are almost exactly the same, and contain all five key frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' As the coefficients of the attention pattern weights have the opposite non-constant components (Table 2, Figure 10), their attention scores sum almost exactly to 1 across all inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' This implies that heads 1 and 3 are used to output the first order terms sin (wk) , cos (wk) in the five key frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We speculate that this is because of weight decay encouraging the embeddings WE to be small, causing the network to allocate two of its attention heads to effectively increasing the size of WE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Bringing it all together, this implies that attention heads 0 and 2 are approximately computing a degree 2 polynomial of cosines and sines of a single frequency each, while heads 1 and 3 amplify the key frequencies in the residual stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 16 arXiv preprint 0 10 20 30 40 50 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 3 cos sin Fourier components of OV circuit for head 0 Frequency k Norm of Fourier Component 0 10 20 30 40 50 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 4 cos sin Fourier components of OV circuit for head 1 Frequency k Norm of Fourier Component 0 10 20 30 40 50 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 cos sin Fourier components of OV circuit for head 2 Frequency k Norm of Fourier Component 0 10 20 30 40 50 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 cos sin Fourier components of OV circuit for head 3 Frequency k Norm of Fourier Component Figure 11: We plot the output of the OV circuit W j OW j V x(0) in the Fourier basis for each of the four heads j ∈ {0, 1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' As with the attention pattern weights Cj in Figure 10, we observe that the only components with significant norm are those corresponding to key frequencies, and that the largest component corresponds to the frequencies of the attention patterns of the attention heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' As attention pattern of heads 1 and 3 are sum to one, but their OV circuits are almost exactly the same and consist of all five key frequencies, this implies that heads 1 and 3 are used to increase the magnitude of key frequencies in the residual stream (Section C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 0 50 100 100 80 60 40 20 0 0 50 100 0 50 100 0 50 100 0 1 2 3 4 Neuron Activations for Additional Neurons a a a a b Figure 12: Plots of neuron activations for MLP neurons 1, 2, 3 and 4, for inputs a, b ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=', 112}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' As with Neuron 0, all of the activation patterns are periodic in both inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 17 海arXiv preprint 0 10k 20k 30k 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='8 1 Train Accuracy Test Accuracy Restricted Accuracy (Train) Restricted Accuracy (Test) Pure Restricted Accuracy Epoch Figure 13: Accuracy when restricting Fourier Components to the five key frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' As with restricted loss, this shows that the model figures out how to generalize modulo deleting noise before it removes the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 0 5k 10k 15k 20k 25k 30k 0 5 10 15 20 25 30 Freq 14 35 41 42 52 Coefficients of cos(w k (a+b-c)) in the Logits Epoch Figure 14: The coefficients of cos(w(a + b − c)) in the logits over the model’s training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' As with the metrics in the paper, this shows a nice interpolation and growth of each cosine term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='3 PERIODICITY IN THE ACTIVATIONS OF ADDITIONAL NEURONS In Figure 12, we display the activations of four more MLP neurons, as a function of the inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' As with neuron 0, the activations of these neurons are also periodic in the inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4 ADDITIONAL GROKKING FIGURES FOR MAINLINE RUN In Figure 13, we display the accuracy of the model when restricting the model to use only the five key frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' As with restricted loss, this improves model performance during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' In Figure 14, we show the coefficients of the five key frequencies in the logits, calculated by regress- ing the logits against the five cos (wk(a + b − c)) terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' In Figure 15, we plot the excluded loss if we exclude each of the five key frequencies (as opposed to all five key frequencies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' All three of these figures have inflection points corresponding to the relevant phases of grokking, discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 18 arXiv preprint 0 5k 10k 15k 20k 25k 30k 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='8 1 Train accuracy Test Accuracy k=14 k=35 k=41 k=42 k=52 Accuracy when Excluding Key Frequencies Epoch Accuracy 0 5k 10k 15k 20k 25k 30k 1μ 100μ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='01 1 Train loss Test Loss k=14 k=35 k=41 k=42 k=52 Loss when Excluding Key Frequencies Epoch Loss Figure 15: The excluded accuracy (left) and loss (right) if we exclude each of the five key frequencies for our mainline model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' As with the excluded loss results in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1, this shows that the model interpolates between memorising and generalising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2 ADDITIONAL RESULTS FROM DIFFERENT RUNS In this section, we plot relevant figures from other runs, either with the same architecture (Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1) or with different architectures or experimental setups (Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Note that in general, while all models learn to use variants of the modular arithmetic algorithm, they use a varying number of different key frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' In order to find the key frequencies to calculate the excluded and restricted loss, we perform a DFT on the neuron-logit map WL, then take the frequencies with nontrivial coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='3 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1 ADDITIONAL RESULTS FOR DIFFERENT RUNS WITH THE SAME ARCHITECTURE In this section, we provide evidence that all 4 other runs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=', random seeds) using the experimental setup of our mainline model also use the Fourier multiplication algorithm, and then confirm that the same phases of grokking also occur on these runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Confirming that the other seeds use the Fourier Multiplication Algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' In Figure 16, we show the norms of the Fourier components of the embedding matrix WE for each of the 4 other random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' As with the mainline model, the matrices are sparse in the Fourier basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' In Figure 17, we show the norms of the Fourier components of the neuron-logit map WL for the 4 other random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' The matrices are sparse in the Fourier basis, enabling us to identify 3 or 4 key frequencies for each of the seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Again, note that these are different frequencies per seed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Using the key frequencies identified in the neuron-logit map, we repeat the experiment in Sec- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2, where we “read off” the MLP activations in the 6 or 8 directions corresponding to the key frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' As with our mainline model, this lets us identify the trigonometric identities for cos (wk(a + b)) and sin (wk(a + b)) being computed at the MLP layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We confirm that the trigono- metric identities are a good approximation by approximating the activations with a single term of the form cos (wk(a + b)) or sin (wk(a + b))—as with the mainline model, the fraction of variance explained is consistently close to 100%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Next, we ablate the key frequencies from the logits as in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4 and report the results in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' As with the mainline model, ablating all of the key frequencies reduces performance to worse than chance, while ablating everything but the key frequencies improves test performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Progress measures and grokking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Finally, we confirm the progress measure and grokking results from the mainline model on other runs with the same architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' In Figure 18, we display the train, test, and restricted loss for each of the four other random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' In Figure 19, we display the Gini coefficients of the Fourier components of the embedding matrix WE and the neuron-logit map WL for each of the four other random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' The shape of the curves are very similar to those of the mainline model, allowing us to divide grokking on these models into the same three phases identified in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Interestingly, while all of the models complete memorization by around 1400 epochs, circuit formation and cleanup occur at different times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 3One method for getting a general (model-independent) progress measure for this task is to compute the excluded loss for each of the 56 unique frequencies and then take the max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We omit the plots for this variant of the excluded loss as they are broadly similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 19 arXiv preprint 0 10 20 30 40 50 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 cos sin Embedding Matrix (Seed 1) Frequency k Norm of Fourier Component 0 10 20 30 40 50 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 cos sin Embedding Matrix (Seed 2) Frequency k Norm of Fourier Component 0 10 20 30 40 50 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 cos sin Embedding Matrix (Seed 3) Frequency k Norm of Fourier Component 0 10 20 30 40 50 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 cos sin Embedding Matrix (Seed 4) Frequency k Norm of Fourier Component Figure 16: The norms of the Fourier components in the embedding matrix WE for each of four other random seeds for the original (1 layer) architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' As discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1 and Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1, the sparsity of WE in the Fourier basis is evidence that the network is operating in a Fourier basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 0 10 20 30 40 50 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 cos sin Neuron-Logit Map (Seed 1) Frequency k Norm of Fourier Component 0 10 20 30 40 50 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 3 cos sin Neuron-Logit Map (Seed 2) Frequency k Norm of Fourier Component 0 10 20 30 40 50 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 3 cos sin Neuron-Logit Map (Seed 3) Frequency k Norm of Fourier Component 0 10 20 30 40 50 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 cos sin Neuron-Logit Map (Seed 4) Frequency k Norm of Fourier Component Figure 17: The norms of the direction corresponding to sine and cosine waves in the neuron-logit map weights WL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' As with the mainline model discussed in the main body and discussed in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1, WL is consistently sparse, providing is evidence that all four are operating in a Fourier basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 20 arXiv preprint WL Component Fourier components of uT k MLP(a, b) or vT k MLP(a, b) FVE cos (w2c) 147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4 cos (w2a) cos (w2b) − 145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='8 sin (w2a) sin (w2b) ≈ 146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='6 cos (w2(a + b)) 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2% sin (w2c) 145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 cos (w2a) sin (w2b) + 145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='6 sin (w2a) cos (w2b) ≈ 145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 sin (w2(a + b)) 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1% cos (w9c) 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='3 cos (w9a) cos (w9b) − 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='0 sin (w9a) sin (w9b) ≈ 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='6 cos (w9(a + b)) 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4% sin (w9c) 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='6 cos (w9a) sin (w9b) + 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 sin (w9a) cos (w9b) ≈ 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 sin (w9(a + b)) 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='7% cos (w19c) 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='0 cos (w19a) cos (w19b) − 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='3 sin (w19a) sin (w19b) ≈ 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2 cos (w19(a + b)) 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4% sin (w19c) 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='3 cos (w19a) sin (w19b) + 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4 sin (w19a) cos (w19b) ≈ 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4 sin (w19(a + b)) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='9% cos (w31c) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4 cos (w31a) cos (w31b) − 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4 sin (w31a) sin (w31b) ≈ 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4 cos (w31(a + b)) 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4% sin (w31c) 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2 cos (w31a) sin (w31b) + 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1 sin (w31a) cos (w31b) ≈ 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2 sin (w31(a + b)) 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='7% (a) Seed 1 WL Component Fourier components of uT k MLP(a, b) or vT k MLP(a, b) FVE cos (w40c) 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='0 cos (w40a) cos (w40b) − 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4 sin (w40a) sin (w40b) ≈ 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2 cos (w40(a + b)) 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='3% sin (w40c) 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='3 cos (w40a) sin (w40b) + 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='3 sin (w40a) cos (w40b) ≈ 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='3 sin (w40(a + b)) 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='7% cos (w44c) 309.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1 cos (w44a) cos (w44b) − 338.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='7 sin (w44a) sin (w44b) ≈ 323.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='9 cos (w44(a + b)) 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5% sin (w44c) 327.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='3 cos (w44a) sin (w44b) + 327.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2 sin (w44a) cos (w44b) ≈ 327.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='3 sin (w44(a + b)) 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='9% cos (w53c) 192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1 cos (w53a) cos (w53b) − 192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2 sin (w53a) sin (w53b) ≈ 192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1 cos (w53(a + b)) 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='3% sin (w53c) 166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='7 cos (w53a) sin (w53b) + 166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='8 sin (w53a) cos (w53b) ≈ 166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='8 sin (w53(a + b)) 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='7% (b) Seed 2 WL Component Fourier components of uT k MLP(a, b) or vT k MLP(a, b) FVE cos (w31c) 156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1 cos (w31a) cos (w31b) − 156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 sin (w31a) sin (w31b) ≈ 156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='3 cos (w31(a + b)) 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='3% sin (w31c) 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='7 cos (w31a) sin (w31b) + 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='7 sin (w31a) cos (w31b) ≈ 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='7 sin (w31(a + b)) 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='9% cos (w45c) 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 cos (w45a) cos (w45b) − 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='8 sin (w45a) sin (w45b) ≈ 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='6 cos (w45(a + b)) 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='9% sin (w45c) 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='7 cos (w45a) sin (w45b) + 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='6 sin (w45a) cos (w45b) ≈ 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='6 sin (w45(a + b)) 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='6% cos (w49c) 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='9 cos (w49a) cos (w49b) − 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 sin (w49a) sin (w49b) ≈ 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='7 cos (w49(a + b)) 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='0% sin (w49c) 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='8 cos (w49a) sin (w49b) + 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='8 sin (w49a) cos (w49b) ≈ 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='8 sin (w49(a + b)) 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='9% cos (w52c) 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='6 cos (w52a) cos (w52b) − 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1 sin (w52a) sin (w52b) ≈ 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='9 cos (w52(a + b)) 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5% sin (w52c) 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='7 cos (w52a) sin (w52b) + 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='7 sin (w52a) cos (w52b) ≈ 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='7 sin (w52(a + b)) 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='9% (c) Seed 3 WL Component Fourier components of uT k MLP(a, b) or vT k MLP(a, b) FVE cos (w17c) 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='0 cos (w17a) cos (w17b) − 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 sin (w17a) sin (w17b) ≈ 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='8 cos (w17(a + b)) 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4% sin (w17c) 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4 cos (w17a) sin (w17b) + 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4 sin (w17a) cos (w17b) ≈ 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4 sin (w17(a + b)) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='9% cos (w32c) 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='7 cos (w32a) cos (w32b) − 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4 sin (w32a) sin (w32b) ≈ 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 cos (w32(a + b)) 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2% sin (w32c) 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='0 cos (w32a) sin (w32b) + 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='0 sin (w32a) cos (w32b) ≈ 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='0 sin (w32(a + b)) 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='3% cos (w42c) 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4 cos (w42a) cos (w42b) − 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='0 sin (w42a) sin (w42b) ≈ 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2 cos (w42(a + b)) 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='9% sin (w42c) 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2 cos (w42a) sin (w42b) + 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1 sin (w42a) cos (w42b) ≈ 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1 sin (w42(a + b)) 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='6% cos (w51c) 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='0 cos (w51a) cos (w51b) − 116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2 sin (w51a) sin (w51b) ≈ 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1 cos (w51(a + b)) 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='0% sin (w51c) 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='3 cos (w51a) sin (w51b) + 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2 sin (w51a) cos (w51b) ≈ 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2 sin (w51(a + b)) 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5% (d) Seed 4 Table 3: For each of the directions in the neuron-logit map WL of the final models from 4 other ran- dom seeds (Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1), we project the MLP activations in that direction then perform a Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' For brevity, we omit terms with coefficients less than 15% of the largest coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We then compute the fraction of variance explained (FVE) if we replace the projection with a multiple of a single term of the form cos (wk(a + b)) or sin (wk(a + b)), and find that this is consistently close to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Seed Test Loss Loss (Key frequencies removed) Loss (All other frequencies removed) 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='07 · 10−7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 · 100 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='7 · 10−8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1 · 10−7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1 · 101 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2 · 10−8 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='05 · 10−7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='7 · 100 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 · 10−8 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='33 · 10−7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='8 · 100 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='0 · 10−8 Table 4: As discussed in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1, ablating the key frequencies for each of the networks re- duces performance to worse than chance, while ablating all other frequencies improves performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 21 arXiv preprint 0 5k 10k 15k 20k 25k 10n 1μ 100μ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='01 1 Train loss Test loss Restricted loss Restricted Loss for Seed 1 Epoch Loss 0 5k 10k 15k 20k 25k 1μ 100μ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='01 1 Train loss Test loss Restricted loss Restricted Loss for Seed 2 Epoch Loss 0 5k 10k 15k 20k 25k 10n 1μ 100μ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='01 1 Train loss Test loss Restricted loss Restricted Loss for Seed 3 Epoch Loss 0 5k 10k 15k 20k 25k 30k 1μ 100μ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='01 1 Train loss Test loss Restricted loss Restricted Loss for Seed 4 Epoch Loss Figure 18: The train, test, and restricted loss for each of the four other random seeds described in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' The lines delineate the 3 phases of training: memorization, circuit formation, and cleanup (and a final stable phase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' As with the mainline model, restricted loss consistently declines prior to train loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Note that while the shapes of the loss curves are similar to each other and those of the mainline model, the exact time that grokking occurs (and thus the dividers between the phases of grokking) differ by random seed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Interestingly, memorization is complete by around 1400 steps for all five runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 22 arXiv preprint 0 5k 10k 15k 20k 25k 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='8 Gini Coefficients, Seed 1 Epoch Gini coefficient 0 5k 10k 15k 20k 25k 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='8 Gini Coefficients, Seed 2 Epoch Gini coefficient 0 5k 10k 15k 20k 25k 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='8 Gini Coefficients, Seed 3 Epoch Gini coefficient 0 5k 10k 15k 20k 25k 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='8 Gini Coefficients, Seed 4 Epoch Gini coefficient Figure 19: The Gini coefficients (a measure of sparsity) of the Fourier components of the embedding matrix WE and the neuron-logit map WL for each of the four other random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' The lines delineate the 3 phases of training: memorization, circuit formation, and cleanup (and a final stable phase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' As with the mainline model, sparsity increases slowly during memorization and circuit formation, and then quickly during cleanup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2 RESULTS FOR OTHER EXPERIMENTAL SETUPS In this section, we provide further evidence that small transformers grok on the modular addition task, by varying the size of the network, the amount of training data, and the size of the prime P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 1-Layer Transformers with Varying Fractions of Training Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We find that grokking occurs for the modular addition task with P = 113 for many data fractions (that is, the fraction of the 113 · 113 pairs of inputs that the model sees during training), as shown in Figure 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Smaller amount lead to slower grokking, but sufficiently large fractions of data (≥ 60%) lead to immediate generalization, as shown in Figures 20 and 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' As with the results in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1, all of the 1-layer transformers in this section also converge to using the Fourier multiplication algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 2-Layer Transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' As shown in Figure 22, 2-layer transformers also exhibit some degree of grokking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' However, this is complicated by the slingshot mechanism (Thilak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We display the excluded loss of a 2-layer transformer in Figure 23 and find it shows a similar pattern to the mainline 1-layer transformer, in that it improves relatively smoothly before grokking occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Smaller and larger primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We also examined smaller and larger prime moduli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' For P = 53 (Figure 24), we explored a variety of weight decays to observe grokking in the small prime case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' With the original weight decay setting of λ = 1, we found that the models never generalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' However, increasing the weight decay to λ = 5 does allow the model to grok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We speculate that this is because the memorization solution is significantly smaller (since there are only 53 · 53 total pairs), thereby requiring more aggressive weight decay for the generalizing solution to be favored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' For P = 109, we saw exactly the same behavior as with the mainline model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' For P = 401 (Figure 25), we could not get grokking, even by varying the weight decay parame- ter λ ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5, 1, 3, 5, 8}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Instead, the model immediately learns the generalizing solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We 23 arXiv preprint 0 5k 10k 15k 20k 1μ 100μ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='01 1 100 Train loss Test loss Data Fraction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1 Epoch Loss 0 5k 10k 15k 20k 1μ 100μ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='01 1 Train loss Test loss Data Fraction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2 Epoch Loss 0 5k 10k 15k 20k 1μ 100μ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='01 1 Train loss Test loss Data Fraction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='3 Epoch Loss 0 1000 2000 3000 4000 1μ 100μ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='01 1 Train loss Test loss Data Fraction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4 Epoch Loss 0 1000 2000 3000 4000 1μ 100μ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='01 1 Train loss Test loss Data Fraction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 Epoch Loss 0 1000 2000 3000 4000 1μ 100μ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='01 1 Train loss Test loss Data Fraction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='6 Epoch Loss 0 1000 2000 3000 4000 1μ 100μ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='01 1 Train loss Test loss Data Fraction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='7 Epoch Loss 0 1000 2000 3000 4000 1μ 100μ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='01 1 Train loss Test loss Data Fraction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='8 Epoch Loss 0 1000 2000 3000 4000 1μ 100μ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='01 1 Train loss Test loss Data Fraction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='9 Epoch Loss Figure 20: Training and test losses for a 1-layer transformer on the modular addition task with P = 113, with varying fractions of the 113 · 113 pairs of possible inputs used in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Grokking occurs when between 30 − 50% of the dataset is used during training and lower fractions of data lead to slower grokking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Using ≥ 60% data leads to immediate generalization, while using 10% or 20% of the data doesn’t lead to grokking even after 40k epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Note the different x-axes: we only show 5k epochs for the runs with data fraction ≥ 40% for more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='9 2k 4k 6k 8k 10k 12k 14k 16k 18k Train loss Test loss Training Epochs until <1e-06 Loss Fraction of train data Number of steps Figure 21: Number of steps for train/test loss to be < 10−6, as a function of the amount of training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' While train loss immediately converges to below 10−6 for all data fractions, generalization takes significantly longer with lower fractions of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Note that the plots for other thresholds are also qualitatively similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 24 arXiv preprint Figure 22: Training and test loss for a 2-layer version of the original architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Average across 5 random seeds is in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Figure 23: Training, test, and full excluded loss for a 2-layer version of the original architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' One random seed chosen for readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Figure 24: The training and test losses for P = 53 and all other hyperparameters except weight decay (γ = 5) the same as the main training run discussed in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' The averages are bold, and all contributing runs are partially transparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Note that grokking occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 25 AverageTrain Loss Average TestLoss Loss 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='01 Log 100u 1μ 10n 0 5k 10k 15k 20k EpochExcluded Loss Over All Frequencies 100 Excluded Loss Train Loss TestLoss 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='01 LOSS 100μ 1μ 10n 0 5k 10k 15k 20k 25k 30k 35k Epoch-Average Train Loss Average Test Loss Log Loss 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='01 100μ 1μ 0 5k 10k 15k 20k EpocharXiv preprint Figure 25: The training and test losses for P = 401 and all other hyperparameters the same as the main training run discussed in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Grokking doesn’t occur (the model generalizes immedi- ately), even across a variety of weight decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' believe this is because the amount of data seen by the model is greatly increased compared to the P = 113 case (from 30% of 113 · 113 pairs to 30% of 401 · 401 pairs), thereby favoring the gen- eralizing solution from the start.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We then trained 3 models each using 5%, 10%, 20% of the pairs of training data with λ = 1, and found that the models trained on 5% and 10% of the data imme- diately overfit and never generalized, while the models trained on 20% of the data also generalized immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='3 GENERALIZING MODELS CONSISTENTLY USE THE FOURIER MULTIPLICATION ALGORITHM For each of the models in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2 that achieve low test loss, we repeated the analysis per- formed in the mainline model, and summarize the results in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We list their key frequencies, Gini coefficients, and relevant FVEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We find that every model trained with weight decay and that generalizes correctly implements some variation of the Fourier multiplication algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Interestingly, the embedding and unembedding matrices of the models trained with dropout are not sparse in the Fourier basis, and the logits for the p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2 models are not as well explained by a sum of cosines as the other models (likely because the p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2 models are simply worse at the task).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We speculate that this is likely due to a combination of insufficient training epochs (as dropout models seem to take much longer to grok) and the inherent need for redundancy for networks trained via dropout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' As with the mainline model, we ignore the final skip connection (around the final MLP), as all of the generalizing models studied do not suffer significant performance penalties if the skip connection is zero or mean ablated (Table 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' D ADDITIONAL RESULTS ON GROKKING D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1 BOTH REGULARIZATION AND LIMITED DATA ARE NECESSARY FOR GROKKING As discussed in Section 7 and Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2, the weight decay and the amount of data seem to have a strong effect on whether grokking occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' To confirm this, we experiment with removing weight decay and varying the amount of data on 1-layer transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' In Figure 26, we give the training, test, and full excluded loss for a typical training run with λ = 0 (no weight decay).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' As the figure shows, no grokking occurs, and excluded loss does not increase, suggesting that the model does not form the circuit for generalizing algorithm at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 26 Train Loss Tost Loss Log Loss 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='01 100μ 1μ 0 5k 10k 15k 20k EpocharXiv preprint Model Test Loss Gini(WE) Gini(WL) Key Frequencies Logit FVE MLP FVE 40% Training Data 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='98 · 10−7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='79 [17, 43, 49, 55] 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='9% 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='3% [26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1%] 50% Training Data 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='68 · 10−7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='77 [2, 17, 31, 41, 44] 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2% 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2% [28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2%] 60% Training Data 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='23 · 10−7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='84 [2, 23, 34, 51] 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4% 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='7% [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4%] 70% Training Data 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='85 · 10−8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='91 [14, 15, 26] 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='0% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='9% [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4%] 80% Training Data 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='83 · 10−7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='80 [38, 41] 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='9% 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1% [2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5%] 90% Training Data 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='11 · 10−7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='88 [3, 26, 34, 43] 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='6% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='7% [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='3%] 2 Layer Transformer 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='54 · 10−7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='59 0.' metadata={'source': 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35, 49] 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1% 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2% [6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4%] 2 Layer Transformer 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='50 · 10−2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='80 [4, 9, 28] 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5% 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4% [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='9%] 2 Layer Transformer 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='18 · 10−2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='76 [4, 5, 15, 54] 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4% 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2% [17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='8%] 2 Layer Transformer 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='75 · 10−2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='71 [3, 4, 13, 30, 38] 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='0% 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='9% [19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5%] P = 53 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='00 · 10−4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='68 [6, 9, 16, 21] 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2% 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2% [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='8%] P = 53 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='03 · 10−4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='72 [4, 13, 16] 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='8% 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1% [6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4%] P = 53 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='21 · 10−5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='79 [13, 22, 23] 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2% 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='6% [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='9%] P = 53 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='95 · 10−6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='74 [3, 14, 15] 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5% 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='8% [4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='6%] P = 53 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='56 · 10−6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='80 [10, 14, 22] 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='3% [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='6%] P = 109 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='02 · 10−7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='83 [6, 7, 22, 25] 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='0% 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='3% [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='9%] P = 109 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='95 · 10−7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='82 [8, 14, 29, 32, 41] 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2% 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='7% [2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='3%] P = 109 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='66 · 10−7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='86 [13, 23, 39, 45] 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5% 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='6% [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='9%] P = 109 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='50 · 10−7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='82 [8, 13, 32, 41] 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='8% 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5% [2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='3%] P = 109 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='77 · 10−7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='85 [29, 37, 38, 49] 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='9% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1% [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='8%] Dropout p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='65 · 10−1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='46 [1, 4, 7, 17, 22, 33, 40, 49, 55] 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='3% 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='0% [17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5%] Dropout p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='52 · 10−1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='46 [3, 8, 19, 28, 32, 34, 40, 44] 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='3% 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4% [10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='7%] Dropout p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='03 · 10−1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='45 [4, 5, 32, 38, 41, 44, 49, 50] 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2% 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1% [10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='6%] Dropout p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 < 10−8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='56 [1, 4, 26, 46, 47, 55] 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4% 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='9% [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5%] Dropout p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='01 · 10−2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='49 [16, 21, 35, 47, 53] 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4% 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='4% [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='0%] Dropout p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 < 10−8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='54 [1, 4, 7, 19, 29, 31, 42] 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1% 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='6% [4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='0%] Table 5: For each of the models in Appendices C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='3 and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1 that generalizes to test data, we report the test loss, the Gini coefficients of the norms of the Fourier components of WE and WL (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1), the key frequencies of the network, and the fraction of variance in logits explained by a weighted sum of cos (wk(a + b − c))s over the key frequencies (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' In addition, we find the components uk, vk of WL that correspond to cosines and sines of the key frequencies, and then report the average fraction of variance of uT k MLP(a, b) and vT k MLP(a, b) explained by a single term of form cos (wk(a + b)) or sin (wk(a + b)) respectively (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Numbers in square brackets represent the standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' For 2 Layer models, we use the final layer MLP activations for MLP(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We omit test accuracy because every model on this list except for the dropout p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2 mod- els achieves > 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='95% test accuracy, while the dropout p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2 models achieve around 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='6% test accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Model Type Loss Accuracy Ablated Loss Ablated Acuracy Varying Data Fraction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='83 · 10−7 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='65 · 10−7) 100% 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='74 · 10−7 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='74 · 10−7) 100% 2 Layer Transformer 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='97 · 10−2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='41 · 10−2 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='6% 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='63 · 10−2 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='72 · 10−2 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='7% P = 53 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='96 · 10−5 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='91 · 10−5) 100% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 · 10−4 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='70 · 10−4) 100% P = 109 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='94 · 10−7 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='74 · 10−8) 100% 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='53 · 10−7 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='41 · 10−7) 100% Dropout p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='215 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='091) 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='7% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='205 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='075) 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='7% Dropout p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='68 · 10−3 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='11 · 10−3) 100% 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='6 · 10−3 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='82 · 10−3) 100% Table 6: We confirm that the skip connection around the final MLP layer is not important for perfor- mance by mean ablating the skip connection and computing loss and accuracy over the entire dataset for each problem, averaged over all runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' (We report the standard deviation of loss over the runs in parentheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=') While loss does increase a small amount, accuracy remains consistently high and the loss of the ablated model remains low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Results with zero ablations are also similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 27 arXiv preprint Figure 26: Training, test, and full excluded loss for a 1-layer version of the original architecture without weight decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' One random seed chosen for readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Note that not having weight decay prevents grokking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 0 5k 10k 15k 20k 25k 10n 1μ 100μ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='01 1 Weight Decay 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='3 Epoch Log Loss 0 5k 10k 15k 20k 25k 1μ 10μ 100μ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1 1 10 Average Train Loss Average Test Loss Weight Decay 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='0 Epoch Log Loss Loading [MathJax]/extensions/MathMenu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='js Figure 27: The train and test loss over the course of training with weight decay λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='3 (left) and λ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='0 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Less aggressive weight decay leads to slower grokking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' In Figure 20, we show the test loss curves for models trained with weight decay λ = 1 and on various fractions of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Though all the train losses are approximately the same—that is, they memorize at the same rate, models trained on smaller fractions of data take longer to grok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' In Figure 27, we display the test and train loss of models trained with λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='3 and λ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Smaller amounts of weight decay lead to slower grokking, while larger amounts of weight decay lead to faster grokking—on average, it takes around 3k epochs for models to grok with weight decay λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='3, 5-10k epochs for the models to grok with weight decay λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='0, and 20k epochs for the models to grok with weight decay λ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Finally, we test whether other forms of regularization can also induce grokking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We replaced weight decay with the following types of regularization while keeping all other hyperpameters the same: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Dropout We add dropout Srivastava et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' (2014) to the MLP neurons, with p ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='8}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' That is, for each individual neuron, we set it to 0 with probability p during training, and also multiply the outputs of the other neurons by 1 1−p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' ℓ1 Regularization We add an ℓ1 penalty to the loss term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We use λ ∈ {1, 10, 100}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Note that we do not decouple the updates with respect to the ℓ1 penalty from optimization steps done with respect to the log loss (as is done for ℓ2 regularization via AdamW Loshchilov & Hutter (2017)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' In each case, we ran three random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We show the results in Figure 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' While grokking did not occur with ℓ1 regularization, we found that it does occur for all three seeds using dropout with p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2 or p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We speculate that this is because both dropout and weight decay encourage the network to spread out computation (which is required for the Fourier multiplication algorithm), while ℓ1 regularization encourages the network to become more sparse in the neuron basis and thus 28 Excluded Loss Over All Frequencies 100 Train Loss Test Loss ExcludedLoss 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='01 Loss 100μ 1μ 10n 100p 0 5k 10k 15k 20k 25k 30k 35k EpocharXiv preprint 0 10k 20k 30k 40k 1μ 10μ 100μ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1 1 10 100 Average Train Loss Average Test Loss Dropout p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2 Epoch Log Loss 0 10k 20k 30k 40k 100p 10n 1μ 100μ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='01 1 100 Average Train Loss Average Test Loss L1 penalty, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='0 Epoch Log Loss 0 10k 20k 30k 40k 1μ 10μ 100μ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1 1 10 100 Average Train Loss Average Test Loss Dropout p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 Epoch Log Loss 0 10k 20k 30k 40k 100p 10n 1μ 100μ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='01 1 100 Average Train Loss Average Test Loss L1 penalty, 10 Epoch Log Loss 0 10k 20k 30k 40k 1μ 10μ 100μ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1 1 10 100 Average Train Loss Average Test Loss Dropout p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='8 Epoch Log Loss 0 10k 20k 30k 40k 100p 10n 1μ 100μ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='01 1 100 Average Train Loss Average Test Loss L1 penalty, 100 Epoch Log Loss Figure 28: The train and test loss over the course of training with two types of regularization, dropout and ℓ1 regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Grokking occurs with some runs for dropout but never for ℓ1 regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' less sparse in the Fourier basis, preventing the network from learning the Fourier Multiplication Algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2 THE SLINGSHOT MECHANISM OFTEN OCCURS, BUT IS UNNECESSARY FOR GROKKING As noted in Section C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2, our 2-layer transformers exhibit significant slingshots (Thilak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=', 2022) during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We speculate that this is due to how gradients of different scale interact with adaptive optimizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We were even able to induce slingshots on a 1-layer by reducing the precision of the loss calculations (as this causes many gradients to round to 0 and thus greatly increases the differences in scale of gradients).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' However, as many of our 1-layer models do not exhibit slingshots but nonetheless grok, the slingshot mechanism is unnecessary for grokking to occur, in the presence of weight decay or other regular- ization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We speculate that the slingshots of Thilak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' (2022) (which co-occur with grokking for training runs without weight decay) serve as an implicit regularization mechanism that favors the simpler, generalizing solution over the more complicated 29 arXiv preprint 0 500 1000 1500 2000 2500 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 Train/Test Loss for 5 Digit Addition w/ Infinite Data Steps Loss 0 500 1000 1500 2000 2500 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 Token 0 loss Token 1 loss Token 2 loss Token 3 loss Token 4 loss Token 5 loss Per Token Train/Test Loss for 5 Digit Addition w/ Infinite Data Steps Loss Figure 29: (Top) The training/test loss for 5 Digit Addition trained on randomly generated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Note that training and test loss coincide, as the model does not see repeated pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' (Bottom) The train/test loss per token for 5 Digit Addition, trained with randomly generated data at each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Note that phase changes in the average loss correspond to phase changes in individual tokens, though one phase change (token 1, around step 270) is not visible on the averaged loss as it overlaps with the end of the first phase change (token 0, starting around step 150).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 30 arXiv preprint 0 5k 10k 15k 0 2 4 6 8 Train Loss Test Loss Train/Test Loss for 5 Digit Addition with 700 Data Points Epochs Loss Figure 30: The train and test loss for 5 Digit Addition trained on 700 data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Unlike the infinite, randomly generated data case, this shows both a sharp phase change and clear train test divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='3 ADDITIONAL EVIDENCE FROM OTHER ALGORITHMIC TASKS We now provide addition analysis of grokking phenomena on 3 additional algorithmic tasks and confirm that limited data is an important part of grokking: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 5 digit addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We sample pairs of random 5 digit numbers and have the model predict their sum 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Predicting repeated subsequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We take a uniform random sequence of tokens, ran- domly choose a subsequence to repeat, and train the model to predict the repeated tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Skip trigram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We feed in a sequence of tokens from 0 to 19, of which exactly one is greater than or equal to 10, and the model needs to output the token that is ≥ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' This can be solved with learning 10 skip trigrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We use a 1-layer full transformer for 5-digit addition, a 2-layer attention only transformer for pre- dicting repeated subsequences, and a 1-layer attention only transformer for the skip trigram task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Otherwise, we use the same hyperparameters as in the mainline model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 5 Digit Addition We first consider the case where we train on the approximately infinite data regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' For each minibatch, we randomly new sample 5 digit numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We report the results in Figure 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Train loss coincides with test loss, so grokking does not occur, as the model almost never sees the same pair of 5 digit numbers twice, with 1010 such pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Interestingly, the various small bumps in Figure 29 correspond to the model learning how to calculate each of the 6 tokens in the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' However, grokking does occur when we restrict the model to only see 700 data points, as shown in Figure 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Repeated subsequence As with the 5-digit addition task, we find that restricting the amount of data is necessary and sufficient for grokking on the repeated subsequence task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' In Figure 31, the model sees new data at every step exhibits no grokking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' In contrast, clear grokking occurs when we restrict the model to only see 512 data points in Figure 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Skip trigram As with the previous tasks, we find that restricting the amount of data is necessary and sufficient for grokking on the skip trigram task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' The model that sees new data at every step exhibits no grokking in Figure 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Meanwhile, the model restricted to only see 512 data points exhibits clear grokking in Figure 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Taken together, these results echo the importance of limited data for grokking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 31 arXiv preprint 0 1000 2000 3000 4000 5000 0 1 2 3 4 5 Repeated Subsequence Prediction w/ Infinite data Step Loss Figure 31: The training/test loss for repeated subsequences trained on randomly generated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Note that training and test loss coincide, as the model does not see repeated pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' There sharp phase change corresponds to the model forming induction heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' (Olsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=', 2022) 0 50k 100k 150k 200k 0 2 4 6 8 10 12 14 16 Train Loss Test Loss Repeated Subsequence Prediction w/ 512 Data Points Epoch Loss Figure 32: The train and test loss for the repeated subsequence task, trained on 512 data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Unlike the infinite, randomly generated data case, this shows both a sharp phase change and clear train test divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 0 500 1000 1500 2000 2500 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 Train/Test Loss for Skip Trigram Task w/ Infinite Data Epoch Loss Figure 33: The training/test loss for the skip trigram task, trained on randomly generated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Note that training and test loss coincide, as the model does not see repeated pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' The sharp phase change corresponds to the network learning all of the skip trigrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 32 arXiv preprint 0 1000 2000 3000 4000 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='5 Train Loss Test Loss Train/Test Loss for Skip Trigram Task w/ 50 Data Points Epoch Loss Figure 34: The train and test loss for the skip trigram task, trained on 512 data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Unlike the infinite, randomly generated data case, this shows both a sharp phase change and clear train test divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' E FURTHER SPECULATIONS ON GROKKING E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='1 AN INTUITIVE EXPLANATION OF GROKKING In this section, we speculate on what might be happening “under the hood” when a model groks and explore why this phenomena happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' The evidence is only suggestive, so this a promising direction for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Grokking occurs when models, trained on algorithmic tasks with certain hyperparameters, initially overfit the training data where train loss significantly improves while test loss worsens and the two diverge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' But later in training, there is a sudden improvement in test loss, so test and train loss converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' In contrast to Power et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' (2022) but in line with Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' (2022), grokking does not occur when both train and test loss improve together without the initial divergence, as shown in many of the figures in this paper, for example Figures 2 and 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' The core issue is that the model has two possible solutions: memorization (with low train loss and high test loss) and a generalization (with low train loss and low test loss).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' In our case, the Fourier Multiplication Algorithm is the generalization solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Intuitively, with very little training data, the model will overfit and memorize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' With more training data, the model must generalize or suffer poor performance on both train and test loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Since neural networks have an inductive bias favoring “simpler” solutions, memorization complexity scales with the size of the training set, whereas generalization complexity is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' The two must cross at some point!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Yet, the surprising aspect of grokking is the abrupt shift during training, when the model switches from memorization to generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' The other component of grokking is phase transitions - the phenomena where models trained on a certain task develop a specific capability fairly rapidly during a brief period of training, as shown for the case of induction heads forming in transformer language models in Olsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' (2022) and our results in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' That is, rather than slowly forming that capability over training, the model rapidly goes from being bad at it to being good at it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' One interpretation of a phase transition is that there’s some feature of the loss landscape that makes the generalising solution harder to reach rather than a smooth gradient for the model to follow, it instead initially finds it difficult to make progress, but then crosses some threshold where it can rapidly make progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Therefore, grokking occurs with phase transitions, limited data, and regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Models exhibit phase transitions despite having enough training data to avoid overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Regularization (weight decay in our case) favors simpler solutions over complex ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' The model has enough data to marginally prefer generalization over memorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' The phase transition indicates that generaliza- tion is “hard to reach” while the model has no problems with memorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' But as it memorizes, the network becomes more complex until the weight decay prevents further memorization then moves towards equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' The gradient to memorize balances the gradient towards smaller weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' With 33 arXiv preprint generalization, the model is incentivized to both memorize and simplify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Strikingly, it is capable of both while maintaining a somewhat constant training performance in this circuit formation phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Next, as the model approaches generalization, the memorization weights are removed in the cleanup phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' The cost from complexity outweighs the benefit from lower loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Due to the phase transition during this training period, as model’s progress towards generalization accelerates, the cleanup rate sharpens as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' A model that learns a perfect solution and is trained with weight decay has competing incentives: larger weights (for more extreme logits and thus lower loss) and smaller weights (from weight decay).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' So for any solution and any level of weight decay, there will always be a level of train loss where these two forces equilibrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Thus, memorization is not necessarily a “simpler” solution than generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' The key is that generalization will have smaller weights holding train loss fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' In fact, weight decay should be expected to equilibrate at a slightly lower train loss in generalization, since the base solution is simpler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' This matches what we observe in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 4 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='2 HYPOTHESIS: PHASE TRANSITIONS ARE INHERENT TO COMPOSITION A promising line of work in the growing field of mechanistic interpretability suggests that models form circuits (Cammarata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=', 2020) – clean interpretable algorithms formed by subnetworks of the model, such as curve detectors (Cammarata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=', 2020) in image classification networks and induction heads (Elhage et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Olsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=', 2022) in LLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' This is surprisingly true!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' A circuit represents the model learning an algorithm, a fundamentally discrete thing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' each step in the algorithm only makes sense if the other steps are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' But neural networks are fundamentally continuous, trained to follow gradients towards lower loss and struggle to jump to new optima with- out following a smooth gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' So how can a model learn a discrete algorithm?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' As a concrete example, let’s consider the case of induction heads in LLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' There is a subnetwork of a next-token prediction autoregressive language model that learns to continue repeated subse- quences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' It detects whether the current token occurred earlier in the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' If so, it predicts the same token after that previous occurrence will also come next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' The circuit consists of a previous token head, which attends to each previous token and copies the context of the previous token to the current token, and an induction head which attends to the token after a previous occurrence of the current token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' The induction head composes with the previous token head by forming a query vector representing the current token and a key vector representing the previous token head’s output using K-Composition, the context of the previous token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' It attends to a token where this query and key match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' This circuit significantly improves loss but only in the context of the other heads present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Before either head is present, no gradient encourages the formation of either head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' At initialization, we have neither head, so gradient descent should never discover this circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Naively, we might predict that neural networks will only produce circuits analogous to linear regression, where each weight will marginally improve performance as it continuously trains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' And yet in practice, neural networks indeed form such sophisticated circuits, involving several parts interacting in non-trivial, algorithmic ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' So how can this be?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' A few possible explanations: Lottery tickets (Frankle & Carbin, 2018): Initially, each layer of the network is the superposition of many partial circuit components, and the output of each layer is the average of the output of each component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' The full output of the network is the average of many different circuits, with significant interference from non-linear interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Some of these circuits are systematically useful to reducing loss, but most aren’t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Gradients for useless circuits will have zero mean, while gradients for useful circuits will have non-zero mean, with a lot of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' SGD reinforces relevant circuits and suppresses useless ones, so circuits will gradually form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 4One subtlety: the grokking phenomena is often incorrectly summarized as “the model learned to generalize even after achieving zero loss.” Zero loss does not exist with cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Although the model achieves perfect accuracy, it is trained to optimize loss not accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' This means the model is always incentivized to further improve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' In particular, the easiest way to improve performance with perfect accuracy is by scaling up the logits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' This lowers the temperature and pushes the softmax closer to an argmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 34 arXiv preprint Random walk: The network wanders randomly around the loss landscape until it encoun- ters a half-formed previous token head and induction head that somewhat compose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' This half-formed circuit becomes useful for reducing loss, so gradient descent completes the circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Evolution: A similar mystery arises from how organisms develop sophisticated machinery, like the human eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Each part is only useful in the context of other parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' A compelling explanation is a component first developed that was somewhat useful in its own right, like a light-detecting membrane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' It was reinforced as a useful component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Then, later compo- nents developed depending on the first, like the lens of the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Evolution is a natural explanation, However, based on our toy tasks, it cannot be the whole story.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' In the repeated subsequence task, we have a sequence of uniform randomly generated tokens, apart from a repeated subsequence at an arbitrary location, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 7 2 8 3 1 9 3 8 3 1 9 9 2 5 END.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' This means all pairs of tokens are independent, apart from pairs of equal tokens in the repeated subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' In particular, this means that a previous token head can never reduce loss for the current token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' The previous token will always be independent of the next token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' So a previous token head is only useful in the context of an induction-like head that completes the circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Likewise, an induction head relies on K-composition with a previous token head and so cannot be useful on its own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Yet the model eventually forms an induction circuit!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' A priori, the random walk seems insufficient on its own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' An induction circuit is relatively compli- cated, representing a small region in model space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' So a random walk is unlikely to stumble upon it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Concretely, in our modular addition case, progress measures show significant hidden progress pre-grokking, indicating the model did not stumble upon the solution by chance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Thus, the lottery ticket hypothesis seems the most explanatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' An induction head is useless without a previous token head but might be slightly useful when composing with a head that uniformly attends to prior tokens, since part of its output will include the previous token!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Nevertheless, we suspect that all explanations contribute to the entire picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' This seems most plausible if the uniform head just so happens to attend a bit more to the previous token via a random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Returning to phase transitions, the lottery ticket-style explanation suggests that we might expect phase transitions as circuits form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Early in circuit formation, each part of the circuit is rough, so the effect on the loss of improving any individual component is weak, meaning gradients will be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' As each component develops, other components will become more useful, meaning all gradients will increase together non-linearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' As the circuit nears completion, we should expect an acceleration in the loss curve for this circuit, resulting in a phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' F FURTHER DISCUSSION ON USING MECHANISTIC INTERPRETABILITY AND PROGRESS MEASURES FOR STUDYING EMERGENT PHENOMENA While we find approach of using mechanistic interpretability to define progress measures rela- tively promising, there remains significant uncertainty as to how scalable existing mechanistic inter- pretability approaches really are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Broadly speaking, depending on the success of future mechanistic interpretability work, we think there are three methods through which mechanistic interpretability and progress measures can help with understanding and predicting emergent phenomena: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' If mechanistic interpretability can be scaled to large models to the level where we can un- derstand the mechanisms behind significant portions of their behavior, we could perform the same style of analysis as was done in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We believe it’s currently unclear as to whether or not mechanistic interpretability will successfully scale to large models to this extent (or even if there exist human-understandable explanations for all of their sophisti- cated behavior).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' That being said, in cases where mechanistic interpretability does recover human-understandable mechanisms, we could simply use the parts of the mechanism as progress measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' If future mechanistic interpretability can only recover parts of the mechanism of larger models (as in Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' (2022)) and can only generate comprehensive understanding of the mechanisms of smaller models, we might still be able to use our understanding from smaller models to guide the development measures that track parts of the behavior of 35 arXiv preprint the larger model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' We find this scenario relatively plausible, as existing mechanistic inter- pretability work already allows us to recover fragments of large model behavior and un- derstand these fragments by analogy to smaller models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' For example, Olsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' (2022) use this approach to understand the emergence of in-context learning in medium-sized lan- guage transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Even if mechanistic interpretability fails to recover understandable mechanisms at all on large models, we might still be able to derive progress measures that don’t require human understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' For example, if we end up with automated mechanistic interpretability (that nonetheless still fails to recover human-understandable mechanisms), we might be able to use the outputs of those opaque processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' Another approach is task-independent progress measures: if we can discover progress mea- sures that don’t depend on the task, perhaps using many small, interpretable models as testbeds, we might be able to apply these progress measures to large models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' That being said, we think the future work outlined in Section 6 is necessary to successfully apply our approach to predict and understand emergent behavior in existing large language models, and so remain cautiously optimistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} +page_content=' 36' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf'} diff --git a/k9AyT4oBgHgl3EQfYPfO/content/tmp_files/2301.00201v1.pdf.txt b/k9AyT4oBgHgl3EQfYPfO/content/tmp_files/2301.00201v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ec445e540d886c9dddf63a45ba94efce56e7af43 --- /dev/null +++ b/k9AyT4oBgHgl3EQfYPfO/content/tmp_files/2301.00201v1.pdf.txt @@ -0,0 +1,1829 @@ +EXPLORING SINGULARITIES IN POINT CLOUDS WITH +THE GRAPH LAPLACIAN: AN EXPLICIT APPROACH +MARTIN ANDERSSON AND BENNY AVELIN +Abstract. We develop theory and methods that use the graph Lapla- +cian to analyze the geometry of the underlying manifold of point clouds. +Our theory provides theoretical guarantees and explicit bounds on the +functional form of the graph Laplacian, in the case when it acts on func- +tions defined close to singularities of the underlying manifold. We also +propose methods that can be used to estimate these geometric properties +of the point cloud, which are based on the theoretical guarantees. +1. Introduction +High dimensional data is common in many research problems across aca- +demic fields. It is often assumed that a data set X = {Xi}n +i ⊂ RN lies on +a lower-dimensional set Ω and is in fact a sample from a probability distri- +bution over Ω. It is also often assumed that Ω can be represented as the +union of several manifolds Ωi, where each Ωi represents a different class in a +classification problem. For instance, if a data set contains two classes, i and +j, class i might be contained in Ωi and class j in Ωj, with the two classes +potentially being disjoint. However, classification is not always so clear-cut: +For instance, in the MNIST dataset, where handwritten digits of ”1” ∈ Ω1 +and ”7” ∈ Ω7 can appear very similar, suggesting that Ω1 ∩ Ω7 ≠ ∅. There- +fore, understanding geometric situations such as intersections is of interest +in classification problems. +In the manifold model of data, an intersection between two different man- +ifolds Ωi,Ωj is either represented just as such, or it can be viewed as a sin- +gularity if we consider Ω = Ωi ∪ Ωj as a single manifold. Other regions in Ω +that can be viewed as singular, such as boundaries and edges, may also be +of interest as they can signify important features in the data. +To study such singularities, we use the graph Laplacian Ln,t. This op- +erator, which depends on the number of data points n and a parameter t, +can act on functions defined on the data set X. As n tends to infinity and +t tends to 0, Ln,t converges to the Laplace-Beltrami operator in the interior +of a single manifold [1]. In this work, we primarily study the behavior of +x → Ln,tf(x) for functions f, when x is close to singular points. +Our contribution in this paper is primarily an extension and reframing of +work done in [2]. At the same time, we also focus on the specific case when +the function f is assumed to be of the form f(x) = v ⋅ x, where v is a unit +vector. We also consider more restricted classes of manifolds. +2020 Mathematics Subject Classification. Primary 58K99; Secondary 68R99, 60B99. +Key words and phrases. Graph Laplacian, geometry, singularities. +1 +arXiv:2301.00201v1 [stat.ML] 31 Dec 2022 + +2 +ANDERSSON AND AVELIN +Since Ln,t converges to Laplace-Beltrami, a second order differential op- +erator, in the interior of Ω, we expect that for f as above Ln,tf(x) ≈ 0. +However, for singular points like intersections, the limit operator is of first +order [2], and Ln,tf(x) ≠ 0, which can be seen in Fig. 1. +Our results show how x → Ltf(x) and, through a finite-sample bound, +how x → Ln,tf(x) behaves. +More specifically, given x0 ∈ Ωi near some +singularity, and x in the ball BR(x0), including the case when x /∈ Ωi, we +show how the function x → Ln,tf(x) deviates from being constantly 0 and +has specific functional forms. These forms depend on the type of singularity. +In [2] they showed what these forms are, up to some asymptotically defined +error term, as t → 0, We build on this to get explicit expressions of Ltf(x) +when t is fixed. +Overview of results: First, in Section 4.1 we consider the case that Ω +is flat manifold of dimension d, and where we have a geometric situation +similar to Fig. 3. +To set up the results, we start with an x0 ∈ Ω, and let x ∈ BR(x0), where +R = +√ +tr0 > 0, and use ˆx to denote the projection of x to Ω. We also define +vn,Ω as the projection of v onto x − ˆx, and vn,∂Ω is the projection of v onto +the outwards normal of ∂Ω. Then we show the following: +● In Theorem 1, we let ∥x − x0∥ = r +√ +t and θ is the angle between +vectors x − x0 and ˆx − x0. If x is not close to ∂Ω, then +Ltf(x) = A(x)t +d+1 +2 vn,Ω sin(θ)re− sin2(θ)r2 + t +d+1 +2 B(x). +The function A is close to being constantly equal to πd/2, and B can +be made, uniformly, arbitrarily small. Both functions have explicit +bounds. +● Theorem 2 shows what happens when x is close to ∂Ω: +Ltf(x) = ̂ +A1(x)t +d+1 +2 vn,Ω sin(θ)re− sin2(θ)r2 ++ ̂ +A2(x)t +d +2 vn,∂Ωe− sin2(θ)r2 ++ B(x)t +d+1 +2 e−r2 +0, +where functions ̂ +A1, ̂ +A2 and B have explicitly computable bounds. +In Section 4.2 and Section 4.3 we prove more general results: +● In Theorem 3 we relax the conditions on Ω, considering non-flat +manifolds, and prove a weaker version of Theorem 1. +● In Theorem 4 we relax the conditions further, and allow for noise +when sampling from Ω. +To connect Lt to Ln,t, in Section 4.4 we prove two finite-sample bounds. +Finally, in Section 5.1 we propose methods to find intersections in data +and estimate the angle of such intersections, which are motivated by the +aforementioned theorems and Corollary 4.5. We also provide numerical ex- +periments, in Section 5, to test these methods. +2. Earlier work +The framework of assuming an underlying low-dimensional manifold of +data, in conjunction with graph-related tools and in particular the graph + +3 +Figure 1. Graph Laplacian Ln,t acting on a linear function +f. Purple color showing positive, and green color negative +values of Ln,tf, where lack of color indicates values near 0 +Laplacian, has been used extensively. Some examples include work in clus- +tering [3, 4, 5, 6, 7], dimensionality reduction [8, 9], and semi-supervised +learning [10]. +Several of the approaches to study data sets that use the graph Lapla- +cian leverage that if the manifold is smooth enough and well-behaved, then +the graph Laplacian approximates some well-understood operator (for in- +stance the Laplace-Beltrami operator [11]), which has useful mathematical +properties. +Therefore, the question of convergence properties of the graph Laplacian +is useful and important, and it has partly been explicated in [1, 12, 13, 9]. +In particular, and highly influential of this paper, is what the asymptotic +convergence looks like near singularities of the manifold, which was shown +in [2]. +3. Basic mathematical objects and theory +In this section, we provide more precise definitions and introduce the basic +mathematical theory we will be using to present and prove our results. This +is similar to the problem setup in [2]. +3.1. Conditions on manifolds. We will consider sets of the form Ω = +∪m +i Ωi, where each Ωi is a smooth and compact d-dimensional Riemannian +submanifold of RN. We will assume that if Ωi,Ωj, and i ≠ j have a non- +empty intersection, then this intersection will have dimension lower than +d. +Associated to Ω will be a probability measure with density p ∶ Ω → R such +that the restriction of p to Ωi is smooth, and there are constants a and b +such that 0 < a ≤ p ≤ b. +If x ∈ Ωi, we can consider the tangent space TΩi,x ≃ Rd, which we will +identify as a subspace of the ambient space RN. More precisely, given open + +0.50 +0.50 +0.25 +0.25 +0.00 +0.00 +0.25 +-0.25 +Z +-0.50 +-0.50 +0.25 +0.2 +0.50 +0.50 +0.75 +0.75 ++4 +ANDERSSON AND AVELIN +subsets U ⊂ Rd and W ⊂ Ωi (W is open in the subspace topology of Ωi), +and a coordinate chart α ∶ U → W such that α(0) = x, we define TΩi,x as +the image of Rd under the action of the Jacobian. We denote the Jacobian +Dα ∶ U → RN×d, evaluated at 0, by Dα(0). The best linear approximation +to u ↦ α(u) is of course given by u ↦ x + Dα(0)u, and x + TΩi,x is the best +flat approximation to Ωi around x. +The definition of Ω implies that a point x ∈ Ω can have more than one +associated tangent space. For example, if x ∈ Ωi ∩ Ωj and i ≠ j, then both +TΩi,x and TΩj,x exist, and they can be different. +A note on notation is that we will denote the interior of a manifold Ωi by +IntΩi, and the boundary by ∂Ωi. +3.2. Types of singularities. The following are what we will refer to as +singular points, which will be of four different kinds. Given x ∈ Ω = ∪Ωi, we +have the following types: +(Type 1) There is a submanifold Ωi such that x ∈ ∂Ωi. +(Type 2) There are submanifolds Ωi ≠ Ωj such that x ∈ IntΩi ∩ IntΩj. +(Type 3) There are submanifolds Ωi ≠ Ωj such that x ∈ ∂Ωi ∩ IntΩj. +(Type 4) There are submanifolds Ωi ≠ Ωj such that x ∈ ∂Ωi ∩ ∂Ωj. +The different types above can of course have non-empty intersection with +each other, and a non-singular point is simply a point x ∈ IntΩi such that +if j ≠ i, then x ≠ Ωj. See Section 3.2 for two examples of singularities. +Figure 2. There is a singularity in the intersection of the +lines above. The left figure shows a point of Type 4, and the +right figure shows a point of Type 2. +3.3. Integration on Ω. We will integrate scalar-valued functions, f ∶ Ω → +R, over Ω. When formulating integration of scalar-valued functions over +submanifolds of RN, we follow the approach in [14]. Because we need some +preliminary results concerning integration on Ω, we make some important +definitions explicit. +First, let x1,...,xk be vectors in RN for k ≤ N. If I = (i1,i2,...,ik) is a +k-tuple of integers such that i1 ≤ i2 ≤ ⋯ ≤ ik, define XI ∈ Rk×k as the k × k +matrix containing only rows i1,...,ik of the matrix X = (x1,...,xk). Now +we can define the volume function V ∶ RN×k → R, by V (X) = +√ +det2(XtX) = +[∑I det2 XI] +1/2, where the I’s span over k-tuples as above, see [14, Theorem +21.4]. +In general, given a coordinate chart α ∶ U → W, where U ⊂ Rd, W ⊂ +Ωi ⊂ RN are open subsets, and Dα is the Jacobian of α, we can express + +=t=5 +integration over W as +∫W f dV = ∫U f ○ α V(Dα). +In the coming proofs, when integrating around a point x ∈ IntΩi, we will +change coordinates to the standard basis in TΩi,x = Rk. With this we mean +that we can find open sets W ⊂ Ωi around x such that the projection map +π ∶ W → B ⊂ x+TΩi,x is a diffeomorphism, where x+TΩi,x ∶= {x+y ∣ y ∈ TΩi,x }. +To integrate over TΩi,x we use the map π−1 precomposed with an inclusion +map. +More specifically and without loss of generality, by translation and an +orthonormal coordinate change, we can assume that TΩi,x = Rd × {0}n−d. In +this coordinate system we can write +α ∶ U +i�→ x + TΩi,x +π−1 +��→ W ⊂ Ωi, +(3.1) +where i is the natural inclusion map and U an open subset in Rk. +3.4. Important bounds. The following bounds will be used later in our +proofs: First, let TΩ,x,U,W,π be as in Section 3.3. Then for any y ∈ W, +∥y − π(y)∥ ≤ O(∥x − π(y)∥2). +(3.2) +This follows since Ωi is smooth and the tangent space represents the best +flat approximation of Ωi around x. +To formulate the second bound, we need the lemma below. +Lemma 3.1. Let U,W,x,y,Ωi,π,i,α be as in Section 3.3. Then the follow- +ing holds for the volume function V : +V (Dα(y)) = 1 + O(∥x − π(y)∥2). +Proof. Since α = π−1○i, and the tangent space is the best flat approximation +of Ω, we can parametrize the W by α(u) = (u,g(u)). It is then easy to see +that for i = 1,...,d we have +∂iα(y) = (ei,∂ig(u)), +where ∂ig(0) = 0 and ∥∂ig(u)∥ = O(∥u∥). Now +detDαI = +⎧⎪⎪⎨⎪⎪⎩ +1 +if I = (1,2...,d) +O(∥u∥) +otherwise. +. +If we Taylor expand x → √x, we get +V (Dα) = (∑ +I +(detDαI)2) +1/2 += 1 + O(∥u∥2), +and by applying the above on (u,0) = x − π(y) we are finished. +□ +Further, since we have a finite union Ω = ∪iΩi and each Ωi is compact, +(3.2), the previous lemma implies that we can find a uniform bound L such +that for all tuples (U,W,x,y,π,Ωi) +∥y − π(y)∥ ≤ L∥x − π(y)∥2 +(3.3) +and +∣V (Dα) − 1∣ ≤ L∥x − π(y)∥2 +(3.4) + +6 +ANDERSSON AND AVELIN +holds. +3.4.1. (L,r)-regular manifolds. To formulate our results we will need some +measure of how regular, with regard to curvature, our set Ω is. The following +definition captures the necessary information. +Definition 3.2. Let Ω = ∪Ωi be a union of compact submanifolds in RN. +We also let r > 0 be the largest radius such that any point x ∈ IntΩ allows +coordinate charts α ∶ U → Br(x) ∩ Ωi, where U ⊂ Rd and Br(x) ⊂ RN is an +open ball of radius r around x. Further, assume also that conditions (3.3) +and (3.4) hold over all tuples (U,W,x,y,π,Ωi) for some L > 0. Then we say +that Ω is (L,r)-regular. +Example 3.3. Any smooth and compact submanifold is (L,r)-regular. For +instance the graph of the function x → x2 over the compact interval [−1,1] +is (1,1)-regular. +3.5. Graph Laplacian. In this section we introduce the graph Laplacian +and how it acts on real-valued functions defined on RN. +Given n i.i.d. random samples X = {X1,...,Xn} from the distribution +with density p on Ω, we build a weighted fully connected graph G = (V,E) as +follows: We let each sample Xi represent a vertex i, and for vertices i,j ∈ V +the weight on (i,j) ∈ E is given by +Wn,t(i,j) ∶= Wn,t(Xi,Xj) = 1 +nKt(Xi,Xj) = 1 +ne− +∥Xi−Xj∥2 +t +. +The function Wn,t is naturally viewed as an n × n matrix, and the variable +t is in the literature often referred to as the bandwidth of the kernel Kt. +Remark 3.4. In the limit analysis as n → ∞, it is useful also normalize by +1 +td/2+1/2 . But, since a priori we do not know the dimension d, we will work +without this normalization. +We define the diagonal weighted degree matrix as +Dn,t(i,i) = ∑ +j +Wn,t(i,j), +and the graph Laplacian Ln,t as +Ln,t = Dn,t − Wn,t. +Remark 3.5. This is often referred to as the unnormalized graph Laplacian. +There are other normalizations of this matrix which are used, for example, +in [6, 7, 13]. One difference between these normalizations are their limit +properties. +Given the fully connected graph G = (E,V ), the graph Laplacian above +can be seen as an operator acting on arbitrary functions f ∶ V → R in the +following way: +Ln,tf(Xi) = 1 +n ∑ +j +Kt(Xi,Xj)(f(Xi) − f(Xj)), +(Xi,Xj) ∈ E. + +7 +We extend this operator to acting on functions f ∈ Cc(RN,R), by the canon- +ical choice +Ln,tf(x) = 1 +n ∑ +j +Kt(x,Xj)(f(x) − f(Xj)), +x ∈ RN. +(3.5) +Our main results will be stated in terms of the expected operator: +Ltf(x) = Ep[Ln,tf(x)] = ∫Ω Kt(x,y)(f(x) − f(y))p(y)dy. +(3.6) +That this is well-defined follows from the assumptions that X1,...,Xn are +i.i.d., that f is continuous and that Ω is compact. +One immediate consequence of the linearity of the integral is that +Ltf(x) = ∫Ω Kt(x,y)(f(x) − f(y))dy += ∑ +i ∫Ωi +Kt(x,y)(f(x) − f(y))p(y)dy. +(3.7) +In our approach it is useful to work with the restricted Laplacian Li +t, which +is defined by +Li +tf(x) = ∫Ωi +Kt(x,y)(f(x) − f(y))p(y)dy. +(3.8) +3.6. Gamma functions. In the proofs of several of our results we will +need to handle the Gamma function Γ(⋅), and both the lower and upper +incomplete gamma functions, γ(⋅,⋅) and Γ(⋅,⋅) respectively. These are well- +known and are defined by the equations +Γ(a) = ∫ +∞ +0 +ta−1e−t dt, +γ(a,x) = ∫ +x +0 +ta−1e−t dt, +Γ(a,x) = ∫ +∞ +x +ta−1e−t dt. +In this paper both a and x are non-negative real numbers. +We will need the following bounds: First, if a ≥ 1, then ta−1 ≥ xa−1 and +Γ(a,x) ≥ xa−1 ∫ +∞ +x +e−x dt = xa−1e−x. +(3.9) +Secondly, if ex > 2a, then by [15, Theorem 4.4.3], +Γ(a,x) ≤ axa−1e−x. +(3.10) +Finally, we need the lower bound +γ(a,a) ≥ 1 +2Γ(a). +(3.11) +That this holds can be seen by viewing γ(a,x) as an unnormalized version of +the cumulative distribution function of the Gamma distribution, for which +it is well-known that the median ν is less than a. + +8 +ANDERSSON AND AVELIN +4. Main results +Now that we have the necessary definitions and mathematical background, +we are ready to present and prove our main results. +Before stating the +theorems, we will provide a brief section that explains the geometry of some +terms that will be used in the theorem statements. This will help make the +theorems easier to understand. +Remark 4.1. Some of our results are given in the particular case when +Ω = ∪Ωi is such that each Ωi is flat. This is easier to analyze and gives +better bounds, but it is also motivated by a particular use-case: Sets of the +form +Ω∶ = {W ∈ Rk ∶ ∣fW (x) − g(x)∣ = 0, +x ∈ D}, +where fW is a neural network with weights W and ReLU activation func- +tions. Here g is a target function, and D some dataset. That is, the zero sets +of the optimization problem which one tries to minimize during training of +a common type of neural network. +General structure of results. By (3.7) it is enough to understand the +restricted Laplacian, Li +t defined in (3.8). Because of this, our results are for- +mulated to show the behavior of Li +t. Depending on what type of singularity +being examined, it is easy to extend the results to the full Laplacian. In +Corollary 4.5 we give one example of how to extend the results to the sum +∑2 +i=1 Li +t when one is close to an intersection of two manifolds. +Geometry and notation for Section 4.1. We will in several theorems also +formulate the function x → Li +tf(x) partly in terms of new coordinates (r,θ). +Here r is defined by the relation ∥x − x0∥ = +√ +tr, and given the projection +ˆx of x to a plane Ωi, we define θ ∈ [0,π/2] to be the angle between vectors +x0 − x and ˆx − x, as the schematic in Fig. 3. By simple geometry, it also +follows that ∥ˆx − x∥ = r sinθ. +Given a vector v ∈ RN, we will have reason to write the expression v⋅(ˆx−x) +as +v ⋅ (ˆx − x) = r +√ +tsin(θ)v ⋅ +ˆx − x +∥ˆx − x∥ = r +√ +tsin(θ)vn,Ωi(x), +where we have defined +vn,Ωi(x) ∶= v ⋅ +ˆx − x +∥ˆx − x∥. +In other words, vn,Ωi is the projection of v onto a unit normal vector of Ωi, +but it depends on x. We define this function to be 0 when x = ˆx, and let +us note that for x ≠ ˆx, this function is constant up to its sign. This implies +that evaluating r +√ +tsin(θ)vn,Ωi(x) is the same as letting vn,Ωi be fixed, but +allowing θ to change sign depending on which side of Ωi x is, i.e. as if we +have fixed the coordinate system in which we measure the angle θ. We will +in our theorem statements suppress the x-dependancy of vn,Ωi, to increase +readability. +Additionally, in Theorem 2 we will have a term vn,∂Ωi that is specific to +that theorem. This will be defined in the case where there is a boundary +close to x. In Fig. 3, this would imply there is a boundary of Ω1 nearby. + +9 +To give the definition if this term, we first let ˆx∂Ωi be the projection of ˆx +to ∂Ωi. We can now define a unit normal at ˆx∂Ωi, denoted by n∂Ωi. Two +choices are natural, a normal pointing either towards, or away from Ωi. We +define n∂Ωi as the latter. Given a vector v ∈ RN, we can define +vn,∂Ω ∶= v ⋅ n∂Ω. +In Theorem 2 we will be close to part of the boundary ∂Ω where n∂Ω is +constant. This implies that, unlike vn,Ωi, vn,∂Ω does not depend on x, but +is (locally) constant. +x0 +x +sinθr +√ +t +ˆx +θ +r +√ +t +Ω1 +Ω2 +r0 +√ +t +Figure 3. Schematic picture of the geometry of Theorem 1, +where Ω1 is the object of interest and x ∈ Ω2 for visualization +purposes. +Geometry and notation for Section 4.2. To help with the geometric picture +for general manifolds, the situation is as explained in Section 4 and Fig. 3: +the terms x,x0, ˆx,θ and vn,Ωi are in the same relation to each other as in +Section 4, but instead of projecting x to a flat manifold Ωi we project x to +the (flat) tangent plane TΩi,x0. In that sense the geometry for more general +manifolds is not more difficult, but handling error terms is more involved. +4.1. Flat manifolds. In this section we assume that Ω = ∪iΩi, where each +Ωi is a flat manifold, which means that each coordinate chart around x ∈ +IntΩi is an isometry between an open neighborhood U of x, where U is a +ball in Rd +In Theorem 1 we give a result concerning the behavior of x → Li +tf(x) +when we are not close to the boundary ∂Ω. This case is easier to prove, +and we give explicit bounds of all terms involved, and express them with +elementary functions. +In Theorem 2 we show what happens when we are close to ∂Ω, but we +have more involved expressions for some terms. +In the following theorems, it is the point x0 one should think of as po- +tentially being a singular point, see Fig. 3, and the theorems show us how +x → Li +tf(x) behaves in a neighborhood around this singular point. By com- +bining Theorem 1 and Theorem 2, it is possible to consider several types of +singularities defined in Section 3.2. + +10 +ANDERSSON AND AVELIN +Theorem 1. Let f(x) = v ⋅ x for some unit vector v ∈ RN and assume +that p is the uniform density over Ω = ∪iΩi. Let x0 ∈ Ωi and assume that +∂Ωi ∩ B2R(x0) = ∅ for R = r0 +√ +t, where r0 > 2. Further, x ∈ BR(x0), and +vn,Ωi, r and θ are as described in Section 4. If t ≤ +R2 +d/2+1, d ≥ 1 and r < 1, +then we have that +Li +tf(x) = td/2+1/2 (A(θ,r0,d)vn,Ωi sinθre− sin2 θr2 + B(x)e−r2 +0), +where A,B are real-valued functions. The function B depends on x and is +uniformly bounded by ∣B(x)∣ ≤ 2 +d+1 +2 rd +0∣Sd−1∣; and A depends on x only through +θ, and is bounded by +max(πd/2,2πd/2 − ∣Sd−1∣2d/2rd−1 +0 +e−r2 +0+1) ≤ A(θ,r0,d) ≤ 2πd/2. +Proof. Since x → Li +tf(x) is translation and rotation invariant, we can with- +out loss of generality assume that Ωi oriented in RN in such a way which +makes it a subset of Rd × {0}N−d. +We want to evaluate +Li +tf(x) = ∫Ωi +Kt(x,y)(f(x) − f(y))pdy. +We begin by splitting the integral above into +∫Ωi +Kt(x,y)(f(x) − f(y))pdy = ∫BR(x)∩Ωi +Kt(x,y)(f(x) − f(y))pdy ++ ∫Ωi∖BR(x) Kt(x,y)(f(x) − f(y))pdy += I1 + I2. +(4.1) +For estimating I2, by translation invariance we can WLOG assume that +x = 0. Now we make a change of variables and rescale y, which allows us to +say that +∣I2∣ = ∣∫Ωi∖BR(0) Kt(0,y)(f(0) − f(y))pdy∣ += ∣∫Ωi∖Br0 +√ +t(0) e−∥y∥2/tv ⋅ (−y)pdy∣ += +����������� +∫( 1 +√ +t Ωi)∖Br0(0) e−∥y∥2v ⋅ (−y +√ +t)td/2pdy +����������� +≤ td/2+1/2 ∫Rd∖Br0 +e−∥y∥2∥y∥pdy. +Now, by first changing to spherical coordinates and integrating out the an- +gular parts, we deduce that +∣I2∣ ≤ td/2+1/2 ∣Sd−1∣p∫ +∞ +r0 +e−s2sdds = ptd/2+1/2 ∣Sd−1∣Γ(d + 1 +2 +,r2 +0). +(4.2) +To finalize the bound of I2, we note that it follows from the assumption +t ≤ +R2 +d/2+1 that r2 +0 > d+1 +2 , and we can use (3.10) and (4.2) to conclude +∣I2∣ ≤ B(x)td/2+1/2e−r2 +0, +(4.3) + +11 +where B(x) is some function such that +B(x) ≤ d + 1 +2 +rd +0p∣Sd−1∣. +To bound I1, we use the following simple geometric fact: +∥x − y∥2 = ∥ˆx − y∥2 + ∥ˆx − x∥2 = ∥ˆx − y∥2 + sin2 θr2t, +which implies that +e−∥x−y∥2/t = e− sin2 θr2e−∥ˆx−y∥2/t. +From the above we can conclude +I1 = e− sin2 θr2 +∫BR(x)∩Ωi +e−∥ˆx−y∥2/tv ⋅ (x − y)pdy += e− sin2 θr2(∫BR(x)∩Ωi +e−∥ˆx−y∥2/tv ⋅ (x − ˆx)pdy ++ ∫BR(x)∩Ωi +e−∥ˆx−y∥2/tv ⋅ (ˆx − y)pdy) += e−r2 sin2 θ(II + III). +(4.4) +It is easier to integrate over ball centered around ˆx, and to this end we +define δ ≥ 0 by +δ = +√ +R2 − tr2 sin2 θ. +(4.5) +Then since ˆx is the orthogonal projection of x, we have that BR(x) ∩ Ωi = +Bδ(ˆx) ∩ Ωi. +Let us focus on II: We use the (4.5) and change to spherical coordinates, +which yields +II = v ⋅ (ˆx − x)td/2 ∫Bδ/ +√ +t(ˆx)∩Ωi +e−∥ˆx−y∥2pdy. += v ⋅ (ˆx − x)td/2∣Sd−1∣p∫ +δ/ +√ +t +0 +e−s2sd−1ds = v ⋅ (ˆx − x)td/2∣Sd−1∣pγ(d/2,δ2/t) += v ⋅ +ˆx − x +∥ˆx − x∥td/2+1/2r sinθ∣Sd−1∣pγ(d/2,δ2/t). +(4.6) +To estimate the RHS of (4.6) we will bound the γ from above and below: +Using r2 +0 ≥ d+2 +2 , r < 1 and the definition of δ, we get +d +2 ≤ r2 +0 − sin2 θr2 = δ2 +t . +By (3.11) we now see that +1 +2Γ(d/2) ≤ γ(d/2,d/2) ≤ γ(d/2,δ2/t). +(4.7) +Further, an application of (3.9) yields +γ(d/2,δ2/t) ≤ γ(d/2,r2 +0) = Γ(d/2) − Γ(d/2,r2 +0) ≤ Γ(d/2) − (r2 +0)d/2−1e−r2 +0 += Γ(d/2) − rd−2 +0 +e−r2 +0. +(4.8) +Now (4.6)–(4.8) together with ∣Sd−1∣ = 2πd/2 +Γ(d/2) finally gives +II = A(d,r0,θ)vΩitd/2+1/2r sinθ, + +12 +ANDERSSON AND AVELIN +where +max(pπd/2,2πd/2p − p∣Sd−1∣rd−2 +0 +e−r2 +0 ≤ A(d,r0,θ) ≤ 2pπd/2. +(4.9) +Finally, III = 0. This follows from that BR(x) ∩ ∂Ωi = ∅, the rotational +symmetry of K, and the fact that the linear function is odd. Collecting +(4.1), (4.3), (4.4) and (4.6) we get +Ltf(x) = td/2+1/2 (A(d,r0,θ)vn,Ωi sinθre− sin2 θr2 + B(x)e−r2 +0). +□ +The following theorem is an extension of Theorem 1 to the case when +the ball BR(x0) ∩ ∂Ωi ≠ ∅, which gives rise to an additional term in the +expression of Li +tf(x). We again refer to the schematic picture of Fig. 3 and +comments in Section 4 for explanation of the coordinates (r,θ), function +vn,Ωi and constant vn,∂Ωi. +Theorem 2. Let f(x) = v ⋅ x for some unit vector v ∈ RN, and assume +that p is the uniform density over Ω = ∪iΩi. Let x0 ∈ Ωi and assume that +∂Ωi ∩ B2R(x0) is part of a d − 1 dimensional plane for R = r0 +√ +t, where +r0 > 2. Further, x ∈ BR(x0), and vn,Ωi, vn,∂Ωi, r and θ are as described in +Section 4. If t ≤ +R2 +d/2+1, d ≥ 1 and r < 1, then we have that +Li +tf(x) = ̂ +A1(x)t +d+1 +2 vn,Ωi sin(θ)re− sin2(θ)r2 + ̂ +A2(x)t +d +2 vn,∂Ωie− sin2(θ)r2 ++ B(x)t +d+1 +2 e−r2 +0, +for explicitly computable function ̂ +A2, and with explicitly computable bounds +of function ̂ +A1. The function B has the same bounds as in Theorem 1. +Remark 4.2. The function ̂ +A1 is bounded by +1 +2δ0 +⎛ +⎝e−k2 +0γ (d − 1 +2 +,δ2 +0 − k2 +0) − 2(δ2 +0 − k2 +0) +d−1 +2 +d − 1 +⎞ +⎠ ≤ ̂ +A1 ≤ Γ(d − 1 +2 +)√π +and ̂ +A2 is given by +̂ +A2 = ∣Sd−2∣ +2 +(e−δ2 +0 (δ2 +0 − k2 +0)(d−1)/2 +d − 1 ++ 1 +2e−k2 +0γ (d − 1 +2 +,δ2 +0 − k2 +0).) +To define k0 and δ0, we recall the geometric picture of Section 4. Then K +is the projection of (ˆx − ˆx∂Ωi) to n∂Ωi, k0 = K/ +√ +t, and δ0 = +√ +r2 +0 − r2 sin2 θ. +Proof. We will follow the proof of Theorem 1 and modify where needed. Let +I2,II and III be defined as in (4.1) and (4.4). Then, since I2 is bounded +like in (4.3), we only need to find bounds for II and III. +Let δ be defined as in (4.5) and define δ0 = δ/ +√ +t. Recall also the fact that +BR(x) ∩ Ωi = Bδ(ˆx) ∩ Ωi. Now the difference in bounding II and III to the +proof of Theorem 1 is that Bδ(ˆx) ∩ ∂Ωi is nonempty. Since, by assumption, +∂Ωi is part of a d − 1-dimensional flat space, Bδ(ˆx) ∩ Ωi is a d-dimensional +ball, but missing a spherical cap. +We now use cylindrical coordinates (h,ϱ,ϕ) to describe the domain +Bδ/ +√ +t(ˆx) ∩ Ωi. +In these new coordinates we are centered around ˆx, and +(ϱ,ϕ) are coordinates for a d − 1-dimensional ball tangential to ∂Ω, while + +13 +the perpendicular coordinate h is oriented along the outwards normal of +∂Ωi. Let us denote this unit normal by n∂Ω, and the projection of ˆx to ∂Ω +by ˆx∂Ω. We now set K = (ˆx − ˆx∂Ω) ⋅ n∂Ω = +√ +tk0, where −δ0 ≤ k0 ≤ δ0. +Then, with III defined in (4.4) we get +III = ∫ +K +−δ ∫ +√ +δ2−h2 +0 +∫Sd−2 Kt(ˆx,y)v ⋅ (ˆx − y)ϱd−2 dϕdϱdh. +We split v into a normal component vn = (v ⋅ n∂Ω)n∂Ω and a component +vT = v−vn which is tangential to the boundary ∂Ω. Then, since the function +y → vT ⋅ (ˆx − y) is odd as a function centered around ˆx, and the domain of +integration is symmetric around ˆx, we know that the tangential component +of III satisfies +IIIT ∶= ∫ +K +−δ ∫ +√ +δ2−h2 +0 +∫Sd−2 Kt(ˆx,y)vT ⋅ (ˆx − y)ϱd−2 dϕdϱdh = 0. +By definition of vn,∂Ω, we have that vn⋅(ˆx−y) = vn,∂Ω(n∂Ω⋅(ˆx−y)) = vn,∂Ωh, +which implies that +III = vn,∂Ω ∫ +K +−δ ∫ +√ +δ2−h2 +0 +∫Sd−2 Kt(ˆx,y)hϱd−2 dϕdϱdh += vn,∂Ω ∫ +K +−δ ∫ +√ +δ2−h2 +0 +∫Sd−2 e−h2/t−ϱ2/thϱd−2 dϕdϱdh += td/2vn,∂Ω ∫ +k0 +−δ0 +he−h2 +∫ +√ +δ2 +0−h2 +0 +∫Sd−2 e−ϱ2ϱd−2 dϕdϱdh. +Continuing with the two inner integrals, +∫ +√ +δ2 +0−h2 +0 +∫Sd−2 e−ϱ2ϱd−2 dϕdϱ = ∣Sd−2∣ +2 +∫ +δ2 +0−h2 +0 +e−ssd/2−3/2 ds += ∣Sd−2∣ +2 +γ (d − 1 +2 +,δ2 +0 − h2). +Using this expression in the full integral and applying partial integration in +the second equality below yields +III = td/2vn,∂Ω +∣Sd−2∣ +2 +∫ +k0 +−δ0 +e−h2hγ (d − 1 +2 +,δ2 +0 − h2) dh += td/2vn,∂Ω +∣Sd−2∣ +2 +(1 +2 [−e−h2γ (d − 1 +2 +,δ2 +0 − h2)] +k0 +−δ0 +− 1 +2e−δ2 +0 ∫ +k0 +−δ0 +(δ2 − h2)(d−3)/2hdh) += td/2vn,∂Ω +∣Sd−2∣ +2 +(1 +2e−k2 +0γ (d − 1 +2 +,δ2 +0 − k2 +0) + e−δ2 +0 (δ2 +0 − k2 +0)(d−1)/2 +d − 1 +). +Thus, we know that +III = td/2vn,∂Ω +∣Sd−2∣ +2 +(e−δ2 +0 (δ2 +0 − k2 +0)(d−1)/2 +d − 1 ++ 1 +2e−k2 +0γ (d − 1 +2 +,δ2 +0 − k2 +0)). +(4.10) + +14 +ANDERSSON AND AVELIN +We now address the integral II defined in (4.4), which means we need to +calculate +J ∶= ∫BR(x)∩Ωi +e−∥ˆx−y∥2/tpdy. +After a change cylindrical coordinates as for III, we rewrite this integral as +J = ∫ +k0 +−δ0 +e−h2γ (d − 1 +2 +,δ2 +0 − h2) dh. +We can immediately bound J from above by +Γ(d − 1 +2 +)∫ +k0 +−δ0 +e−h2 dh ≤ Γ(d − 1 +2 +)∫ +∞ +−∞ e−h2dx = Γ(d − 1 +2 +)√π. +(4.11) +Now we bound J from below: Since the integrand is positive, we can with- +out loss of generality assume that k0 < 0. +Then a change of variables +h = − +√ +δ2 +0 − y yields that +J ≥ e−δ2 +0 ∫ +δ2 +0−k2 +0 +0 +eyγ (d − 1 +2 +,y) +1 +2 +√ +δ2 +0 − y +dy +≥ e−δ2 +0 1 +2δ0 ∫ +δ2 +0−k2 +0 +0 +eyγ (d − 1 +2 +,y) dy. +Using partial integration above we then get +J ≥ e−δ2 +0 +2δ +⎡⎢⎢⎢⎢⎣ +eyγ (d − 1 +2 +,y) − y +d−1 +2 +d−1 +2 +⎤⎥⎥⎥⎥⎦ +δ2 +0−k2 +0 +0 += e−δ2 +0 +2δ0 +⎛ +⎝eδ2 +0−k2 +0γ (d − 1 +2 +,δ2 +0 − k2 +0) − 2(δ2 +0 − k2 +0) +d−1 +2 +d − 1 +⎞ +⎠. +Simplifying further gives us +J ≥ 1 +2δ0 +⎛ +⎝e−k2 +0γ (d − 1 +2 +,δ2 +0 − k2 +0) − e−δ2 +0 2(δ2 +0 − k2 +0) +d−1 +2 +d − 1 +⎞ +⎠. +(4.12) +Thus, equation (4.10) and the bounds in (4.12) and (4.11) proves the theo- +rem. +□ +4.2. General manifolds. In this section we no longer assume that Ωi is +flat, but more general, as defined in 3.1. +We will also assume that Ω is +(L,r)-regular, see 3.2. +The type of singularity we deal with for a more +general manifold will be a Type 2, and we will assume we are not too close +to any boundary. +Theorem 3 (General manifold). Let f(x) = v⋅x for some unit vector v ∈ RN +and assume that p is the uniform density over a (L,2R)-regular union of +manifolds Ω = ∪Ωi. Let x0 ∈ Ωi and assume that ∂Ωi ∩ B2R(x0) = ∅ for +R = r0 +√ +t, where r0 > 2. Further, x ∈ BR(x0), and vn,Ωi, r and θ are as +described in Section 4. If L4R2 ≤ 1 +2, t ≤ +R2 +d/2+1, d ≥ 1 and r < 1, then we have +that +Li +tf(x) = td/2+1/2 ̂ +A(x)vn,Ωir sinθe−r2 sin2 θ + td/2CL,R(x)4pπd/2 + e−r2 +0D(x). + +15 +In the above, ̂ +A is a function such that +∣A(d,r,θ) − ̂ +A(x)∣ ≤ (1 + 3CL,R)A(d,r,θ) +where A(d,r,θ) as in Theorem 1; CL,R is a function such that +∣CL,R(x)∣ ≤ LR2(1 + 4LR2) + (4LR2)2; +and ∣D(x)∣ ≤ diam(Ω). +Proof. We begin by splitting up the domain Ωi: +Li +tf(x) = ∫Ωi +Kt(x,y)(f(x) − f(y))pdy += ∫Ωi∩BR(x) Kt(x,y)(f(x) − f(y))pdy ++ ∫Ωi∖BR(x) Kt(x,y)(f(x) − f(y))pdy += I + II. +(4.13) +We first note that +II = ∫Ωi∖BR(x) Kt(x,y)(f(x) − f(y))pdy ≤ e−R2/t diam(Ω). +(4.14) +To estimate I we will make a change of variables to the tangent space at x0 +and use arguments similar to those in the proof of Theorem 1. Specifically, +let π ∶ Ωi ∩ BR(x) → TΩi,x ∩ BR(x) be the projection map, and α = π−1 ○ i ∶ +Rd ∩ BR(0) → Ωi ∩ BR(x) a coordinate chart as in (3.1). We will use α to +integrate over TΩi,x0. +To simplify notation, we will use ˆx and ˆy to denote both π(x),π(y) ∈ RN, +and sometimes implicitly assume the projection i−1 such that ˆx, ˆy ∈ Rd. The +space in which these points lie should be clear from context. +Before making the coordinate change, we find bounds relating K(x,y) to +K(x, ˆy): We recall that Kt(x,y) = e +−∥x−y∥2 +t +, and from the triangle inequality +we get +e− ∥x−ˆy∥2 +t +− ∥y−ˆy∥2 +t +≤ e− ∥x−y∥2 +t +≤ e− ∥x−ˆy∥2 +t ++ ∥y−ˆy∥2 +t +. +(4.15) +Since Ωi is (L,2R)-regular, we use (3.3) and the fact that y ∈ B2R(x) to +conclude +∥y − ˆy∥ ≤ L∥x0 − ˆy∥2 ≤ L4R2, +which together with (4.15) yields +e−(L4R2)2Kt(x, ˆy) ≤ Kt(x,y) ≤ e(L4R2)2Kt(x, ˆy). +Furthermore, since L4R2 ≤ 1 +2 we have the bounds +e(L4R2)2 ≤ 1 + (L4R2)2 +and +e−(L4R2)2 ≥ 1 − (L4R2)2. +Thus, +∣Kt(x,y) − Kt(x, ˆy)∣ ≤ (L4R)2Kt(x, ˆy). +(4.16) +Replacing Kt(x,y) with Kt(x, ˆy) in I we get +I = ∫Ωi∩BR(x) Kt(x, ˆy)(f(x) − f(y))pdy + E1, +(4.17) + +16 +ANDERSSON AND AVELIN +and using (4.16) it holds that +∣E1∣ ≤ CL,R ∣∫Ωi∩BR(x) Kt(x, ˆy)(f(x) − f(y))pdy∣. +(4.18) +We now decompose the integral in (4.17) as follows +∫Ωi∩BR(x) Kt(x, ˆy)(f(x) − f(y))pdy = ∫Ωi∩BR(x) Kt(x, ˆy)(f(x) − f(ˆy))pdy ++ ∫Ωi∩BR(x) Kt(x, ˆy)(f(ˆy) − f(y))pdy = I1 + I2 +(4.19) +The quantity I2 will be treated like an error term. Using (3.3) we see that +∣I2∣ ≤ ∫Ωi∩BR(x) Kt(x, ˆy)L∥ˆy − x0∥2 pdy. +Now we make a coordinate change with α and use the bound on the volume +form in (3.4) to get +∫Ωi∩BR(x) Kt(x, ˆy)L∥ˆy − x0∥2 pdy +≤ LR2 ∫TΩi,x0∩BR(x) Kt(x, ˆy)(1 + L∥x0 − ˆy∥2)pdˆy +≤ LR2(1 + L4R2)∫TΩi,x0∩BR(x) Kt(x, ˆy)pdˆy +≤ CL,R ∫TΩi,x0∩BR(x) Kt(x, ˆy)pdˆy. +The RHS of the above display can be handled similarly to (4.6), which means +∣I2∣ ≤ CL,R ∣Sd−1∣td/2pΓ(d/2) = CL,Rtd/22pπd/2. +We proceed now with I1 from (4.19), which we want to estimate as accu- +rately as possible. Using the coordinate change α and (3.4) we write +I1 = e−r2 sin2 θ ∫Ωi∩BR(x) Kt(ˆx, ˆy)(f(x) − f(ˆy))pdy += e−r2 sin2 θ ̂C ∫TΩi,x0∩BR(x) Kt(ˆx, ˆy)(f(x) − f(ˆy))pdˆy, +(4.20) +where ̂C(x) is such that ∣ ̂C − 1∣ ≤ CL,R. +The integral on the right in (4.20) is exactly II from (4.4), which we +compute as in (4.6): +∫TΩi,x0∩BR(x) Kt(ˆx, ˆy)(f(x) − f(ˆy))pdˆy = A(d,r0θ)vΩitd/2+1/2r sinθ, (4.21) +where A(d,r0,θ) is as in (4.9). Now, from (4.19)–(4.21) we have +I1 + I2 = ̂CA(d,r0,θ)vΩitd/2+1/2r sinθe−r2 sin2(θ) + CL,Rtd/22pπd/2. +This combined with the split in (4.17) and (4.18) gives us +I = I1 + I2 + E1 = (1 + CL,R)(I1 + I2) += (1 + CL,R)( ̂CA(d,r0,θ)vΩitd/2+1/2r sinθe−r2 sin2(θ) + CL,Rtd/22pπd/2) + +17 +Defining ̂ +A(x) ∶= (1+CL,R) ̂CA(d,r,θ), and using that since CL,R ≤ 1, C2 +L,R ≤ +CL,R, I can be written as +I = td/2+1/2 ̂ +A(x)r sinθe−r2 sin2 θ + td/2CL,R4pπd/2. +(4.22) +Also, since ∣(1 + CL,R) ̂C∣ ≤ 1 + 3CL,R, we see that +∣A(d,r,θ) − ̂ +A(x)∣ ≤ (1 + 3CL,R)A(d,r,θ). +Finally then, the bounds in (4.22) and (4.14) give us +∫Ωi +K(x,y)(f(x) − f(y))dy = I + II += td/2+1/2 ̂ +A(x)r sinθe−r2 sin2 θ + td/2CL,R4pπd/2 + e−r2 +0D(x). +□ +The next lemma gives useful bounds on Li +tf(x) when x is non-singular. +Lemma 4.3. Given the conditions of Theorem 3 and the additional assump- +tion that x ∈ Ωi, we have that +Li +tf(x) = td/2+1/2 ̂ +A(x)8LR2 + td/2CL,R(x)4pπd/2 + D(x)e−r2 +0, +Proof. First applying Theorem 3 Li +tf(x), and then using the (L,2R) regu- +larity of Ωi, we bound the expression r sinθe−r2 sin2 θ in the following way: +First, by (3.3) we get +∣r sinθ∣ ≤ L∥x0 − ˆx∥2 ≤ L4R2. +Then, after the substitution x = r sinθ, we want to bound a function of the +form h(x) = xe−x2. Taylor expansion of h(x) gives that +∣h(x)∣ ≤ ∣x + 2x2∣ ≤ 2∣x∣, +for x ≤ 1 +2. Thus, for L4R2 ≤ 1 +2, we have that +∣r sinθe−r2 sin2 θ∣ ≤ 8LR2. +The conclusion follows. +□ +Remark 4.4. The result in Lemma 4.3 can be used together with both +Theorem 1 and Theorem 3 to analyze the behavior of the mapping x → +Lf(x) around intersections. +In the proof of the following corollary, the geometry is as in Section 4, +projecting x specifically to the tangent plane TΩ1,x0. +Corollary 4.5. Let f(x) = v ⋅ x for some vector x ∈ RN and assume that +p is the uniform density over a (L,2R)-regular manifold Ω = ∪2 +i=1Ωi. Let +x0 ∈ Ω1 ∩Ω2 and assume that ∂Ωi ∩B2R(x0) = ∅ for i ∈ {1,2} and R = r0 +√ +t, +where r0 > 2. If L4R2 ≤ 1 +2, t ≤ +R2 +d/2+1 and d ≥ 1, then for x ∈ BR(x0)∩Ω2 such +that ∥x − x0∥ = r +√ +t for r < 1, we have that +Ltf(x) = td/2+1/2 ̂ +A(x)vn,Ω1r sinθe−r2 sin2 θ1 + td/2+1/2 ̂ +A(x)8LR2 ++ td/2CL,R(x)8pπd/2 + 2e−r2 +0D(x) + +18 +ANDERSSON AND AVELIN +In the above, θ and vn,Ω1 are as in Section 4, with Ωi = Ω1. +Functions +̂ +A,CL,R and D are as in Theorem 3. +Proof. We apply Theorem 3 to Lt +1f(x) and Lemma 4.3 to L2 +t (x). +Ltf(x) = L1 +t f(x) + L2 +t f(x) += td/2+1/2 ̂ +A(x)vn,Ω1r sinθe−r2 sin2 θ1 + td/2+1/2 ̂ +A(x)8LR2 ++ td/2CL,R(x)8pπd/2 + 2e−r2 +0D(x) +□ +4.3. Manifolds with noise. In the previous results, we assumed that the +samples used to evaluate Ln,tf(x) are taken directly from Ω. +However, +in many applications it is more realistic to expect that the samples only +approximately lie on some manifold. +One way to model this is to assume instead of the operator +Ln,tf(x) = 1 +n +n +∑ +j=1 +Kt(x,Xj)(f(x) − f(Xj)), +we replace Xj by Xj + ϵj, where ϵj ∼ N(0,σ2I): +Ln,t,ϵf(x) = 1 +n +n +∑ +j=1 +Kt(x,Xj + ϵj)(f(x) − f(Xj + ϵj)). +The following theorem gives us the expected value of this operator: +Theorem 4 (Stochastic version). Let Ln,t,ϵ be as above, and the operator +Eϵ[⋅] = E[⋅ ∣ X1,...,XN] be expectation with regard to the random variables +(ϵ1,...,ϵn). Then +EϵLn,t,ϵf(x) = +2tN/2+1 +(2σ2 + t)N/2+1 +1 +n +n +∑ +j=1 +K2σ2+t (x,Xj)(f(x) − f(Xj)). +Proof. To simplify notation, let hj = x − Xj. +EϵLn,tf(x) = 1 +n +n +∑ +j=1 +EϵKt(x,Xj)(f(x) − f(Xj + ϵj)) += 1 +n +n +∑ +j=1 +Eϵe−∥hj−ϵj∥2/tv ⋅ (hj − ϵj) +(4.23) +Let us compute a single term in the sum in (4.23): Since the expectation is +w.r.t ϵ we can treat h = hj as fixed, and then algebraic manipulations give +us for z ∼ N(0,σ2I) +Eze−∥h+z∥2/t(h + z) += (2πσ2)−N/2 ∫RN e−∥h+z∥2/te−∥z∥2/(2σ2)v ⋅ (h + z)dz += (2πσ2)−N/2 ∫RN e−(∥h∥2/t+2⟨h,z⟩/t+∥z∥2/t+∥z∥2/(2σ2))v ⋅ (h + z)dz += (2πσ2)−N/2e− ∥h∥2 +2σ2+t ∫RN e− 1 +kt ∥z+kh∥2v ⋅ (h + z)dz. + +19 +In the second to last step we completed the square and used k = +2σ2 +2σ2+t. This +last integral can be viewed as the expectation +(πkt)−N/2 ∫RN e− 1 +kt ∥z+kh∥2v ⋅ (h + z)dz = EX[(h + X)] = (1 − k)v ⋅ h +where X ∼ N(−kh, I kt +2 ). Then we can conclude that +Eze−∥h+z∥2/tv ⋅ (h + z) = v ⋅ h +tN/2+1 +(2σ2 + t)N/2+1 e− ∥h∥2 +2σ2+t . +□ +The above theorem implies that if t′ = t + σ2 then, up to normalization, +Ln,t,ϵ and Ln,t′ are the same in expectation. This also shows the relationship +between the limit operators of EϵLn,t,ϵ and Ln,t, namely that: +lim +n→∞EϵLn,t,ϵf(x) = lim +n→∞Ln,t′f(x) = Lt′f(x) = Lt+σ2f(x). +4.4. Finite sample bounds. Our next result is a finite-sample bound +based on Hoeffding’s inequality. +This bound quantifies the maximal er- +ror of the operator Ln,t with respect to the limit operator Lt over the entire +manifold, when the operator is evaluated only at the known data points. +We assume that the Xi has a uniform density p over Ω. +Theorem 5. Let f(x) = v ⋅ x for x ∈ Ω, where Ω is flat. Then +P (max +i +∣Ln,tf(Xi) − n − 1 +n +Ltf(Xi)∣ > ϵ) ≤ 2nexp(− +2nϵ2 +(1 + πd/2td/2)2M2 ) +where M = supx,y∈Ω ∥v ⋅ (x − y)∥. +Proof. Using the union bound we get +P(max +i +∣Ln,tf(Xi) − n − 1 +n +Ltf(Xi)∣ > ϵ) +≤ +n +∑ +i=1 +P (∣Ln,tf(Xi) − n − 1 +n +Ltf(Xi)∣ > ϵ). +(4.24) +Using the definitions of Ln,t and Lt, see (3.5) and (3.6), and using that the +random variables X1,...,XN are i.i.d., we can replace each Xi by X1 in +each term in the summand of (4.24). Let Z be an independent copy of X1. +Then each summand in (4.24) equals +P(∣ 1 +n +n +∑ +j=1 +Kt(X1,Xj)(f(X1) − f(Xj)) +− n − 1 +n +EZ[Kt(X1,Z)(f(X1) − f(Z))]∣ > ϵ). +(4.25) +To simplify notation, we denote +Wi(x) = Kt(x,Xi)(f(x) − f(Xi)) +and +Yi(x) = Wi(x) − EXi[Wi(x)]. +We now rewrite (4.25) as +P (∣ +1 +n − 1 +n +∑ +i=2 +Yi(X1)∣ > +n +n − 1ϵ). + +20 +ANDERSSON AND AVELIN +Now by the tower property we have that +P (∣ +1 +n − 1 +n +∑ +i=2 +Yi(X1)∣ > +n +n − 1ϵ) = E[P (∣ +1 +n − 1 +n +∑ +i=2 +Yi(X1)∣ > +n +n − 1ϵ ∣ X1)] +In order to use Hoeffding’s inequality we need to show that Yi(x) is a +bounded random variable for all x ∈ Ω. First +∣Yi∣ ≤ ∣Wi∣+∣E[Wi]∣ ≤ M+∣∫Ω Kt(x,xi)(f(x) − f(xi))pdxi∣ ≤ (1+πd/2td/2p)M, +(4.26) +where M = supx,y∈Ω ∥v⋅(x−y)∥. Now Hoeffdings inequality states that (where +Cn = +n +n−1) +P(∣ +1 +n − 1 +n +∑ +i=2 +Yi∣ > Cnϵ ∣ X1) ≤ 2exp(− +2(n − 1)C2 +nϵ2 +(1 + πd/2td/2p)2M2 ) +and the proof is complete after taking expectations. +□ +Next is an extension to a more general type of manifold. +Corollary 4.6. Let Ω be a d-dimensional (L,R)-regular manifold, +{z1,z2,...,zK} ⊂ Ω, +and B = {BR(zi)}K +i=1 be a set of open balls in RN such that +∪K +i=1BR(xi) ∩ Ω = Ω. +Then the following inequality holds: +P (max +i +∣Ln,tf(Xi) − n − 1 +n +Ltf(Xi)∣ > ϵ) +≤ 2nexp(− +2nϵ2 +(1 + K(1 + LR2)SMtd/2πd/2p), +where M = supx,y∈Ω ∥v ⋅ (x − y)∥. +Proof. We begin proving a simple inequality: Since πzi is a projection to a +plane, implying ˆx − x and ˆy − y are parallel and perpendicular to ˆx − ˆy, we +have that +∥x − y∥2 = ∥ˆx − ˆy∥2 + ∥x − ˆx − (y − ˆy)∥2 . +This implies that ∥x − y∥2 ≥ ∥ˆx − ˆy∥2, and thus e−∥x−y∥2 ≤ e−∥ˆx−ˆy∥2. The last +inequality will be used later. +Now we just need to adapt the proof of Theorem 5, and the only part +that we need to change is the upper bound of +∣∫Ω Kt(x,xi)(f(x) − f(xi))pdxi∣ + +21 +in (4.26). We let πzi be the projection from BR(zi) ∩ Ω to TΩ,zi and denote +ˆx ∶= π(x). +∣∫Ω Kt(x,xi)(f(x) − f(xi))pdxi∣ += ∣ +K +∑ +i=1∫BR(zi)∩Ω Kt(x,xi)(f(x) − f(xi))pdxi∣ +≤ +K +∑ +i=1 +∣∫BR(zi)∩Ω Kt(x,xi)(f(x) − f(xi))pdxi∣. +Focusing now on one term, we use (3.4) and that e−∥x−y∥2 ≤ e∥ˆx−ˆy∥ to +conclude +∣∫BR(zi)∩Ω Kt(x,xi)(f(x) − f(xi))pdxi∣ +≤ (1 + LR2)∣∫BR(zi)∩TΩ,zi +Kt(x,xi)(f(x) − f(xi))pdxi∣ +≤ (1 + LR2)∣∫BR(zi)∩TΩ,zi +Kt(ˆx, ˆxi)(f(x) − f(xi))pdxi∣ +≤ (1 + LR2)M ∣∫BR(zi)∩TΩ,zi +Kt(ˆx, ˆxi)pdxi∣ +≤ (1 + LR2)M ∣∫Rd Kt(ˆx, ˆxi)pdxi∣ +≤ (1 + LR2)Mtd/2πd/2p. +Thus, +∣∫Ω Kt(x,xi)(f(x) − f(xi))pdxi∣ ≤ K(1 + LR2)Mtd/2πd/2p. +The result now follows by the same reasoning as in Theorem 5. +□ +5. Numerical Experiments +5.1. Estimating singularities. In the following experiments we demon- +strate how to estimate the point of intersection and intersecting angle θ of a +union of manifolds Ω = Ω1 ∪ Ω2. We will assume that we both have a set of +samples X ⊂ Ω distributed according to the associated density on Ω, and an +additional set of points Y from curve Γ ⊂ Ωi, for some i ∈ {1,2}. The curve +Γ intersects Ω1 ∩ Ω2, and we assume that no other singularity is very close, +which is a situation like in Fig. 5 and Fig. 8. +5.1.1. Outline of experiments and choice of estimators. Given the set of m +points Y = {y1,...,ym}, where each yi ∈ Γ, we evaluate Ln,tf on Y . This +gives us a set of values P = {p1,...,pm ∶ pi = Ln,tf(yi), pi ∈ Γ}. We know +that these, with enough samples, will be close to +Ltf(xi) = td/2+1/2 (A(d,r0,θ)vn,Ωi sinθirie− sin2 θir2 +i ) + Error(xi,r0,L), (5.1) +where the error term, which depends on xi,r0 and L, can be quantified with +the bounds in Section 4.1 and Section 4.2. We can always, by choosing t +small enough, make r0 however large we want to, and the function A(d,r0,θ) + +22 +ANDERSSON AND AVELIN +−3 +−2 +−1 +0 +1 +2 +3 +−0.4 +−0.2 +0 +0.2 +0.4 +xe−x2 +Figure 4. Graph of y = xe−x2 +can be made arbitrarily close to 2πd/2. The constant L is an upper bound +on how much curvature there is in Ω. +Remark 5.1. In Theorem 3 we have a function ̂ +A(x) instead of A(d,r0,θ), +but ̂ +A(x) can be made arbitrarily close to A(d,r0,θ) by choosing t or L +small enough. +The right-hand side of (5.1) depends on ri = ∥xi − x0∥, and the angle +θi = ∠(xi −x0, ˆxi −x0). Here ˆxi is the projection of xi onto either a manifold +Ωi or a tangent plane, as in Theorem 3, for some point x0. Thus, one can +say firstly that if ∣Ltf(xi)∣ > Error term, then there must be a point x0 +nearby such that ∣vn,Ωi sinθi∣ > 0. This in itself does not allow us to see the +difference between if Ω1 and Ω2 are just close together, or if there really is an +intersection. But if we can find points xi,xj such that Ltf(xi) > ∣Error term∣ +and Ltf(xj) < −∣Error term∣, then we can. This is because vn,Ω sinθi can +only change sign on Γ when passing through an intersection. +Further, looking at g(r,θ) = vn,Ωr sinθe−r2 sin2 θ, we notice that g only +depends on x = r sinθ (up to the sign of vn,Ωi). With some abuse of nota- +tion g(r,θ) = g(x), which is a rescaled (and possibly flipped) version of the +function h(x) = xe−x2. See Fig. 4 for the graph of h. +One easily sees that the minimal and maximal value of h are the points +z1 = − 1 +√ +2 and z2 = +1 +√ +2. The point of intersection will correspond to the +midpoint of these two points. In general then, we can estimate the point s +where Γ intersects Ω1 ∩ Ω2 by the midpoint of the maximum and minimum +value of the set P, as in +ˆs = arg maxxi(P) + arg minxi(P) +2 +We can also get an estimator ˆθ of θ. +First we let ˆrmax be an esti- +mate of the scaled distance from 0 that maximizes g(r,θ), namely ˆrmax = +∥ˆs − arg maxxi P∥/ +√ +t. Then, since maxr g(r,θ) = +1 +√ +2, ˆrmax sinθ ≈ +1 +√ +2, we + +23 +can estimate θ with +ˆθ = arcsin( +1 +√ +2ˆrmax +). +In the following we test these methods of estimation on angles between +hypersurfaces in R3 given by θ ∈ {π/2,π/4,π/8,π/16}, and with t = 10−3. +Remark 5.2. Performing these experiments inside R3 is mainly a conve- +nience here for visualization purposes. The reason for this is that we are +only working with points on the space Ω, which has an intrinsically low di- +mension. However, choosing a function f(x) = v ⋅x for Li +t to act on becomes +more difficult when the dimension is increased. This is because it is harder +to orient v in a way which vn,Ωi makes large: which is especially true of you +choose v randomly, as we do here. +For both flat and curved manifolds we perform 100 runs with random +choices of f(x) = v ⋅ x, and we sample v using the uniform distribution on +S2. For each such run we sample 2 × 104 points from both Ω1 and Ω2, from +a bounded region near the intersection, and evaluate Ln,tf on 103 uniformly +sampled points of Γ. These last evaluations give us our set P. +5.1.2. Flat manifolds. Here we test these methods in the case that Ω1,Ω2 +are flat. Since here we are integrating over two flat manifolds, +Ltf(x) = +2 +∑ +i=1∫Ωi +Kt(x,y)(f(x) − f(y))dy. +Using Theorem 1 together with Lemma 4.3 we get that for xi ∈ Γ +Ltf(xi) = td/2+1/2 (A(d,r0,θi)vn,Ωi sinθirie− sin2 θir2 +i + 2B(xi)e−r2 +0), +where θi and ri are in relation to xi and Ωi as explained in Section 4. For +example in Fig. 5, θi is the angle of the red and green planes. +Remark 5.3. There is some slight abuse of notation here in that we rather +have two different functions B1(xi) and B2(xi), one for each manifold Ω1,Ω2, +and we have implicitly defined a new function B(xi) ∶= B1(xi)+B2(xi) +2 +. +Let us first notice that since the manifolds are flat, the angle θi = θ, where +θ ∈ [0, π +2 ] is fixed. Then, since ∣B(x)∣ ≤ 2 +d+1 +2 rd +0 ∣Sd−1∣, it is sufficient that +max +i +Ln,tf(xi) > 2td/2+1/22 +d+1 +2 rd +0 ∣Sd−1∣e−r2 +0 +and +min +i +Ln,tf(xi) < −2td/2+1/22 +d+1 +2 rd +0 ∣Sd−1∣e−r2 +0, +to be able to say, with some probability that Γ intersects Ω1 ∩ Ω2. +In Fig. 5 we see our samples of Ω and Γ, and in Fig. 6 we see an example +of the values we get in P. Finally, in Fig. 7 we see how well this approach +works in trying to learn both θ and r. + +24 +ANDERSSON AND AVELIN +Figure 5. Samples of Ω = Ω1 ∪ Ω2 and Γ (blue), where Ω1 +(green) and Ω2 (red) have no curvature. +Figure 6. Ln,tf evaluated on Γ. Flat manifolds. + +=π/2 +θ=π/4 +0=π/8 +θ=π/16Ln.tfwith0= +Lntfwith= +2 +2 +2 +1 +0 +0 +-1 +-1 +-2 - +-2 +0 +500 +1000 +0 +500 +1000 +Ln.tfwith= +Lnfwith0= +8 +16 +2 - +2 +0 +-1 +-1 +2. +2 +0 +500 +1000 +0 +500 +100025 +Figure 7. Estimates of θ and s on flat manifolds. +5.1.3. Curved manifolds. Here we test these methods in the case that Ω = +Ω1 ∪Ω2 is (L,2R)-regular, with L = 0.5 and R having no upper bound. The +setup is the same as what we see in Fig. 8. +Using Corollary 4.5 we have that +Ltf(xi) = td/2+1/2 ̂ +A(x)vn,Ω1r sinθe−r2 sin2 θ1 + td/2+1/2 ̂ +A(x)8LR2 ++ td/2CL,R(x)8pπd/2 + 2e−r2 +0D(x). +Let us denote C ∶= LR2(1 + 4LR2) + (4LR2)2 and D ∶= diam(Ω). Then since +̂ +A ≤ 2πd/2, CL,R ≤ C and ̂ +A(x) ≤ (1 + 3C)2πd/2, we need that +max +i +Ln,tf(xi) > 2(td/2+1/2(1 + 3C)8LR2 + td/2C8pπd/2 + 2e−r2 +0) +and +max +i +Ln,tf(xi) < −2(td/2+1/2(1 + 3C)8LR2 + td/2C8pπd/2 + 2e−r2 +0) +to be able to say, with some probability that Γ intersects Ω1∩Ω2. Each term +above can be made arbitrarily by making L small and r0 large enough. +Remark 5.4. Since there is curvature, we cannot expect θi = θ for every i = +1,...,n, or even between any pair of them. However, we can still estimate the +location of the intersection as before, and estimating θ in this way provides +some information about the intersection, even if it is not as strong as in the +case without curvature. The range of possible values for θ, due to curvature, +can be bounded by knowing the curvature constant L. +In Fig. 8 we see our samples of Ω and Γ, in Fig. 9 we see an example of +the values we get in P. Finally, in Fig. 10 we see how well this approach +works in trying to learn both θ and r. + +1.5 +Estimates of A +True value of +1.2 +0.9 +: +0.6 +.8 +0.3 +00 +690 +T/2 +T/4 +T/8 +π/16Estimates of s +0.3 +True value of s +0.2 +0 +. +00 +8 +. +8 +0.1 +60 +8 +0.0 +T/2 +T/4 +T/8 +π/1626 +ANDERSSON AND AVELIN +Figure 8. Samples of Ω = Ω1 ∪ Ω2 and Γ (blue), where Ω1 +(green) and Ω2 (red) have curvature. +Figure 9. Ln,tf evaluated on Γ. Curved manifolds. + +0= π/2 +0=π/4 +=π/8 +θ=π/16Ln,f with 0=" +Lntfwith= +2 +4 +2.5 +2.5 - +0.0 +0.0 +-2.5 - +-2.5 - +0 +500 +1000 +0 +500 +1000 +Lntfwith0= +Lntf with 0=- +8 +16 +2.5 - +2.5 - +0.0 - +0.0 +-2.5- +-2.5 +0 +500 +1000 +0 +500 +100027 +Figure 10. Estimates of θ and s on curved manifolds. +6. Final remarks +In this paper we built upon the work of [2] and developed explicit versions +of their asymptotic analysis of x → Ltf(x). Our results are the strongest +and most useful in the case of flat manifolds, and the motivation to focus +on this scenario comes partly from Remark 4.1. +While the bounds in Theorem 3 are weaker, our numerical experiments +suggest that this approach can be useful for gaining geometric information +about the union of more general manifolds Ω = ∪iΩi. In [2], the authors +mainly considered sets Ω = ∪n +i=1Ωi with n ≤ 2. Our approach of splitting Lt +into components Li +t makes it easy to directly apply our theorems for n ≥ 2, +allowing us to consider a wider range of singularities. For example, we can +extend the framework to examine points that are both of Type 1 and Type +2, or to study intersections of more than two manifolds. A drawback of +this approach is that the error terms are compounded when they are just +added together for each Lt +i, but whether this is a problem will depend on +the specific application. +In our numerical experiments, we assumed that Ω = ∪2 +i=1Ωi and had access +to samples of a continuous curve Γ, which allowed us to estimate geomet- +ric properties near intersections. Future work could involve extending our +framework to other types of singularities and developing similar tests and +estimators. It would also be interesting to explore methods that do not rely +on direct access to such curves. +Similar theorems can be proven for other kernels besides the Gaussian +one, as many ideas used in our proofs are not specific to the Gaussian case, +but rather rely mainly on symmetries of Kt. Investigating the use of other +kernels and comparing their performance in different scenarios is a promising +direction for future research. +Acknowledgments +The first author was supported by the Wallenberg AI, Autonomous Sys- +tems and Software Program (WASP) funded by the Knut and Alice Wallen- +berg Foundation. The second author was supported by the Swedish Research +Council grant dnr: 2019-04098. +References +[1] M. Belkin and P. Niyogi, “Towards a theoretical foundation for laplacian-based man- +ifold methods,” Journal of Computer and System Sciences, vol. 74, no. 8, pp. 1289– +1308, 2008. + +1.5 +Estimates of +True value of +1.2 +0.9 +0.6 +6 +60 +969901 +0.3 +99 +00 +T/2 +π/4 +T/8 +π/16Estimates of s +0.3 +True value of s +90 +0.2 +8 +0 +0609006 +: +0.1 +0009 +: +0.0 +T/2 +T/4 +T/8 +π/1628 +ANDERSSON AND AVELIN +[2] M. Belkin, Q. Que, Y. Wang, and X. Zhou, “Toward understanding complex spaces: +Graph laplacians on manifolds with singularities and boundaries,” in Conference on +learning theory, pp. 36–1, JMLR Workshop and Conference Proceedings, 2012. +[3] M. Belkin and P. Niyogi, “Laplacian eigenmaps and spectral techniques for embedding +and clustering,” Advances in neural information processing systems, vol. 14, 2001. +[4] R. Kannan, S. Vempala, and A. Vetta, “On clusterings: Good, bad and spectral,” +Journal of the ACM (JACM), vol. 51, no. 3, pp. 497–515, 2004. +[5] U. v. Luxburg, O. Bousquet, and M. Belkin, “On the convergence of spectral clus- +tering on random samples: the normalized case,” in International Conference on +Computational Learning Theory, pp. 457–471, Springer, 2004. +[6] J. Shi and J. Malik, “Normalized cuts and image segmentation,” IEEE Transactions +on pattern analysis and machine intelligence, vol. 22, no. 8, pp. 888–905, 2000. +[7] A. Ng, M. Jordan, and Y. Weiss, “On spectral clustering: Analysis and an algorithm,” +Advances in neural information processing systems, vol. 14, 2001. +[8] M. Belkin and P. Niyogi, “Laplacian eigenmaps for dimensionality reduction and data +representation,” Neural computation, vol. 15, no. 6, pp. 1373–1396, 2003. +[9] B. Nadler, S. Lafon, R. R. Coifman, and I. G. Kevrekidis, “Diffusion maps, spectral +clustering and reaction coordinates of dynamical systems,” Applied and Computa- +tional Harmonic Analysis, vol. 21, no. 1, pp. 113–127, 2006. +[10] M. Belkin and P. Niyogi, “Semi-supervised learning on riemannian manifolds,” Ma- +chine learning, vol. 56, no. 1, pp. 209–239, 2004. +[11] M. Belkin and P. Niyogi, “Convergence of laplacian eigenmaps,” Advances in neural +information processing systems, vol. 19, 2006. +[12] O. Bousquet, O. Chapelle, and M. Hein, “Measure based regularization,” Advances +in Neural Information Processing Systems, vol. 16, 2003. +[13] S. S. Lafon, Diffusion maps and geometric harmonics. Yale University, 2004. +[14] J. R. Munkres, Analysis on manifolds. CRC Press, 2018. +[15] W. Gabcke, Neue Herleitung und explizite Restabsch¨atzung der Riemann-Siegel- +Formel. PhD thesis, Georg-August-Universit¨at G¨ottingen, 1979. +Email address: benny.avelin@math.uu.se +Martin Andersson, Department of Mathematics, Uppsala University, S-751 +06 Uppsala, Sweden +Email address: martin.andersson@math.uu.se +Benny Avelin, Department of Mathematics, Uppsala University, S-751 06 +Uppsala, Sweden + diff --git a/k9AyT4oBgHgl3EQfYPfO/content/tmp_files/load_file.txt b/k9AyT4oBgHgl3EQfYPfO/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5f81f6d84624dad2697e444ef4ecb26bb4c82aa6 --- /dev/null +++ b/k9AyT4oBgHgl3EQfYPfO/content/tmp_files/load_file.txt @@ -0,0 +1,821 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf,len=820 +page_content='EXPLORING SINGULARITIES IN POINT CLOUDS WITH THE GRAPH LAPLACIAN: AN EXPLICIT APPROACH MARTIN ANDERSSON AND BENNY AVELIN Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' We develop theory and methods that use the graph Lapla- cian to analyze the geometry of the underlying manifold of point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Our theory provides theoretical guarantees and explicit bounds on the functional form of the graph Laplacian, in the case when it acts on func- tions defined close to singularities of the underlying manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' We also propose methods that can be used to estimate these geometric properties of the point cloud, which are based on the theoretical guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Introduction High dimensional data is common in many research problems across aca- demic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' It is often assumed that a data set X = {Xi}n i ⊂ RN lies on a lower-dimensional set Ω and is in fact a sample from a probability distri- bution over Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' It is also often assumed that Ω can be represented as the union of several manifolds Ωi, where each Ωi represents a different class in a classification problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' For instance, if a data set contains two classes, i and j, class i might be contained in Ωi and class j in Ωj, with the two classes potentially being disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' However, classification is not always so clear-cut: For instance, in the MNIST dataset, where handwritten digits of ”1” ∈ Ω1 and ”7” ∈ Ω7 can appear very similar, suggesting that Ω1 ∩ Ω7 ≠ ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' There- fore, understanding geometric situations such as intersections is of interest in classification problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' In the manifold model of data, an intersection between two different man- ifolds Ωi,Ωj is either represented just as such, or it can be viewed as a sin- gularity if we consider Ω = Ωi ∪ Ωj as a single manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Other regions in Ω that can be viewed as singular, such as boundaries and edges, may also be of interest as they can signify important features in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' To study such singularities, we use the graph Laplacian Ln,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' This op- erator, which depends on the number of data points n and a parameter t, can act on functions defined on the data set X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' As n tends to infinity and t tends to 0, Ln,t converges to the Laplace-Beltrami operator in the interior of a single manifold [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' In this work, we primarily study the behavior of x → Ln,tf(x) for functions f, when x is close to singular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Our contribution in this paper is primarily an extension and reframing of work done in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' At the same time, we also focus on the specific case when the function f is assumed to be of the form f(x) = v ⋅ x, where v is a unit vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' We also consider more restricted classes of manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Primary 58K99;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Secondary 68R99, 60B99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Graph Laplacian, geometry, singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='00201v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='ML] 31 Dec 2022 2 ANDERSSON AND AVELIN Since Ln,t converges to Laplace-Beltrami, a second order differential op- erator, in the interior of Ω, we expect that for f as above Ln,tf(x) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' However, for singular points like intersections, the limit operator is of first order [2], and Ln,tf(x) ≠ 0, which can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Our results show how x → Ltf(x) and, through a finite-sample bound, how x → Ln,tf(x) behaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' More specifically, given x0 ∈ Ωi near some singularity, and x in the ball BR(x0), including the case when x /∈ Ωi, we show how the function x → Ln,tf(x) deviates from being constantly 0 and has specific functional forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' These forms depend on the type of singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' In [2] they showed what these forms are, up to some asymptotically defined error term, as t → 0, We build on this to get explicit expressions of Ltf(x) when t is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Overview of results: First, in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='1 we consider the case that Ω is flat manifold of dimension d, and where we have a geometric situation similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' To set up the results, we start with an x0 ∈ Ω, and let x ∈ BR(x0), where R = √ tr0 > 0, and use ˆx to denote the projection of x to Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' We also define vn,Ω as the projection of v onto x − ˆx, and vn,∂Ω is the projection of v onto the outwards normal of ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Then we show the following: In Theorem 1, we let ∥x − x0∥ = r √ t and θ is the angle between vectors x − x0 and ˆx − x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' If x is not close to ∂Ω, then Ltf(x) = A(x)t d+1 2 vn,Ω sin(θ)re− sin2(θ)r2 + t d+1 2 B(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' The function A is close to being constantly equal to πd/2, and B can be made, uniformly, arbitrarily small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Both functions have explicit bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Theorem 2 shows what happens when x is close to ∂Ω: Ltf(x) = ̂ A1(x)t d+1 2 vn,Ω sin(θ)re− sin2(θ)r2 + ̂ A2(x)t d 2 vn,∂Ωe− sin2(θ)r2 + B(x)t d+1 2 e−r2 0, where functions ̂ A1, ̂ A2 and B have explicitly computable bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='2 and Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='3 we prove more general results: In Theorem 3 we relax the conditions on Ω, considering non-flat manifolds, and prove a weaker version of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' In Theorem 4 we relax the conditions further, and allow for noise when sampling from Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' To connect Lt to Ln,t, in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='4 we prove two finite-sample bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Finally, in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='1 we propose methods to find intersections in data and estimate the angle of such intersections, which are motivated by the aforementioned theorems and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' We also provide numerical ex- periments, in Section 5, to test these methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Earlier work The framework of assuming an underlying low-dimensional manifold of data, in conjunction with graph-related tools and in particular the graph 3 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Graph Laplacian Ln,t acting on a linear function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Purple color showing positive, and green color negative values of Ln,tf, where lack of color indicates values near 0 Laplacian, has been used extensively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Some examples include work in clus- tering [3, 4, 5, 6, 7], dimensionality reduction [8, 9], and semi-supervised learning [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Several of the approaches to study data sets that use the graph Lapla- cian leverage that if the manifold is smooth enough and well-behaved, then the graph Laplacian approximates some well-understood operator (for in- stance the Laplace-Beltrami operator [11]), which has useful mathematical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Therefore, the question of convergence properties of the graph Laplacian is useful and important, and it has partly been explicated in [1, 12, 13, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' In particular, and highly influential of this paper, is what the asymptotic convergence looks like near singularities of the manifold, which was shown in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Basic mathematical objects and theory In this section, we provide more precise definitions and introduce the basic mathematical theory we will be using to present and prove our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' This is similar to the problem setup in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Conditions on manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' We will consider sets of the form Ω = ∪m i Ωi, where each Ωi is a smooth and compact d-dimensional Riemannian submanifold of RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' We will assume that if Ωi,Ωj, and i ≠ j have a non- empty intersection, then this intersection will have dimension lower than d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Associated to Ω will be a probability measure with density p ∶ Ω → R such that the restriction of p to Ωi is smooth, and there are constants a and b such that 0 < a ≤ p ≤ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' If x ∈ Ωi, we can consider the tangent space TΩi,x ≃ Rd, which we will identify as a subspace of the ambient space RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' More precisely, given open 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='25 Z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='75 +4 ANDERSSON AND AVELIN subsets U ⊂ Rd and W ⊂ Ωi (W is open in the subspace topology of Ωi), and a coordinate chart α ∶ U → W such that α(0) = x, we define TΩi,x as the image of Rd under the action of the Jacobian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' We denote the Jacobian Dα ∶ U → RN×d, evaluated at 0, by Dα(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' The best linear approximation to u ↦ α(u) is of course given by u ↦ x + Dα(0)u, and x + TΩi,x is the best flat approximation to Ωi around x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' The definition of Ω implies that a point x ∈ Ω can have more than one associated tangent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' For example, if x ∈ Ωi ∩ Ωj and i ≠ j, then both TΩi,x and TΩj,x exist, and they can be different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' A note on notation is that we will denote the interior of a manifold Ωi by IntΩi, and the boundary by ∂Ωi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Types of singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' The following are what we will refer to as singular points, which will be of four different kinds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Given x ∈ Ω = ∪Ωi, we have the following types: (Type 1) There is a submanifold Ωi such that x ∈ ∂Ωi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' (Type 2) There are submanifolds Ωi ≠ Ωj such that x ∈ IntΩi ∩ IntΩj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' (Type 3) There are submanifolds Ωi ≠ Ωj such that x ∈ ∂Ωi ∩ IntΩj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' (Type 4) There are submanifolds Ωi ≠ Ωj such that x ∈ ∂Ωi ∩ ∂Ωj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' The different types above can of course have non-empty intersection with each other, and a non-singular point is simply a point x ∈ IntΩi such that if j ≠ i, then x ≠ Ωj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' See Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='2 for two examples of singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' There is a singularity in the intersection of the lines above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' The left figure shows a point of Type 4, and the right figure shows a point of Type 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Integration on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' We will integrate scalar-valued functions, f ∶ Ω → R, over Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' When formulating integration of scalar-valued functions over submanifolds of RN, we follow the approach in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Because we need some preliminary results concerning integration on Ω, we make some important definitions explicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' First, let x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=',xk be vectors in RN for k ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' If I = (i1,i2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=',ik) is a k-tuple of integers such that i1 ≤ i2 ≤ ⋯ ≤ ik, define XI ∈ Rk×k as the k × k matrix containing only rows i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=',ik of the matrix X = (x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=',xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Now we can define the volume function V ∶ RN×k → R, by V (X) = √ det2(XtX) = [∑I det2 XI] 1/2, where the I’s span over k-tuples as above, see [14, Theorem 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' In general, given a coordinate chart α ∶ U → W, where U ⊂ Rd, W ⊂ Ωi ⊂ RN are open subsets, and Dα is the Jacobian of α, we can express =t=5 integration over W as ∫W f dV = ∫U f ○ α V(Dα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' In the coming proofs, when integrating around a point x ∈ IntΩi, we will change coordinates to the standard basis in TΩi,x = Rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' With this we mean that we can find open sets W ⊂ Ωi around x such that the projection map π ∶ W → B ⊂ x+TΩi,x is a diffeomorphism, where x+TΩi,x ∶= {x+y ∣ y ∈ TΩi,x }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' To integrate over TΩi,x we use the map π−1 precomposed with an inclusion map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' More specifically and without loss of generality, by translation and an orthonormal coordinate change, we can assume that TΩi,x = Rd × {0}n−d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' In this coordinate system we can write α ∶ U i�→ x + TΩi,x π−1 ��→ W ⊂ Ωi, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='1) where i is the natural inclusion map and U an open subset in Rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Important bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' The following bounds will be used later in our proofs: First, let TΩ,x,U,W,π be as in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Then for any y ∈ W, ∥y − π(y)∥ ≤ O(∥x − π(y)∥2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='2) This follows since Ωi is smooth and the tangent space represents the best flat approximation of Ωi around x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' To formulate the second bound, we need the lemma below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Let U,W,x,y,Ωi,π,i,α be as in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Then the follow- ing holds for the volume function V : V (Dα(y)) = 1 + O(∥x − π(y)∥2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Since α = π−1○i, and the tangent space is the best flat approximation of Ω, we can parametrize the W by α(u) = (u,g(u)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' It is then easy to see that for i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=',d we have ∂iα(y) = (ei,∂ig(u)), where ∂ig(0) = 0 and ∥∂ig(u)∥ = O(∥u∥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Now detDαI = ⎧⎪⎪⎨⎪⎪⎩ 1 if I = (1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=',d) O(∥u∥) otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' If we Taylor expand x → √x, we get V (Dα) = (∑ I (detDαI)2) 1/2 = 1 + O(∥u∥2), and by applying the above on (u,0) = x − π(y) we are finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' □ Further, since we have a finite union Ω = ∪iΩi and each Ωi is compact, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='2), the previous lemma implies that we can find a uniform bound L such that for all tuples (U,W,x,y,π,Ωi) ∥y − π(y)∥ ≤ L∥x − π(y)∥2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='3) and ∣V (Dα) − 1∣ ≤ L∥x − π(y)∥2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='4) 6 ANDERSSON AND AVELIN holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' (L,r)-regular manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' To formulate our results we will need some measure of how regular, with regard to curvature, our set Ω is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' The following definition captures the necessary information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Let Ω = ∪Ωi be a union of compact submanifolds in RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' We also let r > 0 be the largest radius such that any point x ∈ IntΩ allows coordinate charts α ∶ U → Br(x) ∩ Ωi, where U ⊂ Rd and Br(x) ⊂ RN is an open ball of radius r around x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Further, assume also that conditions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='4) hold over all tuples (U,W,x,y,π,Ωi) for some L > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Then we say that Ω is (L,r)-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Any smooth and compact submanifold is (L,r)-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' For instance the graph of the function x → x2 over the compact interval [−1,1] is (1,1)-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Graph Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' In this section we introduce the graph Laplacian and how it acts on real-valued functions defined on RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Given n i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' random samples X = {X1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=',Xn} from the distribution with density p on Ω, we build a weighted fully connected graph G = (V,E) as follows: We let each sample Xi represent a vertex i, and for vertices i,j ∈ V the weight on (i,j) ∈ E is given by Wn,t(i,j) ∶= Wn,t(Xi,Xj) = 1 nKt(Xi,Xj) = 1 ne− ∥Xi−Xj∥2 t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' The function Wn,t is naturally viewed as an n × n matrix, and the variable t is in the literature often referred to as the bandwidth of the kernel Kt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' In the limit analysis as n → ∞, it is useful also normalize by 1 td/2+1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' But, since a priori we do not know the dimension d, we will work without this normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' We define the diagonal weighted degree matrix as Dn,t(i,i) = ∑ j Wn,t(i,j), and the graph Laplacian Ln,t as Ln,t = Dn,t − Wn,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' This is often referred to as the unnormalized graph Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' There are other normalizations of this matrix which are used, for example, in [6, 7, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' One difference between these normalizations are their limit properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Given the fully connected graph G = (E,V ), the graph Laplacian above can be seen as an operator acting on arbitrary functions f ∶ V → R in the following way: Ln,tf(Xi) = 1 n ∑ j Kt(Xi,Xj)(f(Xi) − f(Xj)), (Xi,Xj) ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' 7 We extend this operator to acting on functions f ∈ Cc(RN,R), by the canon- ical choice Ln,tf(x) = 1 n ∑ j Kt(x,Xj)(f(x) − f(Xj)), x ∈ RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='5) Our main results will be stated in terms of the expected operator: Ltf(x) = Ep[Ln,tf(x)] = ∫Ω Kt(x,y)(f(x) − f(y))p(y)dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='6) That this is well-defined follows from the assumptions that X1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=',Xn are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=', that f is continuous and that Ω is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' One immediate consequence of the linearity of the integral is that Ltf(x) = ∫Ω Kt(x,y)(f(x) − f(y))dy = ∑ i ∫Ωi Kt(x,y)(f(x) − f(y))p(y)dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='7) In our approach it is useful to work with the restricted Laplacian Li t, which is defined by Li tf(x) = ∫Ωi Kt(x,y)(f(x) − f(y))p(y)dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='8) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Gamma functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' In the proofs of several of our results we will need to handle the Gamma function Γ(⋅), and both the lower and upper incomplete gamma functions, γ(⋅,⋅) and Γ(⋅,⋅) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' These are well- known and are defined by the equations Γ(a) = ∫ ∞ 0 ta−1e−t dt, γ(a,x) = ∫ x 0 ta−1e−t dt, Γ(a,x) = ∫ ∞ x ta−1e−t dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' In this paper both a and x are non-negative real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' We will need the following bounds: First, if a ≥ 1, then ta−1 ≥ xa−1 and Γ(a,x) ≥ xa−1 ∫ ∞ x e−x dt = xa−1e−x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='9) Secondly, if ex > 2a, then by [15, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='3], Γ(a,x) ≤ axa−1e−x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='10) Finally, we need the lower bound γ(a,a) ≥ 1 2Γ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='11) That this holds can be seen by viewing γ(a,x) as an unnormalized version of the cumulative distribution function of the Gamma distribution, for which it is well-known that the median ν is less than a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' 8 ANDERSSON AND AVELIN 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Main results Now that we have the necessary definitions and mathematical background, we are ready to present and prove our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Before stating the theorems, we will provide a brief section that explains the geometry of some terms that will be used in the theorem statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' This will help make the theorems easier to understand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Some of our results are given in the particular case when Ω = ∪Ωi is such that each Ωi is flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' This is easier to analyze and gives better bounds, but it is also motivated by a particular use-case: Sets of the form Ω∶ = {W ∈ Rk ∶ ∣fW (x) − g(x)∣ = 0, x ∈ D}, where fW is a neural network with weights W and ReLU activation func- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Here g is a target function, and D some dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' That is, the zero sets of the optimization problem which one tries to minimize during training of a common type of neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' General structure of results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='7) it is enough to understand the restricted Laplacian, Li t defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Because of this, our results are for- mulated to show the behavior of Li t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Depending on what type of singularity being examined, it is easy to extend the results to the full Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' In Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='5 we give one example of how to extend the results to the sum ∑2 i=1 Li t when one is close to an intersection of two manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Geometry and notation for Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' We will in several theorems also formulate the function x → Li tf(x) partly in terms of new coordinates (r,θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Here r is defined by the relation ∥x − x0∥ = √ tr, and given the projection ˆx of x to a plane Ωi, we define θ ∈ [0,π/2] to be the angle between vectors x0 − x and ˆx − x, as the schematic in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' By simple geometry, it also follows that ∥ˆx − x∥ = r sinθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Given a vector v ∈ RN, we will have reason to write the expression v⋅(ˆx−x) as v ⋅ (ˆx − x) = r √ tsin(θ)v ⋅ ˆx − x ∥ˆx − x∥ = r √ tsin(θ)vn,Ωi(x), where we have defined vn,Ωi(x) ∶= v ⋅ ˆx − x ∥ˆx − x∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' In other words, vn,Ωi is the projection of v onto a unit normal vector of Ωi, but it depends on x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' We define this function to be 0 when x = ˆx, and let us note that for x ≠ ˆx, this function is constant up to its sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' This implies that evaluating r √ tsin(θ)vn,Ωi(x) is the same as letting vn,Ωi be fixed, but allowing θ to change sign depending on which side of Ωi x is, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' as if we have fixed the coordinate system in which we measure the angle θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' We will in our theorem statements suppress the x-dependancy of vn,Ωi, to increase readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Additionally, in Theorem 2 we will have a term vn,∂Ωi that is specific to that theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' This will be defined in the case where there is a boundary close to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' 3, this would imply there is a boundary of Ω1 nearby.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' 9 To give the definition if this term, we first let ˆx∂Ωi be the projection of ˆx to ∂Ωi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' We can now define a unit normal at ˆx∂Ωi, denoted by n∂Ωi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Two choices are natural, a normal pointing either towards, or away from Ωi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' We define n∂Ωi as the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Given a vector v ∈ RN, we can define vn,∂Ω ∶= v ⋅ n∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' In Theorem 2 we will be close to part of the boundary ∂Ω where n∂Ω is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' This implies that, unlike vn,Ωi, vn,∂Ω does not depend on x, but is (locally) constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' x0 x sinθr √ t ˆx θ r √ t Ω1 Ω2 r0 √ t Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Schematic picture of the geometry of Theorem 1, where Ω1 is the object of interest and x ∈ Ω2 for visualization purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Geometry and notation for Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' To help with the geometric picture for general manifolds, the situation is as explained in Section 4 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' 3: the terms x,x0, ˆx,θ and vn,Ωi are in the same relation to each other as in Section 4, but instead of projecting x to a flat manifold Ωi we project x to the (flat) tangent plane TΩi,x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' In that sense the geometry for more general manifolds is not more difficult, but handling error terms is more involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Flat manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' In this section we assume that Ω = ∪iΩi, where each Ωi is a flat manifold, which means that each coordinate chart around x ∈ IntΩi is an isometry between an open neighborhood U of x, where U is a ball in Rd In Theorem 1 we give a result concerning the behavior of x → Li tf(x) when we are not close to the boundary ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' This case is easier to prove, and we give explicit bounds of all terms involved, and express them with elementary functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' In Theorem 2 we show what happens when we are close to ∂Ω, but we have more involved expressions for some terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' In the following theorems, it is the point x0 one should think of as po- tentially being a singular point, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' 3, and the theorems show us how x → Li tf(x) behaves in a neighborhood around this singular point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' By com- bining Theorem 1 and Theorem 2, it is possible to consider several types of singularities defined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' 10 ANDERSSON AND AVELIN Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Let f(x) = v ⋅ x for some unit vector v ∈ RN and assume that p is the uniform density over Ω = ∪iΩi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Let x0 ∈ Ωi and assume that ∂Ωi ∩ B2R(x0) = ∅ for R = r0 √ t, where r0 > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Further, x ∈ BR(x0), and vn,Ωi, r and θ are as described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' If t ≤ R2 d/2+1, d ≥ 1 and r < 1, then we have that Li tf(x) = td/2+1/2 (A(θ,r0,d)vn,Ωi sinθre− sin2 θr2 + B(x)e−r2 0), where A,B are real-valued functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' The function B depends on x and is uniformly bounded by ∣B(x)∣ ≤ 2 d+1 2 rd 0∣Sd−1∣;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' and A depends on x only through θ, and is bounded by max(πd/2,2πd/2 − ∣Sd−1∣2d/2rd−1 0 e−r2 0+1) ≤ A(θ,r0,d) ≤ 2πd/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Since x → Li tf(x) is translation and rotation invariant, we can with- out loss of generality assume that Ωi oriented in RN in such a way which makes it a subset of Rd × {0}N−d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' We want to evaluate Li tf(x) = ∫Ωi Kt(x,y)(f(x) − f(y))pdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' We begin by splitting the integral above into ∫Ωi Kt(x,y)(f(x) − f(y))pdy = ∫BR(x)∩Ωi Kt(x,y)(f(x) − f(y))pdy + ∫Ωi∖BR(x) Kt(x,y)(f(x) − f(y))pdy = I1 + I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='1) For estimating I2, by translation invariance we can WLOG assume that x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Now we make a change of variables and rescale y, which allows us to say that ∣I2∣ = ∣∫Ωi∖BR(0) Kt(0,y)(f(0) − f(y))pdy∣ = ∣∫Ωi∖Br0 √ t(0) e−∥y∥2/tv ⋅ (−y)pdy∣ = ����������� ∫( 1 √ t Ωi)∖Br0(0) e−∥y∥2v ⋅ (−y √ t)td/2pdy ����������� ≤ td/2+1/2 ∫Rd∖Br0 e−∥y∥2∥y∥pdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Now, by first changing to spherical coordinates and integrating out the an- gular parts, we deduce that ∣I2∣ ≤ td/2+1/2 ∣Sd−1∣p∫ ∞ r0 e−s2sdds = ptd/2+1/2 ∣Sd−1∣Γ(d + 1 2 ,r2 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='2) To finalize the bound of I2, we note that it follows from the assumption t ≤ R2 d/2+1 that r2 0 > d+1 2 , and we can use (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='10) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='2) to conclude ∣I2∣ ≤ B(x)td/2+1/2e−r2 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='3) 11 where B(x) is some function such that B(x) ≤ d + 1 2 rd 0p∣Sd−1∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' To bound I1, we use the following simple geometric fact: ∥x − y∥2 = ∥ˆx − y∥2 + ∥ˆx − x∥2 = ∥ˆx − y∥2 + sin2 θr2t, which implies that e−∥x−y∥2/t = e− sin2 θr2e−∥ˆx−y∥2/t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' From the above we can conclude I1 = e− sin2 θr2 ∫BR(x)∩Ωi e−∥ˆx−y∥2/tv ⋅ (x − y)pdy = e− sin2 θr2(∫BR(x)∩Ωi e−∥ˆx−y∥2/tv ⋅ (x − ˆx)pdy + ∫BR(x)∩Ωi e−∥ˆx−y∥2/tv ⋅ (ˆx − y)pdy) = e−r2 sin2 θ(II + III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='4) It is easier to integrate over ball centered around ˆx, and to this end we define δ ≥ 0 by δ = √ R2 − tr2 sin2 θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='5) Then since ˆx is the orthogonal projection of x, we have that BR(x) ∩ Ωi = Bδ(ˆx) ∩ Ωi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Let us focus on II: We use the (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='5) and change to spherical coordinates, which yields II = v ⋅ (ˆx − x)td/2 ∫Bδ/ √ t(ˆx)∩Ωi e−∥ˆx−y∥2pdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' = v ⋅ (ˆx − x)td/2∣Sd−1∣p∫ δ/ √ t 0 e−s2sd−1ds = v ⋅ (ˆx − x)td/2∣Sd−1∣pγ(d/2,δ2/t) = v ⋅ ˆx − x ∥ˆx − x∥td/2+1/2r sinθ∣Sd−1∣pγ(d/2,δ2/t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='6) To estimate the RHS of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='6) we will bound the γ from above and below: Using r2 0 ≥ d+2 2 , r < 1 and the definition of δ, we get d 2 ≤ r2 0 − sin2 θr2 = δ2 t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='11) we now see that 1 2Γ(d/2) ≤ γ(d/2,d/2) ≤ γ(d/2,δ2/t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='7) Further, an application of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='9) yields γ(d/2,δ2/t) ≤ γ(d/2,r2 0) = Γ(d/2) − Γ(d/2,r2 0) ≤ Γ(d/2) − (r2 0)d/2−1e−r2 0 = Γ(d/2) − rd−2 0 e−r2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='8) Now (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='6)–(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='8) together with ∣Sd−1∣ = 2πd/2 Γ(d/2) finally gives II = A(d,r0,θ)vΩitd/2+1/2r sinθ, 12 ANDERSSON AND AVELIN where max(pπd/2,2πd/2p − p∣Sd−1∣rd−2 0 e−r2 0 ≤ A(d,r0,θ) ≤ 2pπd/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='9) Finally, III = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' This follows from that BR(x) ∩ ∂Ωi = ∅, the rotational symmetry of K, and the fact that the linear function is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Collecting (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='3), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='4) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='6) we get Ltf(x) = td/2+1/2 (A(d,r0,θ)vn,Ωi sinθre− sin2 θr2 + B(x)e−r2 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' □ The following theorem is an extension of Theorem 1 to the case when the ball BR(x0) ∩ ∂Ωi ≠ ∅, which gives rise to an additional term in the expression of Li tf(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' We again refer to the schematic picture of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' 3 and comments in Section 4 for explanation of the coordinates (r,θ), function vn,Ωi and constant vn,∂Ωi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Let f(x) = v ⋅ x for some unit vector v ∈ RN, and assume that p is the uniform density over Ω = ∪iΩi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Let x0 ∈ Ωi and assume that ∂Ωi ∩ B2R(x0) is part of a d − 1 dimensional plane for R = r0 √ t, where r0 > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Further, x ∈ BR(x0), and vn,Ωi, vn,∂Ωi, r and θ are as described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' If t ≤ R2 d/2+1, d ≥ 1 and r < 1, then we have that Li tf(x) = ̂ A1(x)t d+1 2 vn,Ωi sin(θ)re− sin2(θ)r2 + ̂ A2(x)t d 2 vn,∂Ωie− sin2(θ)r2 + B(x)t d+1 2 e−r2 0, for explicitly computable function ̂ A2, and with explicitly computable bounds of function ̂ A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' The function B has the same bounds as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' The function ̂ A1 is bounded by 1 2δ0 ⎛ ⎝e−k2 0γ (d − 1 2 ,δ2 0 − k2 0) − 2(δ2 0 − k2 0) d−1 2 d − 1 ⎞ ⎠ ≤ ̂ A1 ≤ Γ(d − 1 2 )√π and ̂ A2 is given by ̂ A2 = ∣Sd−2∣ 2 (e−δ2 0 (δ2 0 − k2 0)(d−1)/2 d − 1 + 1 2e−k2 0γ (d − 1 2 ,δ2 0 − k2 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=') To define k0 and δ0, we recall the geometric picture of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Then K is the projection of (ˆx − ˆx∂Ωi) to n∂Ωi, k0 = K/ √ t, and δ0 = √ r2 0 − r2 sin2 θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' We will follow the proof of Theorem 1 and modify where needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Let I2,II and III be defined as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='1) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Then, since I2 is bounded like in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='3), we only need to find bounds for II and III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Let δ be defined as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='5) and define δ0 = δ/ √ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Recall also the fact that BR(x) ∩ Ωi = Bδ(ˆx) ∩ Ωi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Now the difference in bounding II and III to the proof of Theorem 1 is that Bδ(ˆx) ∩ ∂Ωi is nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Since, by assumption, ∂Ωi is part of a d − 1-dimensional flat space, Bδ(ˆx) ∩ Ωi is a d-dimensional ball, but missing a spherical cap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' We now use cylindrical coordinates (h,ϱ,ϕ) to describe the domain Bδ/ √ t(ˆx) ∩ Ωi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' In these new coordinates we are centered around ˆx, and (ϱ,ϕ) are coordinates for a d − 1-dimensional ball tangential to ∂Ω, while 13 the perpendicular coordinate h is oriented along the outwards normal of ∂Ωi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Let us denote this unit normal by n∂Ω, and the projection of ˆx to ∂Ω by ˆx∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' We now set K = (ˆx − ˆx∂Ω) ⋅ n∂Ω = √ tk0, where −δ0 ≤ k0 ≤ δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Then, with III defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='4) we get III = ∫ K −δ ∫ √ δ2−h2 0 ∫Sd−2 Kt(ˆx,y)v ⋅ (ˆx − y)ϱd−2 dϕdϱdh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' We split v into a normal component vn = (v ⋅ n∂Ω)n∂Ω and a component vT = v−vn which is tangential to the boundary ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Then, since the function y → vT ⋅ (ˆx − y) is odd as a function centered around ˆx, and the domain of integration is symmetric around ˆx, we know that the tangential component of III satisfies IIIT ∶= ∫ K −δ ∫ √ δ2−h2 0 ∫Sd−2 Kt(ˆx,y)vT ⋅ (ˆx − y)ϱd−2 dϕdϱdh = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' By definition of vn,∂Ω, we have that vn⋅(ˆx−y) = vn,∂Ω(n∂Ω⋅(ˆx−y)) = vn,∂Ωh, which implies that III = vn,∂Ω ∫ K −δ ∫ √ δ2−h2 0 ∫Sd−2 Kt(ˆx,y)hϱd−2 dϕdϱdh = vn,∂Ω ∫ K −δ ∫ √ δ2−h2 0 ∫Sd−2 e−h2/t−ϱ2/thϱd−2 dϕdϱdh = td/2vn,∂Ω ∫ k0 −δ0 he−h2 ∫ √ δ2 0−h2 0 ∫Sd−2 e−ϱ2ϱd−2 dϕdϱdh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Continuing with the two inner integrals, ∫ √ δ2 0−h2 0 ∫Sd−2 e−ϱ2ϱd−2 dϕdϱ = ∣Sd−2∣ 2 ∫ δ2 0−h2 0 e−ssd/2−3/2 ds = ∣Sd−2∣ 2 γ (d − 1 2 ,δ2 0 − h2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Using this expression in the full integral and applying partial integration in the second equality below yields III = td/2vn,∂Ω ∣Sd−2∣ 2 ∫ k0 −δ0 e−h2hγ (d − 1 2 ,δ2 0 − h2) dh = td/2vn,∂Ω ∣Sd−2∣ 2 (1 2 [−e−h2γ (d − 1 2 ,δ2 0 − h2)] k0 −δ0 − 1 2e−δ2 0 ∫ k0 −δ0 (δ2 − h2)(d−3)/2hdh) = td/2vn,∂Ω ∣Sd−2∣ 2 (1 2e−k2 0γ (d − 1 2 ,δ2 0 − k2 0) + e−δ2 0 (δ2 0 − k2 0)(d−1)/2 d − 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Thus, we know that III = td/2vn,∂Ω ∣Sd−2∣ 2 (e−δ2 0 (δ2 0 − k2 0)(d−1)/2 d − 1 + 1 2e−k2 0γ (d − 1 2 ,δ2 0 − k2 0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='10) 14 ANDERSSON AND AVELIN We now address the integral II defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='4), which means we need to calculate J ∶= ∫BR(x)∩Ωi e−∥ˆx−y∥2/tpdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' After a change cylindrical coordinates as for III, we rewrite this integral as J = ∫ k0 −δ0 e−h2γ (d − 1 2 ,δ2 0 − h2) dh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' We can immediately bound J from above by Γ(d − 1 2 )∫ k0 −δ0 e−h2 dh ≤ Γ(d − 1 2 )∫ ∞ −∞ e−h2dx = Γ(d − 1 2 )√π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='11) Now we bound J from below: Since the integrand is positive, we can with- out loss of generality assume that k0 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Then a change of variables h = − √ δ2 0 − y yields that J ≥ e−δ2 0 ∫ δ2 0−k2 0 0 eyγ (d − 1 2 ,y) 1 2 √ δ2 0 − y dy ≥ e−δ2 0 1 2δ0 ∫ δ2 0−k2 0 0 eyγ (d − 1 2 ,y) dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Using partial integration above we then get J ≥ e−δ2 0 2δ ⎡⎢⎢⎢⎢⎣ eyγ (d − 1 2 ,y) − y d−1 2 d−1 2 ⎤⎥⎥⎥⎥⎦ δ2 0−k2 0 0 = e−δ2 0 2δ0 ⎛ ⎝eδ2 0−k2 0γ (d − 1 2 ,δ2 0 − k2 0) − 2(δ2 0 − k2 0) d−1 2 d − 1 ⎞ ⎠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Simplifying further gives us J ≥ 1 2δ0 ⎛ ⎝e−k2 0γ (d − 1 2 ,δ2 0 − k2 0) − e−δ2 0 2(δ2 0 − k2 0) d−1 2 d − 1 ⎞ ⎠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='12) Thus, equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='10) and the bounds in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='12) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='11) proves the theo- rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' General manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' In this section we no longer assume that Ωi is flat, but more general, as defined in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' We will also assume that Ω is (L,r)-regular, see 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' The type of singularity we deal with for a more general manifold will be a Type 2, and we will assume we are not too close to any boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Theorem 3 (General manifold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Let f(x) = v⋅x for some unit vector v ∈ RN and assume that p is the uniform density over a (L,2R)-regular union of manifolds Ω = ∪Ωi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Let x0 ∈ Ωi and assume that ∂Ωi ∩ B2R(x0) = ∅ for R = r0 √ t, where r0 > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Further, x ∈ BR(x0), and vn,Ωi, r and θ are as described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' If L4R2 ≤ 1 2, t ≤ R2 d/2+1, d ≥ 1 and r < 1, then we have that Li tf(x) = td/2+1/2 ̂ A(x)vn,Ωir sinθe−r2 sin2 θ + td/2CL,R(x)4pπd/2 + e−r2 0D(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' 15 In the above, ̂ A is a function such that ∣A(d,r,θ) − ̂ A(x)∣ ≤ (1 + 3CL,R)A(d,r,θ) where A(d,r,θ) as in Theorem 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' CL,R is a function such that ∣CL,R(x)∣ ≤ LR2(1 + 4LR2) + (4LR2)2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' and ∣D(x)∣ ≤ diam(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' We begin by splitting up the domain Ωi: Li tf(x) = ∫Ωi Kt(x,y)(f(x) − f(y))pdy = ∫Ωi∩BR(x) Kt(x,y)(f(x) − f(y))pdy + ∫Ωi∖BR(x) Kt(x,y)(f(x) − f(y))pdy = I + II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='13) We first note that II = ∫Ωi∖BR(x) Kt(x,y)(f(x) − f(y))pdy ≤ e−R2/t diam(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='14) To estimate I we will make a change of variables to the tangent space at x0 and use arguments similar to those in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Specifically, let π ∶ Ωi ∩ BR(x) → TΩi,x ∩ BR(x) be the projection map, and α = π−1 ○ i ∶ Rd ∩ BR(0) → Ωi ∩ BR(x) a coordinate chart as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' We will use α to integrate over TΩi,x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' To simplify notation, we will use ˆx and ˆy to denote both π(x),π(y) ∈ RN, and sometimes implicitly assume the projection i−1 such that ˆx, ˆy ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' The space in which these points lie should be clear from context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Before making the coordinate change, we find bounds relating K(x,y) to K(x, ˆy): We recall that Kt(x,y) = e −∥x−y∥2 t , and from the triangle inequality we get e− ∥x−ˆy∥2 t − ∥y−ˆy∥2 t ≤ e− ∥x−y∥2 t ≤ e− ∥x−ˆy∥2 t + ∥y−ˆy∥2 t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='15) Since Ωi is (L,2R)-regular, we use (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='3) and the fact that y ∈ B2R(x) to conclude ∥y − ˆy∥ ≤ L∥x0 − ˆy∥2 ≤ L4R2, which together with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='15) yields e−(L4R2)2Kt(x, ˆy) ≤ Kt(x,y) ≤ e(L4R2)2Kt(x, ˆy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Furthermore, since L4R2 ≤ 1 2 we have the bounds e(L4R2)2 ≤ 1 + (L4R2)2 and e−(L4R2)2 ≥ 1 − (L4R2)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Thus, ∣Kt(x,y) − Kt(x, ˆy)∣ ≤ (L4R)2Kt(x, ˆy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='16) Replacing Kt(x,y) with Kt(x, ˆy) in I we get I = ∫Ωi∩BR(x) Kt(x, ˆy)(f(x) − f(y))pdy + E1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='17) 16 ANDERSSON AND AVELIN and using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='16) it holds that ∣E1∣ ≤ CL,R ∣∫Ωi∩BR(x) Kt(x, ˆy)(f(x) − f(y))pdy∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='18) We now decompose the integral in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='17) as follows ∫Ωi∩BR(x) Kt(x, ˆy)(f(x) − f(y))pdy = ∫Ωi∩BR(x) Kt(x, ˆy)(f(x) − f(ˆy))pdy + ∫Ωi∩BR(x) Kt(x, ˆy)(f(ˆy) − f(y))pdy = I1 + I2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='19) The quantity I2 will be treated like an error term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='3) we see that ∣I2∣ ≤ ∫Ωi∩BR(x) Kt(x, ˆy)L∥ˆy − x0∥2 pdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Now we make a coordinate change with α and use the bound on the volume form in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='4) to get ∫Ωi∩BR(x) Kt(x, ˆy)L∥ˆy − x0∥2 pdy ≤ LR2 ∫TΩi,x0∩BR(x) Kt(x, ˆy)(1 + L∥x0 − ˆy∥2)pdˆy ≤ LR2(1 + L4R2)∫TΩi,x0∩BR(x) Kt(x, ˆy)pdˆy ≤ CL,R ∫TΩi,x0∩BR(x) Kt(x, ˆy)pdˆy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' The RHS of the above display can be handled similarly to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='6), which means ∣I2∣ ≤ CL,R ∣Sd−1∣td/2pΓ(d/2) = CL,Rtd/22pπd/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' We proceed now with I1 from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='19), which we want to estimate as accu- rately as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Using the coordinate change α and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='4) we write I1 = e−r2 sin2 θ ∫Ωi∩BR(x) Kt(ˆx, ˆy)(f(x) − f(ˆy))pdy = e−r2 sin2 θ ̂C ∫TΩi,x0∩BR(x) Kt(ˆx, ˆy)(f(x) − f(ˆy))pdˆy, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='20) where ̂C(x) is such that ∣ ̂C − 1∣ ≤ CL,R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' The integral on the right in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='20) is exactly II from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='4), which we compute as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='6): ∫TΩi,x0∩BR(x) Kt(ˆx, ˆy)(f(x) − f(ˆy))pdˆy = A(d,r0θ)vΩitd/2+1/2r sinθ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='21) where A(d,r0,θ) is as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Now, from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='19)–(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='21) we have I1 + I2 = ̂CA(d,r0,θ)vΩitd/2+1/2r sinθe−r2 sin2(θ) + CL,Rtd/22pπd/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' This combined with the split in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='17) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='18) gives us I = I1 + I2 + E1 = (1 + CL,R)(I1 + I2) = (1 + CL,R)( ̂CA(d,r0,θ)vΩitd/2+1/2r sinθe−r2 sin2(θ) + CL,Rtd/22pπd/2) 17 Defining ̂ A(x) ∶= (1+CL,R) ̂CA(d,r,θ), and using that since CL,R ≤ 1, C2 L,R ≤ CL,R, I can be written as I = td/2+1/2 ̂ A(x)r sinθe−r2 sin2 θ + td/2CL,R4pπd/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='22) Also, since ∣(1 + CL,R) ̂C∣ ≤ 1 + 3CL,R, we see that ∣A(d,r,θ) − ̂ A(x)∣ ≤ (1 + 3CL,R)A(d,r,θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Finally then, the bounds in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='22) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='14) give us ∫Ωi K(x,y)(f(x) − f(y))dy = I + II = td/2+1/2 ̂ A(x)r sinθe−r2 sin2 θ + td/2CL,R4pπd/2 + e−r2 0D(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' □ The next lemma gives useful bounds on Li tf(x) when x is non-singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Given the conditions of Theorem 3 and the additional assump- tion that x ∈ Ωi, we have that Li tf(x) = td/2+1/2 ̂ A(x)8LR2 + td/2CL,R(x)4pπd/2 + D(x)e−r2 0, Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' First applying Theorem 3 Li tf(x), and then using the (L,2R) regu- larity of Ωi, we bound the expression r sinθe−r2 sin2 θ in the following way: First, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='3) we get ∣r sinθ∣ ≤ L∥x0 − ˆx∥2 ≤ L4R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Then, after the substitution x = r sinθ, we want to bound a function of the form h(x) = xe−x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Taylor expansion of h(x) gives that ∣h(x)∣ ≤ ∣x + 2x2∣ ≤ 2∣x∣, for x ≤ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Thus, for L4R2 ≤ 1 2, we have that ∣r sinθe−r2 sin2 θ∣ ≤ 8LR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' The conclusion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' The result in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='3 can be used together with both Theorem 1 and Theorem 3 to analyze the behavior of the mapping x → Lf(x) around intersections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' In the proof of the following corollary, the geometry is as in Section 4, projecting x specifically to the tangent plane TΩ1,x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Let f(x) = v ⋅ x for some vector x ∈ RN and assume that p is the uniform density over a (L,2R)-regular manifold Ω = ∪2 i=1Ωi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Let x0 ∈ Ω1 ∩Ω2 and assume that ∂Ωi ∩B2R(x0) = ∅ for i ∈ {1,2} and R = r0 √ t, where r0 > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' If L4R2 ≤ 1 2, t ≤ R2 d/2+1 and d ≥ 1, then for x ∈ BR(x0)∩Ω2 such that ∥x − x0∥ = r √ t for r < 1, we have that Ltf(x) = td/2+1/2 ̂ A(x)vn,Ω1r sinθe−r2 sin2 θ1 + td/2+1/2 ̂ A(x)8LR2 + td/2CL,R(x)8pπd/2 + 2e−r2 0D(x) 18 ANDERSSON AND AVELIN In the above, θ and vn,Ω1 are as in Section 4, with Ωi = Ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Functions ̂ A,CL,R and D are as in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' We apply Theorem 3 to Lt 1f(x) and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='3 to L2 t (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Ltf(x) = L1 t f(x) + L2 t f(x) = td/2+1/2 ̂ A(x)vn,Ω1r sinθe−r2 sin2 θ1 + td/2+1/2 ̂ A(x)8LR2 + td/2CL,R(x)8pπd/2 + 2e−r2 0D(x) □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Manifolds with noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' In the previous results, we assumed that the samples used to evaluate Ln,tf(x) are taken directly from Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' However, in many applications it is more realistic to expect that the samples only approximately lie on some manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' One way to model this is to assume instead of the operator Ln,tf(x) = 1 n n ∑ j=1 Kt(x,Xj)(f(x) − f(Xj)), we replace Xj by Xj + ϵj, where ϵj ∼ N(0,σ2I): Ln,t,ϵf(x) = 1 n n ∑ j=1 Kt(x,Xj + ϵj)(f(x) − f(Xj + ϵj)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' The following theorem gives us the expected value of this operator: Theorem 4 (Stochastic version).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Let Ln,t,ϵ be as above, and the operator Eϵ[⋅] = E[⋅ ∣ X1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=',XN] be expectation with regard to the random variables (ϵ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=',ϵn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Then EϵLn,t,ϵf(x) = 2tN/2+1 (2σ2 + t)N/2+1 1 n n ∑ j=1 K2σ2+t (x,Xj)(f(x) − f(Xj)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' To simplify notation, let hj = x − Xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' EϵLn,tf(x) = 1 n n ∑ j=1 EϵKt(x,Xj)(f(x) − f(Xj + ϵj)) = 1 n n ∑ j=1 Eϵe−∥hj−ϵj∥2/tv ⋅ (hj − ϵj) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='23) Let us compute a single term in the sum in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='23): Since the expectation is w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='t ϵ we can treat h = hj as fixed, and then algebraic manipulations give us for z ∼ N(0,σ2I) Eze−∥h+z∥2/t(h + z) = (2πσ2)−N/2 ∫RN e−∥h+z∥2/te−∥z∥2/(2σ2)v ⋅ (h + z)dz = (2πσ2)−N/2 ∫RN e−(∥h∥2/t+2⟨h,z⟩/t+∥z∥2/t+∥z∥2/(2σ2))v ⋅ (h + z)dz = (2πσ2)−N/2e− ∥h∥2 2σ2+t ∫RN e− 1 kt ∥z+kh∥2v ⋅ (h + z)dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' 19 In the second to last step we completed the square and used k = 2σ2 2σ2+t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' This last integral can be viewed as the expectation (πkt)−N/2 ∫RN e− 1 kt ∥z+kh∥2v ⋅ (h + z)dz = EX[(h + X)] = (1 − k)v ⋅ h where X ∼ N(−kh, I kt 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Then we can conclude that Eze−∥h+z∥2/tv ⋅ (h + z) = v ⋅ h tN/2+1 (2σ2 + t)N/2+1 e− ∥h∥2 2σ2+t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' □ The above theorem implies that if t′ = t + σ2 then, up to normalization, Ln,t,ϵ and Ln,t′ are the same in expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' This also shows the relationship between the limit operators of EϵLn,t,ϵ and Ln,t, namely that: lim n→∞EϵLn,t,ϵf(x) = lim n→∞Ln,t′f(x) = Lt′f(x) = Lt+σ2f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Finite sample bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Our next result is a finite-sample bound based on Hoeffding’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' This bound quantifies the maximal er- ror of the operator Ln,t with respect to the limit operator Lt over the entire manifold, when the operator is evaluated only at the known data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' We assume that the Xi has a uniform density p over Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Let f(x) = v ⋅ x for x ∈ Ω, where Ω is flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Then P (max i ∣Ln,tf(Xi) − n − 1 n Ltf(Xi)∣ > ϵ) ≤ 2nexp(− 2nϵ2 (1 + πd/2td/2)2M2 ) where M = supx,y∈Ω ∥v ⋅ (x − y)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Using the union bound we get P(max i ∣Ln,tf(Xi) − n − 1 n Ltf(Xi)∣ > ϵ) ≤ n ∑ i=1 P (∣Ln,tf(Xi) − n − 1 n Ltf(Xi)∣ > ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='24) Using the definitions of Ln,t and Lt, see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='5) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='6), and using that the random variables X1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=',XN are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=', we can replace each Xi by X1 in each term in the summand of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Let Z be an independent copy of X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Then each summand in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='24) equals P(∣ 1 n n ∑ j=1 Kt(X1,Xj)(f(X1) − f(Xj)) − n − 1 n EZ[Kt(X1,Z)(f(X1) − f(Z))]∣ > ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='25) To simplify notation, we denote Wi(x) = Kt(x,Xi)(f(x) − f(Xi)) and Yi(x) = Wi(x) − EXi[Wi(x)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' We now rewrite (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='25) as P (∣ 1 n − 1 n ∑ i=2 Yi(X1)∣ > n n − 1ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' 20 ANDERSSON AND AVELIN Now by the tower property we have that P (∣ 1 n − 1 n ∑ i=2 Yi(X1)∣ > n n − 1ϵ) = E[P (∣ 1 n − 1 n ∑ i=2 Yi(X1)∣ > n n − 1ϵ ∣ X1)] In order to use Hoeffding’s inequality we need to show that Yi(x) is a bounded random variable for all x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' First ∣Yi∣ ≤ ∣Wi∣+∣E[Wi]∣ ≤ M+∣∫Ω Kt(x,xi)(f(x) − f(xi))pdxi∣ ≤ (1+πd/2td/2p)M, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='26) where M = supx,y∈Ω ∥v⋅(x−y)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Now Hoeffdings inequality states that (where Cn = n n−1) P(∣ 1 n − 1 n ∑ i=2 Yi∣ > Cnϵ ∣ X1) ≤ 2exp(− 2(n − 1)C2 nϵ2 (1 + πd/2td/2p)2M2 ) and the proof is complete after taking expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' □ Next is an extension to a more general type of manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Let Ω be a d-dimensional (L,R)-regular manifold, {z1,z2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=',zK} ⊂ Ω, and B = {BR(zi)}K i=1 be a set of open balls in RN such that ∪K i=1BR(xi) ∩ Ω = Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Then the following inequality holds: P (max i ∣Ln,tf(Xi) − n − 1 n Ltf(Xi)∣ > ϵ) ≤ 2nexp(− 2nϵ2 (1 + K(1 + LR2)SMtd/2πd/2p), where M = supx,y∈Ω ∥v ⋅ (x − y)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' We begin proving a simple inequality: Since πzi is a projection to a plane, implying ˆx − x and ˆy − y are parallel and perpendicular to ˆx − ˆy, we have that ∥x − y∥2 = ∥ˆx − ˆy∥2 + ∥x − ˆx − (y − ˆy)∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' This implies that ∥x − y∥2 ≥ ∥ˆx − ˆy∥2, and thus e−∥x−y∥2 ≤ e−∥ˆx−ˆy∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' The last inequality will be used later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Now we just need to adapt the proof of Theorem 5, and the only part that we need to change is the upper bound of ∣∫Ω Kt(x,xi)(f(x) − f(xi))pdxi∣ 21 in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' We let πzi be the projection from BR(zi) ∩ Ω to TΩ,zi and denote ˆx ∶= π(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' ∣∫Ω Kt(x,xi)(f(x) − f(xi))pdxi∣ = ∣ K ∑ i=1∫BR(zi)∩Ω Kt(x,xi)(f(x) − f(xi))pdxi∣ ≤ K ∑ i=1 ∣∫BR(zi)∩Ω Kt(x,xi)(f(x) − f(xi))pdxi∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Focusing now on one term, we use (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='4) and that e−∥x−y∥2 ≤ e∥ˆx−ˆy∥ to conclude ∣∫BR(zi)∩Ω Kt(x,xi)(f(x) − f(xi))pdxi∣ ≤ (1 + LR2)∣∫BR(zi)∩TΩ,zi Kt(x,xi)(f(x) − f(xi))pdxi∣ ≤ (1 + LR2)∣∫BR(zi)∩TΩ,zi Kt(ˆx, ˆxi)(f(x) − f(xi))pdxi∣ ≤ (1 + LR2)M ∣∫BR(zi)∩TΩ,zi Kt(ˆx, ˆxi)pdxi∣ ≤ (1 + LR2)M ∣∫Rd Kt(ˆx, ˆxi)pdxi∣ ≤ (1 + LR2)Mtd/2πd/2p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Thus, ∣∫Ω Kt(x,xi)(f(x) − f(xi))pdxi∣ ≤ K(1 + LR2)Mtd/2πd/2p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' The result now follows by the same reasoning as in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Numerical Experiments 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Estimating singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' In the following experiments we demon- strate how to estimate the point of intersection and intersecting angle θ of a union of manifolds Ω = Ω1 ∪ Ω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' We will assume that we both have a set of samples X ⊂ Ω distributed according to the associated density on Ω, and an additional set of points Y from curve Γ ⊂ Ωi, for some i ∈ {1,2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' The curve Γ intersects Ω1 ∩ Ω2, and we assume that no other singularity is very close, which is a situation like in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' 5 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Outline of experiments and choice of estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Given the set of m points Y = {y1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=',ym}, where each yi ∈ Γ, we evaluate Ln,tf on Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' This gives us a set of values P = {p1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=',pm ∶ pi = Ln,tf(yi), pi ∈ Γ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' We know that these, with enough samples, will be close to Ltf(xi) = td/2+1/2 (A(d,r0,θ)vn,Ωi sinθirie− sin2 θir2 i ) + Error(xi,r0,L), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='1) where the error term, which depends on xi,r0 and L, can be quantified with the bounds in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='1 and Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' We can always, by choosing t small enough, make r0 however large we want to, and the function A(d,r0,θ) 22 ANDERSSON AND AVELIN −3 −2 −1 0 1 2 3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='4 xe−x2 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Graph of y = xe−x2 can be made arbitrarily close to 2πd/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' The constant L is an upper bound on how much curvature there is in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' In Theorem 3 we have a function ̂ A(x) instead of A(d,r0,θ), but ̂ A(x) can be made arbitrarily close to A(d,r0,θ) by choosing t or L small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' The right-hand side of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='1) depends on ri = ∥xi − x0∥, and the angle θi = ∠(xi −x0, ˆxi −x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Here ˆxi is the projection of xi onto either a manifold Ωi or a tangent plane, as in Theorem 3, for some point x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Thus, one can say firstly that if ∣Ltf(xi)∣ > Error term, then there must be a point x0 nearby such that ∣vn,Ωi sinθi∣ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' This in itself does not allow us to see the difference between if Ω1 and Ω2 are just close together, or if there really is an intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' But if we can find points xi,xj such that Ltf(xi) > ∣Error term∣ and Ltf(xj) < −∣Error term∣, then we can.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' This is because vn,Ω sinθi can only change sign on Γ when passing through an intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Further, looking at g(r,θ) = vn,Ωr sinθe−r2 sin2 θ, we notice that g only depends on x = r sinθ (up to the sign of vn,Ωi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' With some abuse of nota- tion g(r,θ) = g(x), which is a rescaled (and possibly flipped) version of the function h(x) = xe−x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' 4 for the graph of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' One easily sees that the minimal and maximal value of h are the points z1 = − 1 √ 2 and z2 = 1 √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' The point of intersection will correspond to the midpoint of these two points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' In general then, we can estimate the point s where Γ intersects Ω1 ∩ Ω2 by the midpoint of the maximum and minimum value of the set P, as in ˆs = arg maxxi(P) + arg minxi(P) 2 We can also get an estimator ˆθ of θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' First we let ˆrmax be an esti- mate of the scaled distance from 0 that maximizes g(r,θ), namely ˆrmax = ∥ˆs − arg maxxi P∥/ √ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Then, since maxr g(r,θ) = 1 √ 2, ˆrmax sinθ ≈ 1 √ 2, we 23 can estimate θ with ˆθ = arcsin( 1 √ 2ˆrmax ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' In the following we test these methods of estimation on angles between hypersurfaces in R3 given by θ ∈ {π/2,π/4,π/8,π/16}, and with t = 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Performing these experiments inside R3 is mainly a conve- nience here for visualization purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' The reason for this is that we are only working with points on the space Ω, which has an intrinsically low di- mension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' However, choosing a function f(x) = v ⋅x for Li t to act on becomes more difficult when the dimension is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' This is because it is harder to orient v in a way which vn,Ωi makes large: which is especially true of you choose v randomly, as we do here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' For both flat and curved manifolds we perform 100 runs with random choices of f(x) = v ⋅ x, and we sample v using the uniform distribution on S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' For each such run we sample 2 × 104 points from both Ω1 and Ω2, from a bounded region near the intersection, and evaluate Ln,tf on 103 uniformly sampled points of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' These last evaluations give us our set P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Flat manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Here we test these methods in the case that Ω1,Ω2 are flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Since here we are integrating over two flat manifolds, Ltf(x) = 2 ∑ i=1∫Ωi Kt(x,y)(f(x) − f(y))dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Using Theorem 1 together with Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='3 we get that for xi ∈ Γ Ltf(xi) = td/2+1/2 (A(d,r0,θi)vn,Ωi sinθirie− sin2 θir2 i + 2B(xi)e−r2 0), where θi and ri are in relation to xi and Ωi as explained in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' For example in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' 5, θi is the angle of the red and green planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' There is some slight abuse of notation here in that we rather have two different functions B1(xi) and B2(xi), one for each manifold Ω1,Ω2, and we have implicitly defined a new function B(xi) ∶= B1(xi)+B2(xi) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Let us first notice that since the manifolds are flat, the angle θi = θ, where θ ∈ [0, π 2 ] is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Then, since ∣B(x)∣ ≤ 2 d+1 2 rd 0 ∣Sd−1∣, it is sufficient that max i Ln,tf(xi) > 2td/2+1/22 d+1 2 rd 0 ∣Sd−1∣e−r2 0 and min i Ln,tf(xi) < −2td/2+1/22 d+1 2 rd 0 ∣Sd−1∣e−r2 0, to be able to say, with some probability that Γ intersects Ω1 ∩ Ω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' 5 we see our samples of Ω and Γ, and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' 6 we see an example of the values we get in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Finally, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' 7 we see how well this approach works in trying to learn both θ and r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' 24 ANDERSSON AND AVELIN Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Samples of Ω = Ω1 ∪ Ω2 and Γ (blue), where Ω1 (green) and Ω2 (red) have no curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Ln,tf evaluated on Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Flat manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' =π/2 θ=π/4 0=π/8 θ=π/16Ln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='tfwith0= Lntfwith= 2 2 2 1 0 0 1 1 2 - 2 0 500 1000 0 500 1000 Ln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='tfwith= Lnfwith0= 8 16 2 - 2 0 1 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' 2 0 500 1000 0 500 100025 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Estimates of θ and s on flat manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Curved manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Here we test these methods in the case that Ω = Ω1 ∪Ω2 is (L,2R)-regular, with L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='5 and R having no upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' The setup is the same as what we see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Using Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='5 we have that Ltf(xi) = td/2+1/2 ̂ A(x)vn,Ω1r sinθe−r2 sin2 θ1 + td/2+1/2 ̂ A(x)8LR2 + td/2CL,R(x)8pπd/2 + 2e−r2 0D(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Let us denote C ∶= LR2(1 + 4LR2) + (4LR2)2 and D ∶= diam(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Then since ̂ A ≤ 2πd/2, CL,R ≤ C and ̂ A(x) ≤ (1 + 3C)2πd/2, we need that max i Ln,tf(xi) > 2(td/2+1/2(1 + 3C)8LR2 + td/2C8pπd/2 + 2e−r2 0) and max i Ln,tf(xi) < −2(td/2+1/2(1 + 3C)8LR2 + td/2C8pπd/2 + 2e−r2 0) to be able to say, with some probability that Γ intersects Ω1∩Ω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Each term above can be made arbitrarily by making L small and r0 large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Since there is curvature, we cannot expect θi = θ for every i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=',n, or even between any pair of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' However, we can still estimate the location of the intersection as before, and estimating θ in this way provides some information about the intersection, even if it is not as strong as in the case without curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' The range of possible values for θ, due to curvature, can be bounded by knowing the curvature constant L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' 8 we see our samples of Ω and Γ, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' 9 we see an example of the values we get in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Finally, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' 10 we see how well this approach works in trying to learn both θ and r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='5 Estimates of A True value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='9 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='3 00 690 T/2 T/4 T/8 π/16Estimates of s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='3 True value of s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='2 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' 00 8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='1 60 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='0 T/2 T/4 T/8 π/1626 ANDERSSON AND AVELIN Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Samples of Ω = Ω1 ∪ Ω2 and Γ (blue), where Ω1 (green) and Ω2 (red) have curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Ln,tf evaluated on Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Curved manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' 0= π/2 0=π/4 =π/8 θ=π/16Ln,f with 0=" Lntfwith= 2 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='5 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='5 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='5 - 0 500 1000 0 500 1000 Lntfwith0= Lntf with 0=- 8 16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='5 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='5 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='0 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='5- 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='5 0 500 1000 0 500 100027 Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Estimates of θ and s on curved manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Final remarks In this paper we built upon the work of [2] and developed explicit versions of their asymptotic analysis of x → Ltf(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Our results are the strongest and most useful in the case of flat manifolds, and the motivation to focus on this scenario comes partly from Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' While the bounds in Theorem 3 are weaker, our numerical experiments suggest that this approach can be useful for gaining geometric information about the union of more general manifolds Ω = ∪iΩi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' In [2], the authors mainly considered sets Ω = ∪n i=1Ωi with n ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Our approach of splitting Lt into components Li t makes it easy to directly apply our theorems for n ≥ 2, allowing us to consider a wider range of singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' For example, we can extend the framework to examine points that are both of Type 1 and Type 2, or to study intersections of more than two manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' A drawback of this approach is that the error terms are compounded when they are just added together for each Lt i, but whether this is a problem will depend on the specific application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' In our numerical experiments, we assumed that Ω = ∪2 i=1Ωi and had access to samples of a continuous curve Γ, which allowed us to estimate geomet- ric properties near intersections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Future work could involve extending our framework to other types of singularities and developing similar tests and estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' It would also be interesting to explore methods that do not rely on direct access to such curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Similar theorems can be proven for other kernels besides the Gaussian one, as many ideas used in our proofs are not specific to the Gaussian case, but rather rely mainly on symmetries of Kt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Investigating the use of other kernels and comparing their performance in different scenarios is a promising direction for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Acknowledgments The first author was supported by the Wallenberg AI, Autonomous Sys- tems and Software Program (WASP) funded by the Knut and Alice Wallen- berg Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' The second author was supported by the Swedish Research Council grant dnr: 2019-04098.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' References [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Belkin and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content=' Niyogi, “Towards a theoretical foundation for laplacian-based man- ifold methods,” Journal of Computer and System Sciences, vol.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='se Martin Andersson, Department of Mathematics, Uppsala University, S-751 06 Uppsala, Sweden Email address: martin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='andersson@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='uu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} +page_content='se Benny Avelin, Department of Mathematics, Uppsala University, S-751 06 Uppsala, Sweden' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfYPfO/content/2301.00201v1.pdf'} diff --git a/ldE4T4oBgHgl3EQfUAxn/content/tmp_files/2301.05012v1.pdf.txt b/ldE4T4oBgHgl3EQfUAxn/content/tmp_files/2301.05012v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b7e1a558fc35c6e0e7e9bfbba1289ce3ab2954c2 --- /dev/null +++ b/ldE4T4oBgHgl3EQfUAxn/content/tmp_files/2301.05012v1.pdf.txt @@ -0,0 +1,941 @@ +Fairly Private: Investigating The Fairness of Visual Privacy Preservation +Algorithms +Sophie Noiret,1 Siddharth Ravi, 2 Martin Kampel, 1 Francisco Florez-Revuelta 2 +1 Vienna University of Technology +2 University of Alicante +snoiret@cvl.tuwien.ac.at.com, siddharth.ravi@gcloud.ua.es, martin.kampel@tuwien.ac.at, francisco.florez@gcloud.ua.es +Abstract +As the privacy risks posed by camera surveillance and fa- +cial recognition have grown, so has the research into privacy +preservation algorithms. Among these, visual privacy preser- +vation algorithms attempt to impart bodily privacy to subjects +in visuals by obfuscating privacy-sensitive areas. While dis- +parate performances of facial recognition systems across phe- +notypes are the subject of much study, its counterpart, pri- +vacy preservation, is not commonly analysed from a fairness +perspective. In this paper, the fairness of commonly used vi- +sual privacy preservation algorithms is investigated through +the performances of facial recognition models on obfuscated +images. Experiments on the PubFig dataset clearly show that +the privacy protection provided is unequal across groups. +1 +Introduction +According to the website IFSEC Global1, the worldwide +video surveillance camera market is expected to almost dou- +ble by 2025 compared to 2019. It is therefore of little sur- +prise that video surveillance is being considered worldwide +as a serious threat to privacy (Kumagai and Cherry 2004). +Surveillance is, however, not always malicious, and could +instead be a necessity. In medical settings, for instance, the +capacity to monitor patients remotely can be crucial to en- +suring their safety. In these cases, while knowing that a per- +son is present in the image is useful, it might not be neces- +sary to know the identity of the person in the image. Visual +privacy preservation algorithms can therefore be deployed to +protect bodily privacy in such settings. Privacy preservation +algorithms (PPA) work by perceptually obfuscating the vi- +sual feed to various degrees depending on the context. Some +of the simplest and most commonly used visual PPAs in- +clude blurring and pixelation. While these algorithms have +been in use for decades, there have been various issues sur- +rounding their use. Simple algorithms are also prone to at- +tacks with which the obfuscation performed can be reversed, +and the original image reconstructed (Korshunov and Ooi +2011; Menon et al. 2020). Facial recognition algorithms can +Copyright © 2023, Association for the Advancement of Artificial +Intelligence (www.aaai.org). All rights reserved. +1https://www.ifsecglobal.com/video-surveillance/whats- +behind-the-global-growth-of-cloud-based-video-surveillance/, last +accessed September 26, 2022 +also be trained to identify subjects present in obfuscated im- +ages. Furthermore, while the biases in facial recognition are +well documented (Buolamwini and Gebru 2018), the effec- +tiveness of obfuscation has not been subjected to the same +amount of scrutiny from a fairness perspective. +The primary questions explored in this work are: +• Is there a racial or gender bias in the degree of privacy +afforded by face obfuscation techniques? +• If so, does this bias depend on the obfuscation method +used? The classifier used for facial recognition? +These questions are examined under the assumption that +people do not wish to be recognized, i.e., that a positive re- +sult is one in which the subject is not correctly identified. In +this work, a system is deemed fair if it achieves equal per- +formance across groups, and one’s privacy is considered to +be protected if their identity is successfully concealed by a +face obfuscation method. +This work studies discrimination based on the protected +attributes of race and gender. By training a facial recognition +system on un-obfuscated images and setting it to predict on +obfuscated images, the capacity of blurring and pixelation to +fairly conceal identities is evaluated. The results show that +both of these techniques yield significantly disparate perfor- +mances across demographics. To quantify experimental re- +sults better, this paper also introduces a new metric called +the personal detection rate. +The rest of this paper is structured as follows. Section 2 +examines related work in the areas of fairness and privacy +in computer vision. Section 3 introduces and motivates the +methodology used for choosing the dataset selected, and the +framework used for analysis in this paper. This section also +presents the experiments conducted, as well as details about +their implementation. Section 4 explains the results obtained +from these experiments and discusses factors of importance +that influence the results. Finally, in Section 5 the paper puts +forth its conclusions, discusses the limitations of the experi- +ments, and proposes various possible avenues for future re- +search. +2 +Related Work +Several works have analysed the intersection of fairness and +algorithmic privacy (Ekstrand, Joshaghani, and Mehrpouyan +2018; Dwork and Mulligan 2013; Cummings et al. 2019). +The existing literature at this intersection relevant to this pa- +arXiv:2301.05012v1 [cs.CV] 12 Jan 2023 + +per can be separated into two categories. These are works +that primarily deal with the issue of fair privacy, and those +dealing with the topic of fairness in facial analysis. +2.1 +Fair Privacy +Prior works have studied the intersection of fairness and pri- +vacy by using sensitive attributes that are not of a visual na- +ture (Ding et al. 2020; Tran, Fioretto, and Van Hentenryck +2021; Ghili, Kazemi, and Karbasi 2019). Research, for ex- +ample, has been conducted on the impact of fairness in the +area of privacy protected data (through differential privacy) +in the domains of voting rights and funds allocation (Pujol +et al. 2020). This highlights the need to consider the fairness +of outcomes when designing privacy algorithms. +Models that provide both privacy and fairness from a +more theoretical standpoint have also been explored. For ex- +ample, the creation of two logistic regression models (PFLR +and PFLR*) that are differentially private and provide fair- +ness have been explored, which at the same time preserves +the utility of the resulting model (Xu, Yuan, and Wu 2019). +2.2 +Fair Facial Analysis +The fairness provided by software performing facial anal- +ysis has come under scrutiny(Phillips et al. 2003). How- +ever, works such as (Wang and Deng 2020) and (Amini +et al. 2019) aim at improving the fairness of facial recog- +nition technology, while this paper analyses a scenario in +which subjects do not wish to be identified. In their work +’Gender Shades’, Buolamwini et al. (Buolamwini and Ge- +bru 2018) evaluate 3 commercial gender classification sys- +tems and show that darker skinned females are the most mis- +classified group. The authors also analyse two facial analy- +sis benchmark datasets, IJB-A (Klare et al. 2015) and Adi- +ence (Eidinger, Enbar, and Hassner 2014), and find that the +composition of these datasets is overwhelmingly made up +by light-skinned individuals. +Methods in the literature aiming to achieve fairness in fa- +cial recognition work by creating fairer facial embeddings. +Alvi et al. (Alvi, Zisserman, and Nellaaker 2018), for exam- +ple, created a facial recognition method that calculates the +cross entropy between the output distributions produced by +classifiers trained on biased data and a uniform distribution. +This is then shown to be a fairer feature representation for +the task of facial recognition. +Neural networks which work by imparting fairness at the +comparison level have also been created for the task of facial +recognition (Terh¨orst et al. 2020). This is achieved by learn- +ing a similarity function model which treats people from +different ethnicities similarly. This produces fair compari- +son scores when presented with biased face embeddings. For +this, a neural network is trained with a loss function which +includes a penalization term prioritizing group fairness or +individual fairness. Although the paper puts forth a method +to improve fairness in facial recognition, it does not provide +results for privacy-protected images. It also assumes a desire +for fairness on the part of the recognition system, which we +do not. +Adversarially trained models that discourage facial recog- +nition networks from encoding information about protected +(a) By race (binary) +(b) By gender +(c) By race and gender +(d) By race (binary), gender +Figure 1: Composition of Dataset +attributes have also been explored (Dhar et al. 2021). This +work, however, also is not tested on privacy-protected facial +images, but rather uses unobfuscated image data to validate +the study. +Masked facial recognition is a field that is of interest to +the topic of this paper. Yu et al. (Yu et al. 2021) create a +method to improve the fairness of masked face recognition +algorithms. Unlike our work, however, the context for anal- +ysis is one in which higher recognition rates are the desired +outcome, and the faces are only partially obfuscated. +This paper, in contrast to these prior works, seeks specifi- +cally to study the fairness of commonly used visual privacy +preservation algorithms. To the best of the authors’ knowl- +edge, such a study has not been attempted. +3 +Methodology and Experiments +The scenario motivating this paper is one in which a surveil- +lance camera is placed in a building. This specific use-case +necessitates to be able to know whether a face is in the pic- +ture or not, but knowing the identity of the person is not +required. Consequently, people’s privacy is protected by ob- +fuscating the face of the person in the image. A bad actor +with access to unobfuscated images of the people in the +building (for instance from employee files) can then train +a model to recognize the faces in the obfuscated images. It +is also impossible to imagine the use of adversarial noise to +thwart machine learning models in a scenario such as this, +mainly due to the lack of computational power in a typical +setup. +3.1 +Dataset +The requirements to conduct this work dictates that the +dataset used contains images annotated for identity and pro- + +175 +150 +125 +100 +coun +75 +50 +25 +0 +White +Non-white120 +100 +80 +Count +60 +40 +20 +0 +Female +MaleMale +100 +Female +80 +Count +60 +40 +20 +0 +WhiteIndianE +Black Asian100 +Male +Female +80 +Count +09 +40 +20 +0 +White +Non-whiteFigure 2: Method overview +tected characteristics, as well as containing several images +per subject. Having a balanced dataset with race and gen- +der information is not essential, as nothing guarantees such +a balance in our scenario. Based on these criteria, the Pub- +Fig dataset (Kumar et al. 2009) is chosen. This dataset con- +tains 58,797 images of 200 people according to the origi- +nal composition and is available as a list of URLs due to +copyright issues. As the dataset is from 2008, many of the +original URLs are broken and the corresponding images are +consequently excluded. Attribute labels are also provided +along with the dataset to facilitate research, containing var- +ious protected attributes such as racial categories and gen- +der. According to the dataset documentation provided, the +attribute labels have been partially acquired through Ama- +zon Mechanical Turk workers and partially generated by an +attribute classifier. As a consequence, the dataset contains +several errors in the attribute labels provided along with the +dataset. This dataset also contains the additional difficulty +of varying picture quality. For this reason, we do not apply +an uniform level of obfuscation. The details are provided in +the next section. Although other datasets are used in face +recognition experiments, but they either lack the necessary +demographic annotations (CelebA (Liu et al. 2015), VG- +GFace (Cao et al. 2018)) or identity information (FairFace +(Karkkainen and Joo 2021)). To create the attribute labels +required for the experiments, the dataset has been cleaned +and aggregated to obtain per-individual data. The original +composition of the dataset can be seen in Fig. 1. Due to +the labelling errors in the original set, protected attributes +have been manually checked and corrected if necessary. As +can be observed, the original composition is severely im- +balanced, with a large majority of the people in the set be- +ing white. While this does not prohibit experimentation, it +leads to some groups (for instance, Indian women) not hav- +ing sufficient representation for analysis. For this reason, the +final attribute labels are chosen to be binary, i.e., race (white +vs non-white) and gender (male vs female).The race groups +chosen as reference to decide white vs non-white individuals +are as defined in the FairFace dataset (Karkkainen and Joo +2021). +3.2 +Process overview +The main steps of the experiment are executed according to +the following pipeline: +1. From the original dataset, faces are detected on each im- +age to only keep images with exactly one face. +2. For each person in the dataset, a random 80/20 split is +performed. +3. 20% of images are obfuscated. On these images, face +detection is performed and a 128-dimensional vector of +face encodings is created for each image. +4. On the remaining 80%, face detection is performed and a +128-dimensional vector of face encodings is created for +each image. +5. A classifier is trained on the encodings of the unobfus- +cated images. +6. This classifier is subsequently used to predict the identity +of the person present in the obfuscated images. + +Yes +Extract Facial Encodings +Foreach image +For each image +Face Detection +Has the +Face Obfuscation: +boundary +been found? +GaussianBlur +. Pixelation +Original +Obfuscated +Image +Image +20% +Original +Pre-processed +Dataset +Dataset +Face +80% +Encodings +For each image +Classifier +Train classifier +Prediction +Train +Encodings +Face +ImageConsidering the varying quality of the pictures present in the +dataset, a different level of obfuscation is required per im- +age on step 3. For the case analysed in this paper, it is nec- +essary to know that a face is in the picture. Consequently, +we choose the maximum level of obfuscation that still al- +lows for face detection. This means that potential biases in +face detection will result in a lower level of obfuscation: the +complete system (face detection and face obfuscation) will +therefore reflect bias in both tasks. To determine this level of +obfuscation, each image goes through the following process: +• A face is detected in the image and obfuscated, and face +detection is run again on the image. +• If the face is not detected, then while no face is detected, +the obfuscation level is cut by half and a new obfuscation +takes place. +• If a face is detected, then while a face is detected, the ob- +fuscation level is doubled and a new obfuscation is per- +formed. +This gives us a range, providing a maximum (when the face +is not detected) and a minimum (when the face is detected) +level of obfuscation. A binary search is then performed on +this interval to find the upper limit of obfuscation that still +allows for face detection in each image. This process can be +seen in Fig. 2. +3.3 +Experiments Conducted +The main steps of the pipeline are implemented through sev- +eral techniques to ascertain their influence on any potential +bias that would emerge. +Pre-processing of dataset. The original dataset includes +a checksum computed using the md5sum command for each +image. In an effort to avoid having the same picture repeated +in the set, only one image per checksum is kept. To obtain +per-image results, face detection is performed on every im- +age in the dataset and only images with exactly one face are +retained for the experiments. +Face Detection. For face detection, the method used is +based on the Histogram of Oriented Gradients (HOG) as im- +plemented in the dlib and face recognition2 libraries. HOG +is a feature descriptor trained on the Labeled Faces in Wild +(Huang et al. 2008) dataset to detect human faces. +Face obfuscation.The two face obfuscation techniques +used are Gaussian blurring and pixelation. Gaussian blur- +ring masks details by considering for each pixel the value +of the pixels in its neighbourhood. The section around each +pixel, called the kernel, is used to modify its value. A bigger +kernel leads to more blurring. Gaussian blur, implemented +in the OpenCV function GaussianBlur, is used for the ex- +periments in this paper. +Pixelation consists of downsizing the image, then re-sizing it +to the original size. When downsizing the image to a smaller +size, pixels are deleted. These pixels are replaced when the +image is restored to its original size by interpolating new +pixels, which are calculated according to an interpolation +method chosen. Pixelation is implemented for the experi- +ments in this paper through the resize function provided in +2Available at https://github.com/ageitgey/face recognition +(a) Original +(b) Blur +(c) Pixelated +Figure 3: Original and modified images +the OpenCV library. These techniques are illustrated in Fig. +3. +Classifier. The four classifiers used to make predictions +in this study are the K-Nearest Neighbour (KNN), the Na¨ıve +Bayes (NB), the Support Vector Classifier (SVC) and the +Multi-Layer Perceptron (MLP). These classifiers are se- +lected because they are commonly used multi-class classi- +fiers. The implementation used is as in the Scikit-learn li- +brary, with parameters either left at default (NB, SVM), or +chosen through the execution of a grid search. +Measuring Algorithmic Fairness. The fairness defini- +tion chosen here is group fairness, i.e., equal or unequal per- +formance across groups. Performance in this case is the ca- +pacity of the privacy-preserving method to obfuscate some- +one’s identity, which means that the favourable outcome is +one where the person is not being recognized. Consequently, +contrary to previous works, low accuracy, precision, recall +and F1-score are the desired outcomes. These metrics are +all reported for completeness; however, they can lead to dif- +ferent conclusions as they place emphasis on different as- +pects. When considering both race and gender, we also re- +port bias, i.e., the biggest gap in performance between any +two groups for each metrics. We make the assumption that +people prefer not to be identified, neither correctly nor incor- +rectly. As a consequence, it is desirable to have low numbers +of both the True Positives (TP, implying a correct identifica- +tion), and False Positives (FP, implying an incorrect identi- +fication of the person). As accuracy reports the proportion +of correct predictions, both TP and True Negatives (TN) in- +crease the accuracy. Additionally, it is sensitive to class im- +balance. Therefore, when accuracy leads to a different con- +clusion than the other metrics, precision and recall are given + +preference. We also report a new metric called Personal De- +tection Rate (PDR), given by the following formula - +PDR = True Positives + False Positives +Number of predictions +(1) +This metric corresponds to the number of times the person +is identified in a picture (correctly or incorrectly), divided by +the number of predictions made. However, it is also sensitive +to class imbalances. +4 +Results and Discussion +The results presented in the following sections are obtained +by using SVC. For reference, we consider that the groups +on which the worst results are obtained are those that score +higher on recognition metrics. Best results are shown in +bold, while the worst results are in italics. +4.1 +Level of Obfuscation Needed +(a) Blurring by race and gender +(b) Blurring by race +(c) Blurring by gender +Figure 4: Distribution of the level of Blurring +Gaussian Blur. The amount of obfuscation performed by +Gaussian blur is based on the size of the kernel. The size +of the kernel is proportional to the blurring performed on +the image. This value is in pixels, so it is normalized by +dividing it by the size of the Region Of Interest (ROI), the +face, in pixels. This aims at reducing the influence of the +quality of the original picture. +We also observe that for non-white people, the level of ob- +fuscation is lower than it is for white people (median value +of 0.109 and 0.133 respectively). This might be because the +dlib library is utilized for its HOG implementation, which +is pre-trained on the LFW dataset. This dataset is shown in +(Karkkainen and Joo 2021) to be imbalanced towards white +faces. The level of obfuscation differs also between men +and women: the median value is 0.132 on men and 0.126 on +women. +Pixelation. Pixelation, as previously mentioned, is done +by downsampling and then subsequently upsampling the re- +gion of interest. The level of obfuscation is the size to which +it is downsampled: the smaller the size, the more pixelated +the result is. This value is also divided by the size of the ROI +to normalize it. The results are presented in Fig 5. To have +a clearer look at the results by race, we present a truncated +version which excludes relative kernel sizes over 0.3. This +excludes 16 images of white people, 12 women and 4 men. +The median level of pixelation is the same between white +and non-white people and between men and women, with a +value of 0.0107 for all. +(a) Pixelation by race +(b) Pixelation by gender +Figure 5: Distribution of the level of Pixelation +4.2 +Bias in Face Recognition Results +The only results presented in the following sections are ob- +tained by using SVC for the sake of brevity. For reference, +we consider that the groups on which the worst results are +obtained are those that score higher on recognition metrics. +Best results are shown in bold, while the worst results are in +italics.The row ”Bias” refers to the biggest gap between two +groups. +The results are shown in Tables 1 and 2. +When using pixelation for face obfuscation, the groups +that get recognized at the lowest rate are women, white peo- +ple, and specifically, white women. When looking at inter- +sectional results, the group that gets the worst results are +non-white women. These results are consistent across all +classifiers. +When using blurring for face obfuscation, the groups that +get recognized at the lowest rate are women, white peo- +ple, and specifically, white women. These results are con- +sistent across all classifiers. When looking at intersectional +results, the groups that get the worst results are non-white +men (SVC, MLP) and non-white women (NB, KNN). + +0.8 +* +0 ++ +20.6 ++ ++ +elative l +0.4 +0.2 +0°0 +White +Non-white0.8 +* ++ +20.6 ++ +0.4 +0.2 +0.0 - +Male +Female35 +White +30 +Non-white +25 +20 +D +15 +10 +5 +0 +0.0 +0.1 +0.2 +0.3 +Relative Kernel Size20 +Male +Female +15 +Density +010 +5 +0 +0.0 +0.1 +0.2 +0.3 +Relative KernelSize0.8 +White + Non-white +0.6 ++ ++ +0.4 +0.2 +0°0 +Male +FemaleBalanced Accuracy +Recall +Precision +F1-score +PDR +Overall +0.359 +0.415 +0.505 +0.404 +White +0.357 +0.416 +0.527 +0.414 +0.887 +Non-White +0.370 +0.407 +0.718 +0.475 +0.113 +Male +0.371 +0.439 +0.567 +0.438 +0.570 +Female +0.342 +0.388 +0.471 +0.385 +0.430 +Non-White Female +0.452 +0.428 +0.767 +0.510 +0.063 +Non-White Male +0.321 +0.389 +0.694 +0.453 +0.051 +White Female +0.323 +0.381 +0.492 +0.388 +0.368 +White Male +0.382 +0.447 +0.588 +0.453 +0.519 +Bias +0.131 +0.066 +0.275 +0.122 +0.012 +Table 1: Pixelation obfuscation, SVC classifier +Balanced Accuracy +Recall +Precision +F1-score +PDR +Overall +0.333 +0.398 +0.556 +0.398 +White +0.300 +0.371 +0.579 +0.393 +0.757 +Non-White +0.508 +0.557 +0.793 +0.613 +0.243 +Male +0.366 +0.438 +0.628 +0.459 +0.531 +Female +0.289 +0.356 +0.516 +0.350 +0.469 +Non-White Female +0.497 +0.514 +0.765 +0.576 +0.138 +Non-White Male +0.514 +0.592 +0.867 +0.669 +0.105 +White Female +0.255 +0.330 +0.543 +0.348 +0.331 +White Male +0.335 +0.409 +0.646 +0.449 +0.427 +Bias +0.259 +0.262 +0.324 +0.321 +0.322 +Table 2: Blurring obfuscation, SVC classifier +These results show a bias in identity obfuscation for both +pixelation and blurring. While women, white people, and +white women get systemically better results, the group that +gets worse results is not systematically non-white men. This +highlights the need for intersectional studies, without which +the bias against non-white women would be concealed. +Figure 6: Bias across classifier, Pixelation +4.3 +Influence of the classifier +As several metrics are used on different groups, it is not pos- +sible to clearly determine which classifier achieves the best +performance: there is not one classifier which consistently +performs better on all metrics and across all groups. More- +over, the goal here is not to determine which classifier is +better, but if trends observed are consistent regardless of the +classifier used. +On performance. When using pixelation for face obfus- +cation, the difference in performance is at most 7%. (be- +tween the precision on non-white females when using SVC +and KNN). On all other metrics, across all different groups, +the difference in performance between classifier is less than +4%. When using blurring for face obfuscation, SVC per- +forms overall slightly better than the other classifiers. How- +ever, even here, the biggest difference in performance (dif- +ference between the precision on non-white males when us- +ing NB and SVC) is only 11%, with the difference on all +other metrics, across all different groups being less than 6%. +Overall, the influence of the classifier on performance is +minimal. +On bias. General trends are independent of classifier. +When controlling for other factors, the groups on which the +best and worst performances are achieved stay the same. As +can be seen in Fig. 6, the reduction of bias depends on which +metric is considered when using the pixelation. However, +the biggest variation in bias (between SVC and NB on pre- +cision) is only 0.045 for pixelation and while SVC performs +better on recall, precision and F1-score, it performs worse +on accuracy. When using blurring as shown in Fig. 7, NB +is overall less biased. However, in the scenario considered, +the choice of classifier is that of the bad actor, and not of the +entity trying to fairly preserve people’s identity. + +Accurac +0.10 +0.05 +0.00 +0.00 +KNN SVC NBMLP +KNN SVC NB +MLP +Precision +F1-score +0.10 +0.2 +0.1 +0.05 +0.0 +0.00 +KNN SVC NB +MLP +KNN SVC NB +MLPFigure 7: Bias across classifier, Gaussian Blur +4.4 +Influence of the face obfuscation methods +The face obfuscation method does not influence the general +trend: the best results are obtained on white people, women +and white women. The worst results are obtained on non- +white people for pixelation, on non-white men (MLP and +SVC) and non-white women (NB) for blurring. However, +as we can see in Fig 7 and Fig 6, using pixelation for face +obfuscation leads to a smaller gap in performance across all +metrics and for all classifiers. +5 +Limitations, Conclusions, and Future +Work +Our work focuses intentionally on an imbalanced dataset. +However, a similar study on a balanced dataset could help +determine the origin of the bias shown in this work. This +has not, however, been attempted because of the low num- +ber of persons in the used dataset, and also the low num- +ber of images present for some persons in the dataset. The +authors note that the creation of a bigger, balanced dataset +with the characteristics of PubFig would be highly beneficial +to push forward research. Additionally, the choice of itera- +tively obfuscating faces to find the limit of face detection is +not one that could easily be implemented in a real-time sys- +tem. The reproducibility of this study is also impeded by the +fact that, like Kumar et al. (Kumar et al. 2009), we are un- +able to distribute image files. However, the code developed +for this study will be made available on Github. +As has been shown, the degree to which the identity is +protected by face detection is subject to racial and gender +bias. This bias is present regardless of the classifier of face +obfuscation technique used, but using pixelation instead of +blurring leads to less bias. +The scope of this study could be broadened in future work +by performing a similar experiment on a dataset balanced +with regard to race and gender, or by considering other pro- +tected attributes and face obfuscation techniques. It could +also be extended to other types of person recognition, such +as whole body recognition, and different types of images, +such as depth data. +6 +Acknowledgements +This work is part of the visuAAL project on Privacy- +Aware and Acceptable Video-Based Technologies and Ser- +vices for Active and Assisted Living (https://www.visuaal- +itn.eu/). This project has received funding from the Euro- +pean Union’s Horizon 2020 research and innovation pro- +gram under the Marie Skłodowska-Curie grant agreement +No 861091. It was also partly supported by the Austrian Re- +search Promotion Agency (FFG) under the grant agreement +No. 878730 and the Wiener Wissenschafts-, Forschungs- +und Technologiefonds (WWTF) under the grant number +ICT20-055. +References +Alvi, M.; Zisserman, A.; and Nellaaker, C. 2018. Turning a +Blind Eye: Explicit Removal of Biases and Variation from +Deep Neural Network Embeddings. In Proceedings of the +European Conference on Computer Vision (ECCV) Work- +shops. +Amini, A.; Soleimany, A. P.; Schwarting, W.; Bhatia, S. N.; +and Rus, D. 2019. Uncovering and Mitigating Algorithmic +Bias through Learned Latent Structure. In Proceedings of +the 2019 AAAI/ACM Conference on AI, Ethics, and Society, +AIES ’19, 289–295. New York, NY, USA: Association for +Computing Machinery. 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In 2021 IEEE/CVF +International Conference on Computer Vision Workshops +(ICCVW), 1531–1540. + diff --git a/ldE4T4oBgHgl3EQfUAxn/content/tmp_files/load_file.txt b/ldE4T4oBgHgl3EQfUAxn/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f3bcb99066f5179a7cc182404022599b55f5878a --- /dev/null +++ b/ldE4T4oBgHgl3EQfUAxn/content/tmp_files/load_file.txt @@ -0,0 +1,730 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf,len=729 +page_content='Fairly Private: Investigating The Fairness of Visual Privacy Preservation Algorithms Sophie Noiret,1 Siddharth Ravi, 2 Martin Kampel, 1 Francisco Florez-Revuelta 2 1 Vienna University of Technology 2 University of Alicante snoiret@cvl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='tuwien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='com, siddharth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='ravi@gcloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='ua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='es, martin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='kampel@tuwien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='at, francisco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='florez@gcloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='ua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='es Abstract As the privacy risks posed by camera surveillance and fa- cial recognition have grown, so has the research into privacy preservation algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Among these, visual privacy preser- vation algorithms attempt to impart bodily privacy to subjects in visuals by obfuscating privacy-sensitive areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' While dis- parate performances of facial recognition systems across phe- notypes are the subject of much study, its counterpart, pri- vacy preservation, is not commonly analysed from a fairness perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' In this paper, the fairness of commonly used vi- sual privacy preservation algorithms is investigated through the performances of facial recognition models on obfuscated images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Experiments on the PubFig dataset clearly show that the privacy protection provided is unequal across groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' 1 Introduction According to the website IFSEC Global1, the worldwide video surveillance camera market is expected to almost dou- ble by 2025 compared to 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' It is therefore of little sur- prise that video surveillance is being considered worldwide as a serious threat to privacy (Kumagai and Cherry 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Surveillance is, however, not always malicious, and could instead be a necessity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' In medical settings, for instance, the capacity to monitor patients remotely can be crucial to en- suring their safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' In these cases, while knowing that a per- son is present in the image is useful, it might not be neces- sary to know the identity of the person in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Visual privacy preservation algorithms can therefore be deployed to protect bodily privacy in such settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Privacy preservation algorithms (PPA) work by perceptually obfuscating the vi- sual feed to various degrees depending on the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Some of the simplest and most commonly used visual PPAs in- clude blurring and pixelation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' While these algorithms have been in use for decades, there have been various issues sur- rounding their use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Simple algorithms are also prone to at- tacks with which the obfuscation performed can be reversed, and the original image reconstructed (Korshunov and Ooi 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Menon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Facial recognition algorithms can Copyright © 2023, Association for the Advancement of Artificial Intelligence (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='aaai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' 1https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='ifsecglobal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='com/video-surveillance/whats- behind-the-global-growth-of-cloud-based-video-surveillance/, last accessed September 26, 2022 also be trained to identify subjects present in obfuscated im- ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Furthermore, while the biases in facial recognition are well documented (Buolamwini and Gebru 2018), the effec- tiveness of obfuscation has not been subjected to the same amount of scrutiny from a fairness perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' The primary questions explored in this work are: Is there a racial or gender bias in the degree of privacy afforded by face obfuscation techniques?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' If so, does this bias depend on the obfuscation method used?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' The classifier used for facial recognition?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' These questions are examined under the assumption that people do not wish to be recognized, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=', that a positive re- sult is one in which the subject is not correctly identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' In this work, a system is deemed fair if it achieves equal per- formance across groups, and one’s privacy is considered to be protected if their identity is successfully concealed by a face obfuscation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' This work studies discrimination based on the protected attributes of race and gender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' By training a facial recognition system on un-obfuscated images and setting it to predict on obfuscated images, the capacity of blurring and pixelation to fairly conceal identities is evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' The results show that both of these techniques yield significantly disparate perfor- mances across demographics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' To quantify experimental re- sults better, this paper also introduces a new metric called the personal detection rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' The rest of this paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Section 2 examines related work in the areas of fairness and privacy in computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Section 3 introduces and motivates the methodology used for choosing the dataset selected, and the framework used for analysis in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' This section also presents the experiments conducted, as well as details about their implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Section 4 explains the results obtained from these experiments and discusses factors of importance that influence the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Finally, in Section 5 the paper puts forth its conclusions, discusses the limitations of the experi- ments, and proposes various possible avenues for future re- search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' 2 Related Work Several works have analysed the intersection of fairness and algorithmic privacy (Ekstrand, Joshaghani, and Mehrpouyan 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Dwork and Mulligan 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Cummings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' The existing literature at this intersection relevant to this pa- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='05012v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='CV] 12 Jan 2023 per can be separated into two categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' These are works that primarily deal with the issue of fair privacy, and those dealing with the topic of fairness in facial analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='1 Fair Privacy Prior works have studied the intersection of fairness and pri- vacy by using sensitive attributes that are not of a visual na- ture (Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Tran, Fioretto, and Van Hentenryck 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Ghili, Kazemi, and Karbasi 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Research, for ex- ample, has been conducted on the impact of fairness in the area of privacy protected data (through differential privacy) in the domains of voting rights and funds allocation (Pujol et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' This highlights the need to consider the fairness of outcomes when designing privacy algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Models that provide both privacy and fairness from a more theoretical standpoint have also been explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' For ex- ample, the creation of two logistic regression models (PFLR and PFLR*) that are differentially private and provide fair- ness have been explored, which at the same time preserves the utility of the resulting model (Xu, Yuan, and Wu 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='2 Fair Facial Analysis The fairness provided by software performing facial anal- ysis has come under scrutiny(Phillips et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' How- ever, works such as (Wang and Deng 2020) and (Amini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' 2019) aim at improving the fairness of facial recog- nition technology, while this paper analyses a scenario in which subjects do not wish to be identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' In their work ’Gender Shades’, Buolamwini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' (Buolamwini and Ge- bru 2018) evaluate 3 commercial gender classification sys- tems and show that darker skinned females are the most mis- classified group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' The authors also analyse two facial analy- sis benchmark datasets, IJB-A (Klare et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' 2015) and Adi- ence (Eidinger, Enbar, and Hassner 2014), and find that the composition of these datasets is overwhelmingly made up by light-skinned individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Methods in the literature aiming to achieve fairness in fa- cial recognition work by creating fairer facial embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Alvi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' (Alvi, Zisserman, and Nellaaker 2018), for exam- ple, created a facial recognition method that calculates the cross entropy between the output distributions produced by classifiers trained on biased data and a uniform distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' This is then shown to be a fairer feature representation for the task of facial recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Neural networks which work by imparting fairness at the comparison level have also been created for the task of facial recognition (Terh¨orst et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' This is achieved by learn- ing a similarity function model which treats people from different ethnicities similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' This produces fair compari- son scores when presented with biased face embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' For this, a neural network is trained with a loss function which includes a penalization term prioritizing group fairness or individual fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Although the paper puts forth a method to improve fairness in facial recognition, it does not provide results for privacy-protected images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' It also assumes a desire for fairness on the part of the recognition system, which we do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Adversarially trained models that discourage facial recog- nition networks from encoding information about protected (a) By race (binary) (b) By gender (c) By race and gender (d) By race (binary), gender Figure 1: Composition of Dataset attributes have also been explored (Dhar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' This work, however, also is not tested on privacy-protected facial images, but rather uses unobfuscated image data to validate the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Masked facial recognition is a field that is of interest to the topic of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' 2021) create a method to improve the fairness of masked face recognition algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Unlike our work, however, the context for anal- ysis is one in which higher recognition rates are the desired outcome, and the faces are only partially obfuscated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' This paper, in contrast to these prior works, seeks specifi- cally to study the fairness of commonly used visual privacy preservation algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' To the best of the authors’ knowl- edge, such a study has not been attempted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' 3 Methodology and Experiments The scenario motivating this paper is one in which a surveil- lance camera is placed in a building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' This specific use-case necessitates to be able to know whether a face is in the pic- ture or not, but knowing the identity of the person is not required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Consequently, people’s privacy is protected by ob- fuscating the face of the person in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' A bad actor with access to unobfuscated images of the people in the building (for instance from employee files) can then train a model to recognize the faces in the obfuscated images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' It is also impossible to imagine the use of adversarial noise to thwart machine learning models in a scenario such as this, mainly due to the lack of computational power in a typical setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='1 Dataset The requirements to conduct this work dictates that the dataset used contains images annotated for identity and pro- 175 150 125 100 coun 75 50 25 0 White Non-white120 100 80 Count 60 40 20 0 Female MaleMale 100 Female 80 Count 60 40 20 0 WhiteIndianE Black Asian100 Male Female 80 Count 09 40 20 0 White Non-whiteFigure 2: Method overview tected characteristics, as well as containing several images per subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Having a balanced dataset with race and gen- der information is not essential, as nothing guarantees such a balance in our scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Based on these criteria, the Pub- Fig dataset (Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' 2009) is chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' This dataset con- tains 58,797 images of 200 people according to the origi- nal composition and is available as a list of URLs due to copyright issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' As the dataset is from 2008, many of the original URLs are broken and the corresponding images are consequently excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Attribute labels are also provided along with the dataset to facilitate research, containing var- ious protected attributes such as racial categories and gen- der.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' According to the dataset documentation provided, the attribute labels have been partially acquired through Ama- zon Mechanical Turk workers and partially generated by an attribute classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' As a consequence, the dataset contains several errors in the attribute labels provided along with the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' This dataset also contains the additional difficulty of varying picture quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' For this reason, we do not apply an uniform level of obfuscation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' The details are provided in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Although other datasets are used in face recognition experiments, but they either lack the necessary demographic annotations (CelebA (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' 2015), VG- GFace (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' 2018)) or identity information (FairFace (Karkkainen and Joo 2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' To create the attribute labels required for the experiments, the dataset has been cleaned and aggregated to obtain per-individual data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' The original composition of the dataset can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Due to the labelling errors in the original set, protected attributes have been manually checked and corrected if necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' As can be observed, the original composition is severely im- balanced, with a large majority of the people in the set be- ing white.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' While this does not prohibit experimentation, it leads to some groups (for instance, Indian women) not hav- ing sufficient representation for analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' For this reason, the final attribute labels are chosen to be binary, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=', race (white vs non-white) and gender (male vs female).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='The race groups chosen as reference to decide white vs non-white individuals are as defined in the FairFace dataset (Karkkainen and Joo 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='2 Process overview The main steps of the experiment are executed according to the following pipeline: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' From the original dataset, faces are detected on each im- age to only keep images with exactly one face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' For each person in the dataset, a random 80/20 split is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' 20% of images are obfuscated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' On these images, face detection is performed and a 128-dimensional vector of face encodings is created for each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' On the remaining 80%, face detection is performed and a 128-dimensional vector of face encodings is created for each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' A classifier is trained on the encodings of the unobfus- cated images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' This classifier is subsequently used to predict the identity of the person present in the obfuscated images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Yes Extract Facial Encodings Foreach image For each image Face Detection Has the Face Obfuscation: boundary been found?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' GaussianBlur .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Pixelation Original Obfuscated Image Image 20% Original Pre-processed Dataset Dataset Face 80% Encodings For each image Classifier Train classifier Prediction Train Encodings Face ImageConsidering the varying quality of the pictures present in the dataset, a different level of obfuscation is required per im- age on step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' For the case analysed in this paper, it is nec- essary to know that a face is in the picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Consequently, we choose the maximum level of obfuscation that still al- lows for face detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' This means that potential biases in face detection will result in a lower level of obfuscation: the complete system (face detection and face obfuscation) will therefore reflect bias in both tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' To determine this level of obfuscation, each image goes through the following process: A face is detected in the image and obfuscated, and face detection is run again on the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' If the face is not detected, then while no face is detected, the obfuscation level is cut by half and a new obfuscation takes place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' If a face is detected, then while a face is detected, the ob- fuscation level is doubled and a new obfuscation is per- formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' This gives us a range, providing a maximum (when the face is not detected) and a minimum (when the face is detected) level of obfuscation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' A binary search is then performed on this interval to find the upper limit of obfuscation that still allows for face detection in each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' This process can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='3 Experiments Conducted The main steps of the pipeline are implemented through sev- eral techniques to ascertain their influence on any potential bias that would emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Pre-processing of dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' The original dataset includes a checksum computed using the md5sum command for each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' In an effort to avoid having the same picture repeated in the set, only one image per checksum is kept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' To obtain per-image results, face detection is performed on every im- age in the dataset and only images with exactly one face are retained for the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Face Detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' For face detection, the method used is based on the Histogram of Oriented Gradients (HOG) as im- plemented in the dlib and face recognition2 libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' HOG is a feature descriptor trained on the Labeled Faces in Wild (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' 2008) dataset to detect human faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Face obfuscation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='The two face obfuscation techniques used are Gaussian blurring and pixelation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Gaussian blur- ring masks details by considering for each pixel the value of the pixels in its neighbourhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' The section around each pixel, called the kernel, is used to modify its value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' A bigger kernel leads to more blurring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Gaussian blur, implemented in the OpenCV function GaussianBlur, is used for the ex- periments in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Pixelation consists of downsizing the image, then re-sizing it to the original size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' When downsizing the image to a smaller size, pixels are deleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' These pixels are replaced when the image is restored to its original size by interpolating new pixels, which are calculated according to an interpolation method chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Pixelation is implemented for the experi- ments in this paper through the resize function provided in 2Available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='com/ageitgey/face recognition (a) Original (b) Blur (c) Pixelated Figure 3: Original and modified images the OpenCV library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' These techniques are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' The four classifiers used to make predictions in this study are the K-Nearest Neighbour (KNN), the Na¨ıve Bayes (NB), the Support Vector Classifier (SVC) and the Multi-Layer Perceptron (MLP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' These classifiers are se- lected because they are commonly used multi-class classi- fiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' The implementation used is as in the Scikit-learn li- brary, with parameters either left at default (NB, SVM), or chosen through the execution of a grid search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Measuring Algorithmic Fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' The fairness defini- tion chosen here is group fairness, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=', equal or unequal per- formance across groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Performance in this case is the ca- pacity of the privacy-preserving method to obfuscate some- one’s identity, which means that the favourable outcome is one where the person is not being recognized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Consequently, contrary to previous works, low accuracy, precision, recall and F1-score are the desired outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' These metrics are all reported for completeness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' however, they can lead to dif- ferent conclusions as they place emphasis on different as- pects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' When considering both race and gender, we also re- port bias, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=', the biggest gap in performance between any two groups for each metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' We make the assumption that people prefer not to be identified, neither correctly nor incor- rectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' As a consequence, it is desirable to have low numbers of both the True Positives (TP, implying a correct identifica- tion), and False Positives (FP, implying an incorrect identi- fication of the person).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' As accuracy reports the proportion of correct predictions, both TP and True Negatives (TN) in- crease the accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Additionally, it is sensitive to class im- balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Therefore, when accuracy leads to a different con- clusion than the other metrics, precision and recall are given preference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' We also report a new metric called Personal De- tection Rate (PDR), given by the following formula - PDR = True Positives + False Positives Number of predictions (1) This metric corresponds to the number of times the person is identified in a picture (correctly or incorrectly), divided by the number of predictions made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' However, it is also sensitive to class imbalances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' 4 Results and Discussion The results presented in the following sections are obtained by using SVC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' For reference, we consider that the groups on which the worst results are obtained are those that score higher on recognition metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Best results are shown in bold, while the worst results are in italics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='1 Level of Obfuscation Needed (a) Blurring by race and gender (b) Blurring by race (c) Blurring by gender Figure 4: Distribution of the level of Blurring Gaussian Blur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' The amount of obfuscation performed by Gaussian blur is based on the size of the kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' The size of the kernel is proportional to the blurring performed on the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' This value is in pixels, so it is normalized by dividing it by the size of the Region Of Interest (ROI), the face, in pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' This aims at reducing the influence of the quality of the original picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' We also observe that for non-white people, the level of ob- fuscation is lower than it is for white people (median value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='109 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='133 respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' This might be because the dlib library is utilized for its HOG implementation, which is pre-trained on the LFW dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' This dataset is shown in (Karkkainen and Joo 2021) to be imbalanced towards white faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' The level of obfuscation differs also between men and women: the median value is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='132 on men and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='126 on women.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Pixelation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Pixelation, as previously mentioned, is done by downsampling and then subsequently upsampling the re- gion of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' The level of obfuscation is the size to which it is downsampled: the smaller the size, the more pixelated the result is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' This value is also divided by the size of the ROI to normalize it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' The results are presented in Fig 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' To have a clearer look at the results by race, we present a truncated version which excludes relative kernel sizes over 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' This excludes 16 images of white people, 12 women and 4 men.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' The median level of pixelation is the same between white and non-white people and between men and women, with a value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='0107 for all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' (a) Pixelation by race (b) Pixelation by gender Figure 5: Distribution of the level of Pixelation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='2 Bias in Face Recognition Results The only results presented in the following sections are ob- tained by using SVC for the sake of brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' For reference, we consider that the groups on which the worst results are obtained are those that score higher on recognition metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Best results are shown in bold, while the worst results are in italics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='The row ”Bias” refers to the biggest gap between two groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' The results are shown in Tables 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' When using pixelation for face obfuscation, the groups that get recognized at the lowest rate are women, white peo- ple, and specifically, white women.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' When looking at inter- sectional results, the group that gets the worst results are non-white women.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' These results are consistent across all classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' When using blurring for face obfuscation, the groups that get recognized at the lowest rate are women, white peo- ple, and specifically, white women.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' These results are con- sistent across all classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' When looking at intersectional results, the groups that get the worst results are non-white men (SVC, MLP) and non-white women (NB, KNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='8 0 + 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='6 + + elative l 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='2 0°0 White Non-white0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='8 + 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='6 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='0 - Male Female35 White 30 Non-white 25 20 D 15 10 5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='3 Relative Kernel Size20 Male Female 15 Density 010 5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='3 Relative KernelSize0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='8 White Non-white 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='6 + + 0.' metadata={'source': 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0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='382 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='447 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='588 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='453 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='519 Bias 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='131 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='066 0.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='331 White Male 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='335 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='409 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='646 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='449 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='427 Bias 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='259 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='262 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='324 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='321 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='322 Table 2: Blurring obfuscation, SVC classifier These results show a bias in identity obfuscation for both pixelation and blurring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' While women, white people, and white women get systemically better results, the group that gets worse results is not systematically non-white men.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' This highlights the need for intersectional studies, without which the bias against non-white women would be concealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Figure 6: Bias across classifier, Pixelation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='3 Influence of the classifier As several metrics are used on different groups, it is not pos- sible to clearly determine which classifier achieves the best performance: there is not one classifier which consistently performs better on all metrics and across all groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' More- over, the goal here is not to determine which classifier is better, but if trends observed are consistent regardless of the classifier used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' On performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' When using pixelation for face obfus- cation, the difference in performance is at most 7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' (be- tween the precision on non-white females when using SVC and KNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' On all other metrics, across all different groups, the difference in performance between classifier is less than 4%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' When using blurring for face obfuscation, SVC per- forms overall slightly better than the other classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' How- ever, even here, the biggest difference in performance (dif- ference between the precision on non-white males when us- ing NB and SVC) is only 11%, with the difference on all other metrics, across all different groups being less than 6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Overall, the influence of the classifier on performance is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' On bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' General trends are independent of classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' When controlling for other factors, the groups on which the best and worst performances are achieved stay the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' As can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' 6, the reduction of bias depends on which metric is considered when using the pixelation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' However, the biggest variation in bias (between SVC and NB on pre- cision) is only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='045 for pixelation and while SVC performs better on recall, precision and F1-score, it performs worse on accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' When using blurring as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' 7, NB is overall less biased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' However, in the scenario considered, the choice of classifier is that of the bad actor, and not of the entity trying to fairly preserve people’s identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Accurac 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='00 KNN SVC NBMLP KNN SVC NB MLP Precision F1-score 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='00 KNN SVC NB MLP KNN SVC NB MLPFigure 7: Bias across classifier, Gaussian Blur 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='4 Influence of the face obfuscation methods The face obfuscation method does not influence the general trend: the best results are obtained on white people, women and white women.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' The worst results are obtained on non- white people for pixelation, on non-white men (MLP and SVC) and non-white women (NB) for blurring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' However, as we can see in Fig 7 and Fig 6, using pixelation for face obfuscation leads to a smaller gap in performance across all metrics and for all classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' 5 Limitations, Conclusions, and Future Work Our work focuses intentionally on an imbalanced dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' However, a similar study on a balanced dataset could help determine the origin of the bias shown in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' This has not, however, been attempted because of the low num- ber of persons in the used dataset, and also the low num- ber of images present for some persons in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' The authors note that the creation of a bigger, balanced dataset with the characteristics of PubFig would be highly beneficial to push forward research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' Additionally, the choice of itera- tively obfuscating faces to find the limit of face detection is not one that could easily be implemented in a real-time sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' The reproducibility of this study is also impeded by the fact that, like Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' (Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' 2009), we are un- able to distribute image files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' However, the code developed for this study will be made available on Github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' As has been shown, the degree to which the identity is protected by face detection is subject to racial and gender bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' This bias is present regardless of the classifier of face obfuscation technique used, but using pixelation instead of blurring leads to less bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' The scope of this study could be broadened in future work by performing a similar experiment on a dataset balanced with regard to race and gender, or by considering other pro- tected attributes and face obfuscation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' It could also be extended to other types of person recognition, such as whole body recognition, and different types of images, such as depth data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' 6 Acknowledgements This work is part of the visuAAL project on Privacy- Aware and Acceptable Video-Based Technologies and Ser- vices for Active and Assisted Living (https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='visuaal- itn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content='eu/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} +page_content=' This project has received funding from the Euro- pean Union’s Horizon 2020 research and innovation pro- gram under the Marie Skłodowska-Curie grant agreement No 861091.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE4T4oBgHgl3EQfUAxn/content/2301.05012v1.pdf'} 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+b, Vitalii Lukichov2 +c, Jannik M¨ahn1 +d and Stefan +K¨opsell1 +e +1Barkhausen Institut gGmbH, W¨urzburger Straße 46, Dresden, Germany +2Department of Information Protection, Vinnytsia National Technical University, Khmelnytske Shosse 95, Vinnytsia, +Ukraine +{yevhen.zolotavkin, jannik.maehn, stefan.koepsell}@barkhauseninstitut.org, {yuriy.baryshev, lukichov.vitalyi}@vntu.edu.ua +Keywords: +privacy, V2X, unlinkability, hidden Markov model, cybersecurity, entropy, obfuscation +Abstract: +In this paper, we develop a new methodology to provide high assurance about privacy in Cooperative Intelli- +gent Transport Systems (C-ITS). Our focus lies on vehicle-to-everything (V2X) communications enabled by +Cooperative Awareness Basic Service. Our research motivation is developed based on the analysis of unlink- +ability provision methods indicating a gap. To address this, we propose a Hidden Markov Model (HMM) +to express unlinkability for the situation where two cars are communicating with a Roadside Unit (RSU) us- +ing Cooperative Awareness Messages (CAMs). Our HMM has labeled states specifying distinct origins of +the CAMs observable by a passive attacker. We then demonstrate that a high assurance about the degree of +uncertainty (e.g., entropy) about labeled states can be obtained for the attacker under the assumption that he +knows actual positions of the vehicles (e.g., hidden states in HMM). We further demonstrate how unlinkability +can be increased in C-ITS: we propose a joint probability distribution that both drivers must use to obfuscate +their actual data jointly. This obfuscated data is then encapsulated in their CAMs. Finally, our findings are +incorporated into an obfuscation algorithm whose complexity is linear in the number of discrete time steps in +HMM. +1 +INTRODUCTION +Due to the intense development of transport systems +over the recent decades, different modes of cooper- +ative intelligence have been incorporated into their +functionalities. +Intelligent transport systems (ITS) +are transport systems in which advanced information, +communication, sensor and control technologies, in- +cluding the Internet, are applied to increase safety, +sustainability, efficiency, and comfort. Cooperative +Intelligent Transport Systems (C-ITS) are a group of +ITS technologies where service provision is enabled +by, or enhanced by, the usage of ‘live’, present situa- +tion related, dynamic data/information from other en- +tities of similar functionality, and/or between different +elements of the transport network, including vehicles +and infrastructure (ISO/TR 17427-7:2015(E), 2015). +Technology allowing a vehicle to exchange ad- +a +https://orcid.org/0000-0002-1875-122X +b +https://orcid.org/0000-0001-8324-8869 +c +https://orcid.org/0000-0002-3423-5436 +d +https://orcid.org/0000-0003-0870-7193 +e +https://orcid.org/0000-0002-0466-562X +ditional information with infrastructure, other vehi- +cles and other stakeholders in the context of C-ITS +is called vehicle-to-everything (V2X). Multiple ad- +vances in modern C-ITS applications, such as col- +laborative forward collision warning and emergency +electronic brake lights, are impossible without V2X. +These advances, however, come at a cost: C-ITS ap- +plications rely on vehicles broadcasting signals to in- +dicate their location, signals which are intended to be +received and processed by a range of other devices. +For example, vehicles may cooperatively broadcast +(with the frequency of 1-10 Hz) geo-spatial informa- +tion to nearby peers using short Cooperative Aware- +ness Messages (CAMs). Hence, V2X raises essen- +tial privacy questions: i) to what degree can specific +vehicles be located and tracked based on such infor- +mation? ii) what are the techniques able to improve +privacy of V2X? To answer these questions, we use +the concept of unlinkability to reason about privacy. +1.1 +Research motivation +Even though the problems of privacy in C-ITS were +acknowledged in several relevant documents (having + +normative and informative character), satisfactory an- +swers have yet to be provided to the privacy questions +mentioned above. For example, the document (ISO +24102-1, 2018) recognizes the importance of unlink- +ing private data from traceable address elements and +identifiers in wireless messages sent by an ITS station +unit (ITS-SU). However, relevant considerations in +this document do not go beyond suggesting that “such +unlinking can be done by means of pseudonyms”: +the sufficiency of these and many similar sugges- +tions remain unaddressed. +In contrast, limitations +of pseudonym changes in CAMs have been recog- +nized by academic authors (Escher et al., 2021; Karim +Emara, 2013; Wiedersheim et al., 2010). In particu- +lar, vehicle tracking becomes possible due to CAM +content being signed but not encrypted: this is de- +manded by the relevant standards in C-ITS(DS/ETSI +EN 302 637-2 V1.4.1, 2019). This is because full en- +cryption of CAMs may impede C-ITS functionalities +that are critical for safety. Nonetheless, recognition of +privacy-affecting issues in C-ITS has yet to result in a +solution where the degree of privacy is correctly mea- +sured, and privacy limitations are eliminated. In this +paper, a methodology to address these challenges is +developed: we provide an assurance for the procedure +estimating the lower bound of unlinkability for CAMs +and an algorithm maximizing this criterion under the +overall constraint of location precision degradation. +1.2 +Research principles +To address V2X privacy questions, it is imperative +to obtain results for which confidence is high. The +methodology developed in this paper rests on the prin- +ciples of robust optimization (RO) (Gorissen et al., +2015; Sniedovich, 2016). The unsuccessfulness of +the previous attempts to answer the mentioned above +‘V2X privacy questions’ is mainly caused by inoppor- +tune privacy assumptions for C-ITS and V2X scenar- +ios in particular. For example, a combination of as- +sumptions that an attacker only observe CAMs con- +taining obfuscated (distorted) geo-position measure- +ments and does not have a precise physical model +for car movements may require reasoning involving +the best-known technique to estimate such a model +(Blackman, 1986). +Because of the computational +complexities associated with these estimations, re- +searchers often rely on heuristic and poorly justifiable +steps: this is unacceptable if high level of privacy as- +surance is demanded (Blackman, 2004; Wiedersheim +et al., 2010). To avoid this situation, we make stronger +assumptions about the information known to the at- +tacker, which allows us to estimate a lower bound of +unlinkability in C-ITS: this estimate has high confi- +dence. This attitude is reflected in the following prin- +ciples shaping our methodology: +• The methodology for privacy risk assessment +should not underestimate the risks: it should pro- +vide clear and quantifiable assessments of how +easy data belonging to the same entity can be ex- +tracted throughout multiple V2X communication +sessions involving more than one user. Therefore, +the methodology should set justifiable bounds for +assessments of such quantity; +• The methodology should propose optimal mod- +ifications of the original user data. +The notion +of ‘optimality’ depends on the kind of modifica- +tion. Reversible modifications (such as encryp- +tion) should be provided with the assurances of +computational infeasibility of such reversions by +illegitimate parties (e.g., the level of such assur- +ance is maximal). For irreversible modifications +(such as noising), the loss of data quality should +be quantified: for each such quantity, there should +be an assurance that no higher degree of unlinka- +bility can be achieved (e.g., the proposed noising +method is optimal). +1.3 +Our contribution +The unique contribution is due to combination of the +study objective (guided by the criterion of unlinkabil- +ity), robust assumptions, and the optimal obfuscation +technique developed in the paper. +• First, the aim of this study is to protect C-ITS +from the threat of linking: this is in contrast with +the numerous obfuscation approaches which aim +at impeding inferencing about the actual location +of ITS-SU (Andr´es et al., 2013; Bordenabe et al., +2014); +• Second, to obtain high confidence in the measured +unlinkability, we assume that an attacker has com- +plete knowledge about the system design, obfus- +cation algorithms, quality degradation (distortion) +constraints, has access to CAMs, stored states ve- +hicles’ geo-positions, and the true states charac- +terizing the geo-positions of the vehicles at any +moment in time; +• Third, we develop an optimal obfuscation algo- +rithm: for a given distortion constraint, it pro- +vides the highest level of uncertainty for the at- +tacker trying to link obfuscated CAMs with their +sources. +The rest of this paper is structured as follows. In +section 2, we set the grounds for the study and provide +basic definitions. It is followed by section 3, where + +we start with a description of a generic information +system. Further specifics of C-ITS are then reflected +using additional sets and relations: as a result, we ob- +tain Hidden Markov Model (HMM), used to study +unlinkability. +In section 4, we formalize assump- +tions, define unlinkability through entropy and opti- +mize joint obfuscation producing observable states in +HMM. Next, section 5 describes a compact and effi- +cient algorithm calculating the unlinkability indicator +in C-ITS and implementing previous findings to im- +prove unlinkability. Finally, we discuss our results, +their importance, novelty, advantages, and limitations +in section 6. +2 +PRELIMINARIES +To justify subsequent modelling steps better, we intro- +duce contextual information supporting our aim, set- +tings and privacy assumptions. +2.1 +Aim of the study +To specify the aim, we analyze relevant cybersecurity +requirements. Here, we use some of the classical def- +initions for privacy in complex systems to specify our +aim with greater precision. Privacy requirements for +C-ITS are often derived from ISO/IEC 15408-2. For +example, (ETSI TS 102 941 V2.1.1, 2021) suggests +that the combination of pseudonymity and unlinka- +bility offers the appropriate sender privacy protection +for basic ITS safety messages (such as CAM). In sim- +ple terms, pseudonymity requires that the identity of a +user is never revealed or inferred. However, one of the +major complications in dealing with pseudonymity is +the following: an attacker may learn the user’s iden- +tity composition based on multiple sessions, events, +or traces. +Unlinkability is the assurance about the +ability to resist learning such a composition (ISO/IEC +15408-2:2022(E), 2022): +Definition 1 (Unlinkability of operations) +Requires that users and/or subjects are unable +to determine whether the same user caused cer- +tain specific operations in the system, or whether +operations are related in some other manner. +In the context of V2X communication in C-ITS +with many users, the cryptographically signed mes- +sages broadcasted by the ITS-SUs (controlled by +these users) should have the property of definition 1 +(Hicks and Garcia, 2020; ETSI TS 102 941 V2.1.1, +2021). Nonetheless, such interpretation has certain +disadvantages, major of which is inflexibility. Indeed, +‘...unable to determine...’ statement can either be false +or true, meaning that the unlinkability of the whole +C-ITS (with many cars and observable during many +hours) is expressed using a binary value. This issue +has been recognized by practitioners and researchers +alike, which is reflected in comments and best prac- +tice recommendationsto ITS engineers and managers. +For example, (ISO/SAE 21434, 2021) contains the ta- +ble ‘Example privacy impact rating criteria’: it in- +cludes Impact rating Criteria interpreting the mean- +ing for the severity degrees (e.g., Negligible, Moder- +ate, Major, Severe) for privacy impact rating indica- +tor. Importantly enough, interpretations in this table +evolve around two aspects: a) the level of sensitivity +of the information about road user; b) how easily it +can be linked to a PII (Personally Identifiable Infor- +mation) principal. Such emphasis on the easiness of +linking motivates us to modify definition 1 in the fol- +lowing manner: +Definition 2 (Unlinkability of operations*) Is +the +degree of inability to determine (by users and/or sub- +jects) whether the same user caused certain specific +operations in the system, or whether operations are +related in some other manner. +To provide convenience in comparing unlinkabil- +ity in C-ITS under different conditions, we use Shan- +non entropy: it is an integral criterion of uncertainty +in a system that fully captures the ‘...degree of inabil- +ity to determine...’ (Wagner and Eckhoff, 2018). +Henceforth, the main aim of our paper is to de- +velop a methodology providing high level of assur- +ance that entropy for CAMs’ origins is high in C-ITS. +2.2 +Settings for the study +In fig. 1, we introduce a general setup for our study. +In fig. 1(a), two cars (ITS-SUs) are driven by Alice +and Bob, respectively. Both ITS-SUs transmit CAMs +with the same frequency, and the roadside unit (RSU) +receives them without losses. The role of the attacker +is played by the RSU, who tries to separate CAMs of +Alice from CAMs of Bob: this allows the attacker to +link CAMs belonging to the same entity. Although +CAMs are signed, in our study we assume that a sig- +nature scheme providing unlinkability is used. There- +fore, the separation is done based on the content of +the CAMs (since the are transmitted unencrpyted) and +the order of arrival of CAMs within each time in- +terval – see fig. 1(b). We consider the ordering of +CAMs’ arrivals to be non-uniform in general: for +example, in one extreme case, the CAM from Alice +arrives first, and Bob’s CAM arrives second at any +time interval i. +If an attacker knows about such a +unique property, he can link CAMs without consider- +ing their payload. However, these cases are unlikely, +meaning that an attacker should also be able to in- + +fer the source (e.g., ‘from Alice’ or ‘from Bob’) of +a CAM based on its content. The requirements for +the content of CAMs can be found in (DS/ETSI EN +302 637-2 V1.4.1, 2019). In particular, we consider +that geo-position, velocity and acceleration are essen- +tial. On the one hand, these parameters are mandatory +for the HighFrequency container in CAM. On the +other hand, numerous techniques using these parame- +ters have been developed for the domain of Multiple- +Target Tracking (MTT): corresponding estimators can +be of great use for reasoning about privacy in C-ITS +(Blackman, 1986; Karim Emara, 2013). +Figure 1: Setting for our study of unlinkability of CAMs. +In this study, CAMs’ content unlinkability is the +core of our attention. We exclude from further con- +sideration the following CAM payload: +1) cryp- +tographically produced proofs of authenticity (e.g., +signatures); 2) categorical data (e.g., vehicle role). +These exclusions are due to substantial attention +to issue ‘1)’ among the members of the crypto- +graphic community. +For example, pseudonym un- +linking solutions were proposed in (Camenisch et +al., 2020; Hicks and Garcia, 2020). +Neverthe- +less, there is a need to complement these efforts +by our study: +the absence of encryption (due to +safety reasons) in CAMs makes pseudonym unlink- +ing necessary but not sufficient for CAMs’ unlink- +ability. +This is because other data, such as geo- +graphic positions, in CAMs may be used for link- +ing. Issue ‘2)’ can be omitted since categorical data +is a part of basicVehicleContainerLowFrequency +and is OPTIONAL in CAMs (DS/ETSI EN 302 637-2 +V1.4.1, 2019). We also exclude VehicleLength and +VehicleWidth (compulsory for the HighFrequency +container), which otherwise are likely to be of great +use in discriminating different vehicles (Escher et al., +2021). For such an exclusion we find justification in +(ETSI TS 102 894-2 V1.3.1, 2018) which allows usage +of the codes 1023 and 62 for the length and width, +respectively, if the corresponding information is un- +available. +Because of the details described above, we will +model CAM as a vector in Rz where z ≥ 1. Such a +step is beneficial: we can apply commonly used dis- +tortion measures such as, for example, Squared Error +(SE). This is a clear and straightforward way to re- +fer to the quality degradation of essential location ser- +vices (Shokri et al., 2016). We, nevertheless, refrain +from further discussions about the chosen distortion +measure in this paper. +2.3 +Privacy assumptions and threats +Here we provide a high-level intuition for the system +and the threat of linkability, while the details will be +introduced in the subsequent sections. Alice and Bob +coordinate their efforts. They distribute the total al- +lowed distortion among N − 1 time steps: as a result, +they know the distortion limit for every time step i. +At the beginning of every time interval i, Alice and +Bob know the true measurements (including position, +speed, acceleration, etc.) of each other. To obfus- +cate data in their CAMs they randomly agree on the +order of their arrival at RSU at every i. For every +i they define a joint distribution according to which +they change (obfuscate) their actual measurements: in +expectation, they remain within the distortion limits. +An attacker who fully controls RSU statistically +infers the source of every pair of CAMs which he ob- +serves during time i: this statistical inference is used +to calculate entropy and aligns with definition 2. For +this, the attacker refers to the joint distribution used by +Alice and Bob during the obfuscation. He also knows +other information, such as the original geo-positions +of the players at every i, and the probabilities for the +order of CAMs’ arrivals. The resulting unlinkability +in the system depends on: i) statistics for the order +of arrival of CAMs from the players; ii) the level of +allowed distortion; iii) how far apart actual measure- +ments of Alice and Bob are at every i. +3 +MATHEMATICAL MODEL +We explain our mathematical model in the follow- +ing sections. To easy the reading, table 1 contains +an overview about our notations. + +RSU +Alice +Bob +a +CAM(B) +CAM(A) +CAM(B) +CAM(A) +0 +2 +1 +N-2 +N-1 +b)Table 1: Notations +Notation +Description +ITS +Intelligent Transport Systems +C-ITS +Cooperative Intelligent Transport Systems +ITS-SU +ITS Station Unit (including installed in vehicles) +V2X +Vehicle-to-Everything +CAM +Cooperative Awareness Message +RSU +Roadside Unit +HMM +Hidden Markov Model +Du +Set of user-related data +Ds +Set of information system-related data +U +Set of information system’s users +P +Set of data processing procedures at the information system +P +Set of players including Alice and Bob +xA +k , 1 ≤ k ≤ µ +A hidden state for Alice +XA = {xA +k } +Set of hidden states for Alice +xB +j , 1 ≤ j ≤ ω +A hidden state for Bob +XB = {xB +j } +Set of hidden states for Bob +X(A,B) +Set of joint hidden states for +� +Alice, Bob +� +X(B,A) +Set of joint hidden states for +� +Bob, Alice +� +R +Index (label) for rose nodes +XR = X(A,B) +Set of all rose nodes +B +Index (label) for blue nodes +XB = X(B,A) +Set of all blue nodes +L = {R ,B} +Set of labels encoding |P|! combinations +Y +Set of joint observable states for Alice and Bob +i ∈ {1,2,...,N − 1} +Time-step in discrete HMM +XA +i +Variable on XA at i +XB +i +Variable on XB at i +Xi +Variable for joint hidden state on step i +ℓi +Variable on L at i +Yi +Variable on Y on step i +Pr(Xi+1 | Xi) +Probability of transition between hidden states +Pr(Yi | Xi) +Conditional probability for observable states +ϕ +Order mixing (label permuting) probability +ρi +Distribution over hidden states on step i +ρi+1|ρi +Conditional distribution over hidden states on step i+ 1 +3.1 +Model setup +We consider the generic case of information systems +with the passive attacker (Eve). An information sys- +tem processes data set Du for certain users, who form +set U. For this, information system executes a set +of data processing algorithms P: data Ds, which de- +scribes configuration and parameters of these pro- +cessing algorithms. Any data processing algorithm +∀p ∈ P is triggered by either a user or information +system’s state described by its configuration and pa- +rameters. At the data processing algorithm p, spe- +cific user data is taken for input that results in cre- +ation of new or altering of existing user’s data as +well as change of information system’s configuration +and parameters. In certain cases data processing al- +gorithms within system can alter even user set U. +Therefore, p is a mapping p : U× D → U× D, where +D = Du ∪ Ds. Thus, information system is described +as a tuple {U,D,P,Eve}. +For uι ∈ U user’s identification information sys- +tem performs special data processing algorithms +identifys(·) ∈ P, so that identifys(duι) = uι ∈ U. +To protect it’s data the information system may +apply obfuscation algorithm ob fuscate(·) ∈ P to +achieve such data alteration d∗ +uι = ob fuscate(duι) that +identifys(d∗ +uι) ̸= uι. +Let’s consider Eve is watching over the data pro- +cessing flow with a certain ability level that allows +her to get access to the data of the information sys- +tem – Eve sees d∗ +u,d∗ +s , where d∗ +u ⊆ Du and d∗ +s ⊆ Ds. +For the unlinkability property of user’s data within the +considered system it is necessary that ∀d∗ +uι ⊆ Du so +that Eve sees d∗ +uι, it is infeasible for Eve to find such +a transformation identifye(duι) that identifye(d∗ +uι) = +uι. +The +following +interpretations +are +possible +in the context of unlinkability for C-ITS. du = +{registrationNumber,manu facturer,...,PATHS}, +where ∀pathι ∈ PATHS, pathι = {⟨xj,yj⟩}. Some +ways of user linking are: +- search(registrationNumber) = uι: +is usually +performed by law enforcement agencies; +- recon(manu facturer,color) = uι: can be per- +formed by private detectives, for instance; +- drivingModelling(pathsι) = uι: +can be per- +formed by gathering of data from roadside units. +The research focuses on the latter way of user link- +ing. Therefore we considering the states of users ve- +hicles at different moments in time (communicated in +CAMs) and assessing unlinkability in C-ITS through +Eve’s ability to infer information using, for instance, +drivingModelling(·) as identifye(·). +3.2 +Markov model for unlinkability +To study unlinkability in V2X we use Hidden Markov +Model which graphical representation is given on +fig. 2. +The following sets are needed to de- +scribe the model. +The set of all players is P = +{Alice, Bob,...}. For each player, there exists a set +of hidden states for his vehicle, e.g., for Alice there +is XA = +� +xA +1,xA +2,...,xA +k ,...,xA +µ +� +and for Bob there is +XB = +� +xB +1,xB +2,...,xB +j ,...,xB +ω +� +. Each state, for example, +xA +1 can be a vector including specific position, veloc- +ity, acceleration and other characteristics applicable to +Alice’s vehicle at certain time. Throughout the paper +we assume that XA ∩XB is in general non-empty. +The system of |P| players is characterized by hid- +den and observable joint states. Transition happens +between hidden states Xi and Xi+1 when time step i +proceeds to i + 1, where joint state Xi = +� +XA +i ,XB +i +� +is +the composition (concatenation) of variables XA +i ∈ XA +and XB +i ∈ XB . As such, ∀k, j(xA +k ,xB +j ) ∈ X(A,B), where +|X(A,B)| = |XA| × |XB| (for simplicity of representa- +tion we further assume |P| = 2, |XA| = µ = 2, |XB| = +ω = 2). +Possible transitions from Xi to Xi+1 are denoted +using indices 1 − 16 (see fig. 2): these transitions +are governed by corresponding probabilities. For ex- +ample, the transition from Xi = +� +XA +i = xA +1,XB +i = xB +1 +� +to Xi+1 = +� +XA +i+1 = xA +2,XB +i+1 = xB +2 +� +is denoted by +index 4. +The probability of such a transition +is Pr +� +XA +i+1 = xA +2,XB +i+1 = xB +2 | XA +i = xA +1,XB +i = xB +1 +� +. +In +practice, these probabilities can be obtained based on + +the well-studied physical models for vehicles (Black- +man, 1986). +For each Xi of the hidden joint states there are +|P|! possible permutations for its concatenated com- +ponents originating from the users. +These permu- +tations are the major cause of uncertainty when an +attacker attempts to label combined CAMs of Alice +and Bob. In practice, this is caused by the unpre- +dictable arrangement of CAMs within each scan (or +session) i. Hence, a permutation should be selected +by randomly following one of the possible transi- +tions. +For example, while the system is in a joint +state +� +XA +i = xA +1,XB +i = xB +1 +� +permutation +� +xA +1,xB +1 +� +(rose +colored node) should be considered if transition with +index 17 takes place, and +� +xB +1,xA +1 +� +(blue coloured +node) should be considered if transition 18 happens +(see fig. 2). We will use notations Xi,R and Xi,B for +rose and blue nodes, respectively, where Xi,R ∈ XR , +Xi,B ∈ XB, and XR = X(A,B), XB = X(B,A). +Fur- +ther in the text, we will refer to the states repre- +sented by the coloured nodes as ‘labelled states’. For +the sake of simplicity and without loss of generality, +for all realizations of hidden states Xi, we consider +Pr +� +Xi,R |Xi +� += ϕ ≤ 0.5, and Pr +� +Xi,B|Xi +� += 1 − ϕ. +1 +2 +3 +4 +17 +18 +6 +5 +7 +8 +10 +11 +9 +12 +14 +15 +16 +13 +19 +20 +21 +23 +22 +24 +25 +27 +26 +28 +� +XA = xA +1 , XB = xB +1 +� +� +xA +1 , xB +1 +� +(ˆy1, ˇy1) +� +XA = xA +1 , XB = xB +2 +� +� +xB +1 , xA +1 +� +(ˇy1, ˆy1) +... +... +� +XA = xA +2 , XB = xB +1 +� +� +xA +2 , xB +2 +� +(ˆy2, ˇy2) +... +� +XA = xA +2 , XB = xB +2 +� +� +xB +2 , xA +2 +� +(ˇy2, ˆy2) +Figure 2: Hidden Markov Model for 2 players sending +CAMs. +To denote the totality of hidden permuted joint +states we use set X{R ,B} = XR ∪ XB, +where +|XR | ≤ |X{R ,B}| ≤ 2|XR |. +For every Xi,R +and +Xi,B there are transitions to observable joint states +Yi ∈ Y, Y = +� +(ˆy1, ˇy1),(ˇy1, ˆy1),...,(ˆyq, ˇyq),(ˇyq, ˆyq) ,..., +(ˆyξ, ˇyξ),(ˇyξ, ˆyξ) +� +. Some of these transitions to observ- +able states are denoted with indices 21 − 28 on fig. 2. +Until proven otherwise, the cardinality of Y is consid- +ered independent on |X{R ,B}|. +Measuring uncertainty about label ℓ ∈ L, L = +{R ,B}, is of our main interest: this is done based +on observable states. +4 +MODEL PROPERTIES +To +formally +express +unlinkability +following +definition +2 +we +will +use +conditional +entropy +H (ℓ1,ℓ2,...|Y1,Y2,...) +for +the +sequence +of +la- +bels ℓ1,ℓ2,...,ℓN−1 given that an attacker observes +Y1,Y2,...,YN−1 (Wagner and Eckhoff, 2018). +4.1 +General expression for unlinkability +For the described HMM, probability of any hidden +state at any time step can be specified using multivari- +ate discrete distribution ρρρ : X(A,B)×{0,1,...,N−1} → +[0,1]N|X(A,B)|. We will further use ρi slices of ρρρ such +that ρρρ = +N−1 +� +i=0 +ρi, where each slice represents a distri- +bution over hidden states at step i. Slice ρ0 defines +distribution over the hidden states before the start of +the system. Because HMM has been previously de- +fined (see fig. 2) using transitional probabilities that +remain unchanged for all time steps, each slice can +be fully determined in a conditioned sequential man- +ner: ρi+1|ρi means that ρi+1 is trivially derived if ρi +is given. +Since an attacker observes Y1,Y2,...,YN−1 and +knows ρρρ analysis of H (ℓ1,ℓ2,... | Y1,Y2,...,ρρρ) is cen- +tral to our reasoning about unlinkability. We state the +following. +Lemma 1 Unlinkability in V2X system (as per fig. 2) +is expressed as: +H +� +ℓ1,ℓ2,... +��Y1,Y2,...,ρρρ +� += +N−2 +∑ +i=0 +H +� +ℓi+1 +��Yi+1,{ρi+1|ρi} +� +. +(1) +Proof: +For simplicity, we consider N = 3 only. +First, it should be noted that +H +� +ℓ1,ℓ2 +��Y1,Y2,ρρρ +� += +H +� +ℓ1,ℓ2,Y1,Y2 +��ρρρ +� +− H +� +Y1,Y2 +��ρρρ +� +. +(2) +We then ponder at the right-hand side of the eq. (2). +Each of these terms can be expressed as: +H +� +ℓ1,ℓ2,Y1,Y2 +��ρρρ +� += +H +� +ℓ2,Y2 +��ℓ1,Y1,ρρρ +� ++ H +� +ℓ1,Y1 +��ρρρ +� +, +(3) +and +H +� +Y1,Y2 +��ρρρ +� += H +� +Y2 +��Y1,ρρρ +� ++ H +� +Y1 +��ρρρ +� +, +(4) +respectively. We point out that H +� +ℓ2,Y2 +��ℓ1,Y1,ρρρ +� += +H +� +ℓ2,Y2 +��ρρρ +� +and +H +� +Y2 +��Y1,ρρρ +� += H +� +Y2 +��ρρρ +� +in +eqs. (2) and (3), respectively. +This follows from +the fact that realizations of ℓi,Yi do not affect +ℓi+1,Yi+1. We finally stress that ρρρ is redundant for + +determining ℓi+1,Yi+1 since only ρi+1|ρi has rele- +vance: H +� +Y1 +��ρρρ +� += H +� +Y1 +��{ρ1|ρ0} +� +, H +� +ℓ1,Y1 +��ρρρ +� += +H +� +ℓ1,Y1 +��{ρ1|ρ0} +� +, +H +� +Y2 +��ρρρ +� += H +� +Y2 +��{ρ2|ρ1} +� +, +H +� +ℓ2,Y2 +��ρρρ +� += H +� +ℓ2,Y2 +��{ρ2|ρ1} +� +. +Hence, eq. (2) +can be rewritten as +H +� +ℓ1,ℓ2 +��Y1,Y2,ρρρ +� += +H +� +ℓ1,Y1 +��{ρ1|ρ0} +� ++ H +� +ℓ2,Y2 +��{ρ2|ρ1} +� +− +� +H +� +Y1 +��{ρ1|ρ0} +� ++ H +� +Y2 +��{ρ2|ρ1} +�� +. +(5) +The latter eq. (5) can be regrouped +H +� +ℓ1,ℓ2 +��Y1,Y2,ρρρ +� += +� +H +� +ℓ1,Y1 +��{ρ1|ρ0} +� +− H +� +Y1 +��{ρ1|ρ0} +�� ++ +� +H +� +ℓ2,Y2 +��{ρ2|ρ1} +� +− H +� +Y2 +��{ρ2|ρ1} +�� += +H +� +ℓ1 +��Y1,{ρ1|ρ0} +� ++ H +� +ℓ2 +��Y2,{ρ2|ρ1} +� +. +(6) +□ +4.2 +Worst-case unlinkability +We aim to obtain a computationally feasible estima- +tion of unlinkability. +Direct utilization of the re- +sults of lemma 1 presupposes computing {ρi+1|ρi} +which has several disadvantages: a) transition prob- +abilities for hidden states need to be specified (which +usually requires studying physical models of move- +ment for the users); b) total computational com- +plexity for defining distributions over the hidden +states is therefore O(Nµ2ω2). +To avoid these +complications, we develop our unlinkability assur- +ance based on a rational lower bound Hr for +H +� +ℓ1,ℓ2,... +��Y1,Y2,...,ρρρ +� +. The concept of the ratio- +nal lower bound is explained through the following +assumptions (Sniedovich, 2016). +Assumption 1 (Worst-case unlinkability) Requires +that an attacker knows sets for the hidden, labelled +and observable states. He knows all the transitions +and the order mixing probability ϕ. +For each ob- +servable state at time i he then defines the worst +possible hidden state(s) which does not contradict his +knowledge. +We nevertheless stress that despite assumption 1 +might be viewed as excessive, the attacker does not +know the labelled state ℓi (and can not force its selec- +tion) at time i. +Assumption 2 (Rational lower bound Hr) +Requires that users are rational and maximize +worst-case unlinkability: +observable states are +obtained through rational obfuscation of the worst +labelled states considered by the attacker. +There are several aspects affecting the task of cal- +culating such Hr: 1) probabilities for transitions be- +tween hidden states (e.g., the probabilities defining +ρρρ); 2) probabilities for transitions from the hidden +states to the labelled states (e.g., ϕ, 1 − ϕ), and from +the labelled states to the observable states. Further, we +consider a situation where the worst case ρρρ (minimiz- +ing entropy) is defined for 1) while the most optimal +probabilities (maximizing entropy) are then specified +for 2) under constraint ˜D on the total distortion over +N − 1 steps. +We use the results of lemma 1 to require the fol- +lowing: +Hr = min +ρρρ +� +H +� +ℓ1,ℓ2,... +��Y1,Y2,...,ρρρ +�� += +N−2 +∑ +i=0 +min +{ρi+1|ρi} +� +H +� +ℓi+1 +��Yi+1,{ρi+1|ρi} +�� +. +(7) +To obfuscate hidden states in the way maximizing +Hr we need to determine properties of +ρmin,i+1 = arg +min +{ρi+1|ρi} +� +H +� +ℓi+1 +��Yi+1,{ρi+1|ρi} +�� +. +(8) +Probabilities +Pr +� +ℓi+1 = R ,Yi+1 +��{ρi+1|ρi} +� +, +Pr +� +ℓi+1 = B,Yi+1 +��{ρi+1|ρi} +� +will be used in our +further derivations. +To simplify notations we +will use Pr(ℓi+1 = R ,Yi+1), +Pr(ℓi+1 = B,Yi+1), +respectively. The probabilities are defined as: +Pr(ℓi+1 = R ,Yi+1) = +∑ +Xi+1,R ∈XR +Pr +� +Yi+1 | Xi+1,R +� +Pr +� +Xi+1,R +� += +∑ +Xi+1,R ∈XR +Pr +� +Yi+1 | Xi+1,R +� +ϕPr(Xi+1) , +(9) +Pr(ℓi+1 = B,Yi+1) = +∑ +Xi+1,B∈XB +Pr +� +Yi+1 | Xi+1,B +� +Pr +� +Xi+1,B +� += +∑ +Xi+1,B∈XB +Pr +� +Yi+1 | Xi+1,B +� +(1 − ϕ)Pr(Xi+1) . +(10) +We then point out that +Pr(ℓi+1 = R | Yi+1) = Pr(ℓi+1 = R ,Yi+1) +Pr(Yi+1) +, +(11) +Pr(ℓi+1 = B | Yi+1) = Pr(ℓi+1 = B,Yi+1) +Pr(Yi+1) +, +(12) +where +Pr(Yi+1) = Pr(ℓi+1 = R ,Yi+1)+ Pr(ℓi+1 = B,Yi+1) . +(13) +The following result establishes an important +property of ρmin,i+1. +Lemma 2 For all i ∈ [1,N − 1] distribution ρmin,i is +degenerate. + +Proof: +We presume that ρmin,i is non-degenerate. +For simplicity and without loss of generality we +consider two-point distribution ρ∗ +min,i : {x1,x2} → +[0,1]2. For instance, realizations x1 = (xA +1,xB +1), x2 = +(xA +2,xB +2) can be used. +Here Pr(Xi = x1) = ξ, and +Pr(Xi = x2) = 1 − ξ. +Minimization of conditional entropy in eq. (8) is +equivalent to the minimization of pℓi,min where +pℓi,min = min +� +Pr(ℓi = R | Yi),Pr(ℓi = B | Yi) +� +, +(14) +and without loss of generality, we assume that +pℓi,min = Pr(ℓi = R | Yi). +To +express +pℓi,min +we then use eqs. (9) to (11) and (13) with +the +following +substitutions +(simplifying +ex- +pressions): +α1 += +ϕPr +� +Yi | Xi,R = (xA +1,xB +1) +� +, +β1 += +ϕPr +� +Yi | Xi,R = (xA +1,xB +1) +� ++ +(1 +− +ϕ)Pr +� +Yi | Xi,B = (xB +1,xA +1) +� +, +α2 += +ϕPr +� +Yi | Xi,R = (xA +2,xB +2) +� +, +β2 += +ϕPr +� +Yi | Xi,R = (xA +2,xB +2) +� ++ +(1 +− +ϕ)Pr +� +Yi | Xi,B = (xB +2,xA +2) +� +. +The +minimization +task is then +min +ξ +pℓi,min = min +ξ +ξα1 + (1 − ξ)α2 +ξβ1 + (1 − ξ)β2 +. +(15) +By analyzing ∂ +∂ξ pℓi,min, we conclude that there are +no local extrema for ξ ∈ (0,1) and, hence, minimum +is obtained in one of the end points, e.g., ξ ∈ {0,1}. +□ +Based on the result of lemma 2, for every Yi there +is one and only worst-case hidden state ˜Xi (because +Pr +� ˜Xi | ρmin,i +� += 1). It implies the following: +Corollary 1 Design of HMM where for every state +(realization) in Y there is one and only transition from +X(A,B) explicitly satisfies assumption 1. +Therefore, we will further adhere to such design +principle and use ˜Xi to denote hidden states. Next, +we will elaborate on: a) what is the optimal number +of different observable states Yi for every ˜Xi? b) how +should we define optimal observable states? c) what +are the probabilities of transition (from the labelled +states to the observable states)? +4.3 +Requirements for the observable +states +Here we provide our analysis from the standpoints of +the system that obfuscates hidden states (e.g., the sys- +tem produces observable states) on behalf of Alice and +Bob, and hence ˜Xi is assumed to be known. The pos- +sibilities of transitions ˜Xi,R → Yi and ˜Xi,B → Yi im- +ply that a non-zero distortion E[Di] takes place: +E[Di] = ∑ +y(i) +j ∈Y(i) +Di,jPr +� +Yi = y(i) +j | ˜Xi +� +, +(16) +where +Di,j = Pr +� +ℓi = R | Yi = y(i) +j +� +d +� +˜Xi,R ,y(i) +j +� ++ +Pr +� +ℓi = B | Yi = y(i) +j +� +d +� +˜Xi,B,y(i) +j +� +. +(17) +Here Y(i) is the set of all observable states to which +transitions exist from the realizations of ˜Xi,R and ˜Xi,B +at time i; y(i) +j +is an element in Y(i); d (·,·) is some +distortion measure (e.g., SE). +The optimization effort is two-fold: i) how shall +we obtain observable states Y(i) in a way that Hr,i +is maximized under constraint ˜Di ≥ E[Di]? ii) how +shall we define ˜Di for every time step i such that +Hr is maximized and the total distortion constraint +˜D ≥ ∑i E[Di] is satisfied? We start with answering +question i), which will assist us in answering question +ii). +For the obfuscation, we utilize the following +principles: every element y(i) +j +in Y(i) can be fully +specified by the realizations of ˜Xi,R , ˜Xi,B, and pa- +rameter λ j. +Probabilities Pr +� +ℓi = R | Yi = y(i) +j +� +, +Pr +� +ℓi = B | Yi = y(i) +j +� +then affect Hr,i,j. All these pa- +rameters affect Di,j. The diagram explaining relations +between all the mentioned parameters is provided on +fig. 3. +In this example, labelled states are ˜Xi,R = +� +xA,xB� +, ˜Xi,B = +� +xB,xA� +; set Y(i) contains only two +elements y(i) +1 = +� +ˆy(i), ˇy(i)� +and y(i) +2 = +� +ˇy(i), ˆy(i)� +. For +example, to specify y(i) +1 +we only need λ1 in addi- +tion to the labelled states (∆ is the distance between +them). To obtain y(i) +2 we should apply a similar pro- +cedure where λ2 is known (in our particular example +λ2 = 1 − λ1). Probability Pr +� +ℓi = R | Yi = y(i) +1 +� +is +denoted as p1: its value affects attacker’s uncertainty +Hr,i,j as well as the distortion Di,j. +Figure 3: Scheme for obfuscation principle. +To maximize Hr,i under ˜Di ≥ E[Di] we consider +realizations of Yi and optimal adjustment of λ: such +adjustment then allows us to increase p1 and 1 − p2. + +We note that Yi shall belong to a line segment (in +a multidimensional space) connecting ˜Xi,R and ˜Xi,B. +This property is trivial (goes without proof) and can +be best understood if triangle △ ˜Xi,R Yi ˜Xi,B is consid- +ered. As a result: +∀Yi +� +Yi ∈ Y(i) =⇒ +� +∃λ ∈ [0,1] +� +∧ +�⃗Yi = +−−→ +˜Xi,R + λ +−−−−−→ +˜Xi,R ˜Xi,B +�� +. +(18) +We then establish the following: +Lemma 3 To minimize Di,j it is required that λ j = +1 − Pr +� +ℓi = R | Yi = y(i) +j +� +. +Proof: +From eq. (17) and eq. (18) we derive that +Di,j = p j∆2 +i λ2 +j + (1 − p j)∆2 +i (1 − λ j)2 , +where p j = Pr +� +ℓi = R | Yi = y(i) +j +� +≤ 0.5, and ∆2 +i = +d +� ˜Xi,R , ˜Xi,B +� += +��� +−−−−−→ +˜Xi,R ˜Xi,B +��� +2 +. +We next analyse +∂ +∂λj Di,j and find that λ j = 1 − p j is the extremum +(minimum) of Di,j. +□ +Corollary 2 Minimal distortion is Di,j = ∆2 +i p j(1 − +p j) ≤ ∆2 +i +4 , where p j = Pr +� +ℓi = R | Yi = y(i) +j +� +, ∆2 +i = +d +� ˜Xi,R , ˜Xi,B +� +. +Corollary 3 For every i, the highest lower bound +(maxmin entropy) is: +Hr,i = −νi log2 νi − (1 − νi)log2 (1 − νi) , +(19) +where νi = min +� +ϕ, ∆i−√ +∆2 +i −4E[Di] +2∆i +� +. +Proof: +It is required: 1) to determine Y(i) and prob- +ability distribution over it; 2) to determine Pr(ℓi | Yi) +for every element in Y(i). For this, we demonstrate +that maximum entropy under distortion constraint on +E[Di] is achieved for +���Y(i)��� ≤ 2: we analyze the case +for Y(i) = +� +y(i) +1 ,y(i) +2 +� +where +Pr +� +ℓi = R | Yi = y(i) +1 +� += Pr +� +ℓi = B | Yi = y(i) +2 +� +. +(20) +To prove the optimality of such settings, we con- +sider several alternative cases where E[Di] = ˜Di is +fixed. Let us first consider an alternative case where +���Y(i)��� = 2 but +Pr +� +ℓi = R | Yi = y(i) +1 +� +̸= Pr +� +ℓi = R | Yi = y(i) +2 +� +; +Pr +� +ℓi = R | Yi = y(i) +1 +� +̸= Pr +� +ℓi = B | Yi = y(i) +2 +� +. +(21) +For simplicity, we use the following notations: +Pr +� +Yi = y(i) +1 | ˜Xi +� += α, and Pr +� +Yi = y(i) +2 | ˜Xi +� += +1 − α; +Pr +� +ℓi = R | Yi = y(i) +1 +� += p1 ≤ 0.5, +and +Pr +� +ℓi = R | Yi = y(i) +2 +� += p2 ≥ 0.5. Taking into ac- +count the expression for conditional entropy, we then +require: +� +max +�Hr,i +� += max +� +αH1 + (1 − α)H2 +� +; +˜Di = αDi,1+ +(1 − α)Di,2 , +(22) +where +H1 += +H +� +ℓi | Yi = y(i) +1 +� +, +H2 += +H +� +ℓi | Yi = y(i) +2 +� +. Based on eq. (21) Di,1 ̸= Di,2. We +now show that H1 and H2 are functions of Di,1 and +Di,2, respectively. For this, we only point out that p1 +(similar results can be obtained for p2) is a mono- +tonically increasing function of Di,1: it follows from +corollary 2 that p1 = ∆i−√ +∆2 +i −4Di,1 +2∆i +. To demonstrate +the fallacy of attaining both eq. (21) and eq. (22) it is +sufficient to show the following (concavity): +αF(x)+ (1 − α)F +� ˜Di − αx +1 − α +� +≤ F( ˜Di) , +(23) +where x = Di,1, and F(x) = −p1(x)log +� +p1(x) +� +− +� +1− +p1(x) +� +log +� +1 − p1(x) +� +. The validity of eq. (23) fol- +lows from +∂ +∂xF(x) = +1 +∆iθ log +� +∆i+θ +∆i−θ +� +≥ 0 ; +∂2 +∂x2 F(x) = − +2 +∆iθ2 +� +1 +∆i+θ + +1 +∆i−θ − 1 +θ log +� +∆i+θ +∆i−θ +�� +≤ 0 , +where θ = +� +∆2 +i − 4x, and x ∈ +� +0, ∆2 +i +4 +� +. +Next, we point out a different case where +���Y(i)��� > +2 and demonstrate that it is non-optimal. For this +we consider +���Y(i)��� = 3 while the conclusions for +���Y(i)��� > 3 can be derived inductively then. Similarly +to eq. (22) we demand +� +max +�Hr,i +� += max +� +αH1+ +βH2 + (1 − α− β)H3 +� +; +˜Di = αDi,1 + βDi,2+ +(1 − α− β)Di,3 . +The +task +is +then +to +show +that +there +is +y(i) +4 +for +which +Di,4 = +αDi,1+βDi,2 +α+β +, +and +maxH4 ≥ max +� +α +α+βH1 + +β +α+βH2 +� +. +We +hence- +forth maintain that +���Y(i)��� ≤ 2 represents optimal +settings. +To obtain max +� +αH1 + (1 − α)H2 +� +in eq. (22) it is +sufficient that H1 = H2 and Di,1 = Di,2 = ˜Di. The lat- +ter requires that either λ1 = λ2 or λ1 = (1 − λ2): the +first condition implies p1 = p2 = 0.5 and leads to a + +trivial situation where y(i) +1 = y(i) +2 = 0.5 +� ˜Xi,R + ˜Xi,B +� +meaning that +���Y(i)��� = 1. The second condition implies +p1 = 1 − p2 and leads to y(i) +1 ̸= y(i) +2 if ˜Di < 0.25∆2 +i . +Requirement α ∈ [0,1] must be consistent with the +order mixing probability ϕ: +αp1 + (1 − α)p2 = ϕ , +(24) +from which we derive α = ϕ+p1−1 +2p1−1 demanding ϕ ≥ p1. +Alternatively, this demand can be understood based +on the fact H (ℓi) ≥ H (ℓi | Yi): setting p1 > ϕ results +in a greater distortion, but this does not increase en- +tropy. +□ +There are several important takeaways from the +proof of corollary 3. First, for every hidden state ˜Xi +there are two observable states that are obtained ac- +cording to eq. (18) where λ(i) +1 = 1 − νi is used to de- +fine realisation y(i) +1 , and λ(i) +2 = 1−λ(i) +1 is used for y(i) +2 . +Second, maximum allowed distortion should be used +at step i meaning that E[Di] = ˜Di. Third, probabili- +ties for transitions from labelled states to observable +states are +Pr +� +Yi = y(i) +1 | ℓi = R +� += +νi +ϕ +ϕ+νi−1 +2νi−1 ; +Pr +� +Yi = y(i) +2 | ℓi = R +� += +1 − Pr +� +Yi = y(i) +1 | ℓi = R +� +; +Pr +� +Yi = y(i) +1 | ℓi = B +� += +1−νi +1−ϕ +ϕ+νi−1 +2νi−1 ; +Pr +� +Yi = y(i) +2 | ℓi = B +� += +1 − Pr +� +Yi = y(i) +1 | ℓi = B +� +. +(25) +4.4 +Optimal obfuscation for N −1 time +steps +For every i we now define ˜Di such that Hr = ∑iHr,i is +maximized under the total distortion constraint ˜D ≥ +∑i ˜Di. +For this reason, we obtain optimal observ- +able states and corresponding transition probabilities +(from the labelled states) for all the time steps. From +the proof of the corollary 3 we use that +∂ +∂ ˜Di Hr,i ≥ 0 and +∂2 +∂ ˜D2 +i Hr,i ≤ 0. To maximize Hr we therefore require + + + + + +∀i +∂ +∂ ˜Di Hr,i = +1 +∆2 +i +√1−κi log +� +1+√1−κi +1−√1−κi +� += C ; +˜D = +N−1 +∑ +i=1 +˜Di = 1 +4 +N−1 +∑ +i=1 +κi∆2 +i , +(26) +where C is some constant, κi = 4 ˜Di +∆2 +i . +We then +solve the system eq. (26) for all κi, i ∈ [1,N − +1], +and according to corollary 3 obtain νi = +min +� +ϕ, 0.5 − √0.25 − 0.25κi +� +. +5 +OBFUSCATION ALGORITHM +Here we represent our aforementioned findings in the +form of obfuscation algorithm (see algorithm 1). It is +practical and can be implemented in real settings: its +complexity (excluding the complexity of solve pro- +cedure) is only O(N). For input, the algorithm ac- +cepts arrays (of size N) XA, XB, and scalars ˜D, ϕ. El- +ements of these arrays are scalar/vector realizations +for XA +i and XB +i characterizing geo-positions of Alice +and Bob, respectively, at time i. In practice, these ar- +rays may contain extrapolations based on historical +data and repetitive patterns. For example, Alice and +Bob may commute to work using the same routes and +roughly at the same time every day. Procedure solve +provides a solution to eq. (26): array κκκ contains ele- +ments κi needed to define realizations for obfuscated +state Yi. It is also needed to calculate the unlink- +ability criterion (entropy) Hr,i dependent on the ob- +fuscation process. Procedure send RSU encapsulates +data obfuscated at time i in accordance with one of the +V2X communication formats and sends it to the near- +est RSU. The output of the algorithm is, therefore, an +array Y containing all the obfuscated records and the +indicator of the total unlinkability in the system over +N − 1 steps, Hr. +Algorithm 1: Obfuscation algorithm +input : XA, XB, ˜D, ϕ ; +output: Y, Hr ; +begin +Hr ← 0, Y ← ∅, κκκ ← solve +� ˜D,XA,XB� +; +for i ← 1 to N −1 do +νi ← min +� +ϕ, 0.5−√0.25−0.25κi +� +, +α ← (ϕ+νi −1)/(2νi −1), +Hr,i ← −νi log(νi)−(1−νi)log(1−νi), +Hr ← Hr +Hr,i, P1,R ← νiα/ϕ, +P1,B ← (1−νi)α/(1−ϕ), +r1 ← UniRand +� +[0,1] +� +, r2 ← UniRand +� +[0,1] +� +, +ΛR ← 0.5+(0.5−νi) sign± +� +P1,R −r2 +� +, +ΛB ← 0.5+(0.5−νi) sign± +� +P1,B −r2 +� +; +if r1 ≤ ϕ then ˆy(i) ← X A +i +ΛR +� +X B +i −X A +i +� +, +ˇy(i) ← X B +i +ΛR +� +X A +i −X B +i +� +, +Yi = concat(ˆy(i), ˇy(i)); +else ˆy(i) ← X A +i +ΛB +� +X B +i −X A +i +� +, +ˇy(i) ← X B +i +ΛB +� +X A +i −X B +i +� +, +Yi = concat(ˇy(i), ˆy(i)); +send RSU(Yi), Y = concat(Y,Yi) ; +6 +DISCUSSION +Here we discuss how well the main aim (see sec- +tion 2.1) – “to develop a methodology providing a +high level of assurance that entropy for CAMs’ ori- +gins is high in C-ITS” – was achieved by our paper. +For this, we provide characteristics about the major +results, their advantages, limitations, and plans for + +further work. +6.1 +Results and their characteristics +In this paper, we combine: (i) the classical definition +of unlinkability and (ii) assumptions about a strong at- +tacker to (iii) measure and improve unlinkability in C- +ITS by developing the optimal joint obfuscation tech- +nique. The academic novelty is due to the combina- +tion of points (i-iii). Next, we discuss the importance +of each point in greater detail. +First, we lean towards the classical definition of +unlinkability demanded by the standards governing +the domain of C-ITS applications (ETSI TS 102 941 +V2.1.1, 2021; ISO 24102-1, 2018). The concept of +this study is, therefore, closer to some early works +on location privacy, such as (Shokri, 2012) relying +on Bayesian inference in HMM and contrasts with +many later works reliant on k-anonymity, differential +privacy, and geo-indistinguishability (Andr´es et al., +2013; Bordenabe et al., 2014; Corser et al., 2016). +In this work, the definition of unlinkability is cap- +tured through Bayesian inference and is further re- +flected by entropy. We also stress on advantages of +such an approach. +Based on definition 2, the at- +tacker’s reasoning about the operations plays the cen- +tral role. Such reasoning may go beyond properties +observable within the object (e.g., cryptographically +signed CAM message): it may additionally rely on +meta-information collected, for example, on a sys- +tem level (e.g., the order of CAMs arrivals at RSU). +This feature accords with many concepts in science +and philosophy. Among others, Leibniz stated that +indiscernible objects have identities (Hacking, 1975): +preferences about these identities can be expressed +statistically (and, hence, used in Bayesian inference). +For instance, in statistics such reasoning may be as- +sisted by means of additional indexing. In contrast, +geo-indistinguishability does not support further rea- +soning about indiscernible pieces of geo-data, mak- +ing this privacy concept less demanding (and, hence, +inferior) compared to unlinkability. To witness the +differences between these concepts, one should ob- +serve order mixing (and corresponding probability ϕ) +in our HMM (see fig. 2): even if XA +i = XB +i , the model +sets (XA +i ,XB +i ) apart from (XB +i ,XA +i ). For that reason, +we agree with the authors of (Montazeri et al., 2017; +Takbiri et al., 2017, 2020), stating that both obfusca- +tion and permutation (order mixing) are required for +strong privacy assurance. +Second, our focus on unlinkability (and not on the +location protection) provides consistency even under +the assumption that an adversary is strong: the at- +tacker knows the actual locations of Alice and Bob +at every moment i. He also knows the probabilistic +obfuscation and order mixing algorithm used by the +players. However, he does not know the outputs of +this probabilistic algorithm. His goal is then to infer +the origin of the digitally signed obfuscated messages. +As a result of applying assumption 1, reasoning about +statistical inference made by the attacker is very much +simplified compared to (Shokri, 2012): information +about HMM’s hidden states (e.g., actual locations, ve- +locities, etc.) and transition probabilities are not re- +quired for such reasoning. The latter detail is benefi- +cial for the privacy assurance since establishing prob- +abilities for transitions in HMM is a laborious and of- +ten imprecise procedure relying on Kalman-like esti- +mators (Blackman, 1986; Lehmann and Pieczynski, +2020). +Third, strong (and simplifying) assumptions as- +sist us in specifying the lower bound of unlinkabil- +ity in C-ITS. For every time step i unlinkability is +expressed through entropy Hr,i: the worst-case in- +ference is made by an attacker meaning that Hr,i is +the lower bound (see assumption 1). +Components +Hr,i are then summed over N − 1 steps to obtain Hr +(see lemma 1). Such summation is a simple and in- +tuitive step. It is, nevertheless, justified because for +any i ∈ {1,2,...,N − 1}, inference about the source +(origin) of arrived CAM is independent from such +inference at i − 1. +An analogy can be established +between entropy (unlinkability) values Hr,i and Hr, +and the concepts of microscopic and macroscopic pri- +vacy, respectively, in (Shokri et al., 2010). Better pro- +tection of macroscopic privacy (e.g., trajectories) re- +quires higher uncertainty about labels ℓi for the lo- +cations reported on the microscopic level (e.g., geo- +graphic points). Higher uncertainty about ℓi can only +be provided at the cost of higher expected distortion +E[Di] of the players’ CAMs. +To maximize uncer- +tainty Hr,i about labels under the constraint on distor- +tions we propose a new simple way to define a joint +distribution that must be followed during the obfus- +cation (to obtain CAMs) conducted cooperatively by +Alice and Bob. Optimality of (multivariate) noise pa- +rameters under various constraints on distortions has +been studied by many authors in the past (Andr´es et +al., 2013; Geng and Viswanath, 2016; Takbiri et al., +2017). Our approach, however, differs: to improve +microscopic privacy at i we insist on joint probabilis- +tic obfuscation of the samples from different play- +ers (see proof of corollary 3). +Because of the lat- +ter feature, our obfuscation approach is clearly data- +dependent (Croft et al., 2019; Guan et al., 2022). We +then analyse how to optimally distribute distortions +over N − 1 time steps if the corresponding distortion +cap is specified for the whole duration of C-ITS ob- + +servation (see eq. (26)). All the findings of this paper +are incorporated in algorithm 1. Procedure solve is +one of the major factors contributing to the time com- +plexity of the algorithm. This, nevertheless, can be +addressed if the obfuscation optimality is slightly sac- +rificed. For example, solve can be pre-computed for +several cases only: each case would produce a dis- +tinct kind of distribution for a random variable ∆2 +i˜D . +Then, the actual input data should be approximated by +the best-matching distribution, and the corresponding +pre-computed outputs of solve should be used for the +obfuscation. Such workaround can also turn our al- +gorithm into a ‘real-time algorithm’: if Alice and Bob +believe that their future data will align well with one +of the pre-computed distributions (e.g., because of ha- +bitual daily commutes) they can obfuscate it ‘on-the- +fly’. Hence, the pre-computed cases for solve can be +treated as profiles that pairs of players agree to use. +6.2 +Limitations and future work +This paper has certain limitations which we plan to +address in our further studies. +Only one particular composition of entropy (to +measure unlinkability) and squared error (to measure +distortion) is considered in our work. +Besides en- +tropy, other uncertainty measures may be useful for +expressing unlinkability in C-ITS (Wagner and Eck- +hoff, 2018). Also, distortion measures other than SE +have been analysed and recommended in the past by +some authors studying Multiple-Target Tracking and +its applications (Gorji et al., 2011). +Only high assurance about minimally achievable +unlinkability (e.g., rational lower bound, see assump- +tions 1 and 2) is studied here. However, to further +improve the practicality of our methodology, indica- +tors obtained for the worst-case scenario (e.g., a very +strong attacker) may be complemented by other in- +dicators relying on less pessimistic scenarios (e.g., a +weaker attacker). +Only 2 players are considered in our model. This +substantially limits the number of permutations for +CAMs: as a result, for every hidden state there are +only 2 labelled states (see fig. 2). Because of that, the +methodology defining observable states is also simple +(see fig. 3). We plan to increase the number of players +in the future. However, for larger numbers of players, +defining optimal observable states and corresponding +probabilities for transitions is a non-trivial task. This +is because more complex structures and transforma- +tions in Rz, z ≥ 1, need to be analyzed to optimize +instant and joint obfuscation of CAMs. +7 +ACKNOWLEDGMENT +This research is co-financed by public funding of the +state of Saxony, Germany. +References +Andr´es, M. E., Bordenabe, N. E., Chatzikoko- +lakis, K., & Palamidessi, C. (2013). Geo- +indistinguishability: Differential privacy for +location-based systems. Proceedings of the +2013 ACM SIGSAC Conference on Com- +puter & Communications Security, 901–914. +Blackman, S. S. (1986, January 1). Multiple-target +tracking with radar applications. +Blackman, S. (2004). Multiple hypothesis tracking +for multiple target tracking. 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Limits of location +privacy under anonymization and obfusca- +tion. 2017 IEEE International Symposium +on Information Theory (ISIT), 764–768. +Takbiri, N., Houmansadr, A., Goeckel, D. L., & +Pishro-Nik, H. (2020). Privacy of Dependent +Users Against Statistical Matching. IEEE +Transactions on Information Theory, 66(9), +5842–5865-5842–5865. +Wagner, I., & Eckhoff, D. (2018). Technical privacy +metrics: A systematic survey. ACM Comput- +ing Surveys (CSUR), 51(3), 1–38-1–38. +Wiedersheim, B., Ma, Z., Kargl, F., & Papadimi- +tratos, P. (2010). Privacy in inter-vehicular +networks: Why simple pseudonym change +is not enough. 2010 Seventh International +Conference on Wireless On-demand Net- +work Systems and Services (WONS), 176– +183. + + +u1 +(1 - X1)△ +V2 +() +() +y +2 +B +r +j= +V2 +3=2 +(1 - >1)△ +()) +V2 +[] +B +p1 +V2ScileP +TeSS +ScienceandTechnologyPublicationsThis figure "orcid.png" is available in "png"� format from: +http://arxiv.org/ps/2301.04130v1 + diff --git a/nNE2T4oBgHgl3EQfzQjl/content/tmp_files/load_file.txt b/nNE2T4oBgHgl3EQfzQjl/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ce75604c97fdb92f02ae4d027bafa23545b97111 --- /dev/null +++ b/nNE2T4oBgHgl3EQfzQjl/content/tmp_files/load_file.txt @@ -0,0 +1,846 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf,len=845 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='04130v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='CR] 10 Jan 2023 Improving unlinkability in C-ITS: a methodology for optimal obfuscation Yevhen Zolotavkin1 a, Yurii Baryshev2 b, Vitalii Lukichov2 c, Jannik M¨ahn1 d and Stefan K¨opsell1 e 1Barkhausen Institut gGmbH, W¨urzburger Straße 46, Dresden, Germany 2Department of Information Protection, Vinnytsia National Technical University, Khmelnytske Shosse 95, Vinnytsia, Ukraine {yevhen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='zolotavkin, jannik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='maehn, stefan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='koepsell}@barkhauseninstitut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='org, {yuriy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='baryshev, lukichov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='vitalyi}@vntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='ua Keywords: privacy, V2X, unlinkability, hidden Markov model, cybersecurity, entropy, obfuscation Abstract: In this paper, we develop a new methodology to provide high assurance about privacy in Cooperative Intelli- gent Transport Systems (C-ITS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Our focus lies on vehicle-to-everything (V2X) communications enabled by Cooperative Awareness Basic Service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Our research motivation is developed based on the analysis of unlink- ability provision methods indicating a gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' To address this, we propose a Hidden Markov Model (HMM) to express unlinkability for the situation where two cars are communicating with a Roadside Unit (RSU) us- ing Cooperative Awareness Messages (CAMs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Our HMM has labeled states specifying distinct origins of the CAMs observable by a passive attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' We then demonstrate that a high assurance about the degree of uncertainty (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=', entropy) about labeled states can be obtained for the attacker under the assumption that he knows actual positions of the vehicles (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=', hidden states in HMM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' We further demonstrate how unlinkability can be increased in C-ITS: we propose a joint probability distribution that both drivers must use to obfuscate their actual data jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' This obfuscated data is then encapsulated in their CAMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Finally, our findings are incorporated into an obfuscation algorithm whose complexity is linear in the number of discrete time steps in HMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' 1 INTRODUCTION Due to the intense development of transport systems over the recent decades, different modes of cooper- ative intelligence have been incorporated into their functionalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Intelligent transport systems (ITS) are transport systems in which advanced information, communication, sensor and control technologies, in- cluding the Internet, are applied to increase safety, sustainability, efficiency, and comfort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Cooperative Intelligent Transport Systems (C-ITS) are a group of ITS technologies where service provision is enabled by, or enhanced by, the usage of ‘live’, present situa- tion related, dynamic data/information from other en- tities of similar functionality, and/or between different elements of the transport network, including vehicles and infrastructure (ISO/TR 17427-7:2015(E), 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Technology allowing a vehicle to exchange ad- a https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='org/0000-0002-1875-122X b https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='org/0000-0001-8324-8869 c https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='org/0000-0002-3423-5436 d https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='org/0000-0003-0870-7193 e https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='org/0000-0002-0466-562X ditional information with infrastructure, other vehi- cles and other stakeholders in the context of C-ITS is called vehicle-to-everything (V2X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Multiple ad- vances in modern C-ITS applications, such as col- laborative forward collision warning and emergency electronic brake lights, are impossible without V2X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' These advances, however, come at a cost: C-ITS ap- plications rely on vehicles broadcasting signals to in- dicate their location, signals which are intended to be received and processed by a range of other devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' For example, vehicles may cooperatively broadcast (with the frequency of 1-10 Hz) geo-spatial informa- tion to nearby peers using short Cooperative Aware- ness Messages (CAMs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Hence, V2X raises essen- tial privacy questions: i) to what degree can specific vehicles be located and tracked based on such infor- mation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' ii) what are the techniques able to improve privacy of V2X?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' To answer these questions, we use the concept of unlinkability to reason about privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='1 Research motivation Even though the problems of privacy in C-ITS were acknowledged in several relevant documents (having normative and informative character), satisfactory an- swers have yet to be provided to the privacy questions mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' For example, the document (ISO 24102-1, 2018) recognizes the importance of unlink- ing private data from traceable address elements and identifiers in wireless messages sent by an ITS station unit (ITS-SU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' However, relevant considerations in this document do not go beyond suggesting that “such unlinking can be done by means of pseudonyms”: the sufficiency of these and many similar sugges- tions remain unaddressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' In contrast, limitations of pseudonym changes in CAMs have been recog- nized by academic authors (Escher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Karim Emara, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Wiedersheim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' In particu- lar, vehicle tracking becomes possible due to CAM content being signed but not encrypted: this is de- manded by the relevant standards in C-ITS(DS/ETSI EN 302 637-2 V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='1, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' This is because full en- cryption of CAMs may impede C-ITS functionalities that are critical for safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Nonetheless, recognition of privacy-affecting issues in C-ITS has yet to result in a solution where the degree of privacy is correctly mea- sured, and privacy limitations are eliminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' In this paper, a methodology to address these challenges is developed: we provide an assurance for the procedure estimating the lower bound of unlinkability for CAMs and an algorithm maximizing this criterion under the overall constraint of location precision degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='2 Research principles To address V2X privacy questions, it is imperative to obtain results for which confidence is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' The methodology developed in this paper rests on the prin- ciples of robust optimization (RO) (Gorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Sniedovich, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' The unsuccessfulness of the previous attempts to answer the mentioned above ‘V2X privacy questions’ is mainly caused by inoppor- tune privacy assumptions for C-ITS and V2X scenar- ios in particular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' For example, a combination of as- sumptions that an attacker only observe CAMs con- taining obfuscated (distorted) geo-position measure- ments and does not have a precise physical model for car movements may require reasoning involving the best-known technique to estimate such a model (Blackman, 1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Because of the computational complexities associated with these estimations, re- searchers often rely on heuristic and poorly justifiable steps: this is unacceptable if high level of privacy as- surance is demanded (Blackman, 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Wiedersheim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' To avoid this situation, we make stronger assumptions about the information known to the at- tacker, which allows us to estimate a lower bound of unlinkability in C-ITS: this estimate has high confi- dence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' This attitude is reflected in the following prin- ciples shaping our methodology: The methodology for privacy risk assessment should not underestimate the risks: it should pro- vide clear and quantifiable assessments of how easy data belonging to the same entity can be ex- tracted throughout multiple V2X communication sessions involving more than one user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Therefore, the methodology should set justifiable bounds for assessments of such quantity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' The methodology should propose optimal mod- ifications of the original user data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' The notion of ‘optimality’ depends on the kind of modifica- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Reversible modifications (such as encryp- tion) should be provided with the assurances of computational infeasibility of such reversions by illegitimate parties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=', the level of such assur- ance is maximal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' For irreversible modifications (such as noising), the loss of data quality should be quantified: for each such quantity, there should be an assurance that no higher degree of unlinka- bility can be achieved (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=', the proposed noising method is optimal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='3 Our contribution The unique contribution is due to combination of the study objective (guided by the criterion of unlinkabil- ity), robust assumptions, and the optimal obfuscation technique developed in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' First, the aim of this study is to protect C-ITS from the threat of linking: this is in contrast with the numerous obfuscation approaches which aim at impeding inferencing about the actual location of ITS-SU (Andr´es et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Bordenabe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=', 2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Second, to obtain high confidence in the measured unlinkability, we assume that an attacker has com- plete knowledge about the system design, obfus- cation algorithms, quality degradation (distortion) constraints, has access to CAMs, stored states ve- hicles’ geo-positions, and the true states charac- terizing the geo-positions of the vehicles at any moment in time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Third, we develop an optimal obfuscation algo- rithm: for a given distortion constraint, it pro- vides the highest level of uncertainty for the at- tacker trying to link obfuscated CAMs with their sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' The rest of this paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' In section 2, we set the grounds for the study and provide basic definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' It is followed by section 3, where we start with a description of a generic information system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Further specifics of C-ITS are then reflected using additional sets and relations: as a result, we ob- tain Hidden Markov Model (HMM), used to study unlinkability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' In section 4, we formalize assump- tions, define unlinkability through entropy and opti- mize joint obfuscation producing observable states in HMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Next, section 5 describes a compact and effi- cient algorithm calculating the unlinkability indicator in C-ITS and implementing previous findings to im- prove unlinkability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Finally, we discuss our results, their importance, novelty, advantages, and limitations in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' 2 PRELIMINARIES To justify subsequent modelling steps better, we intro- duce contextual information supporting our aim, set- tings and privacy assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='1 Aim of the study To specify the aim, we analyze relevant cybersecurity requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Here, we use some of the classical def- initions for privacy in complex systems to specify our aim with greater precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Privacy requirements for C-ITS are often derived from ISO/IEC 15408-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' For example, (ETSI TS 102 941 V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='1, 2021) suggests that the combination of pseudonymity and unlinka- bility offers the appropriate sender privacy protection for basic ITS safety messages (such as CAM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' In sim- ple terms, pseudonymity requires that the identity of a user is never revealed or inferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' However, one of the major complications in dealing with pseudonymity is the following: an attacker may learn the user’s iden- tity composition based on multiple sessions, events, or traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Unlinkability is the assurance about the ability to resist learning such a composition (ISO/IEC 15408-2:2022(E), 2022): Definition 1 (Unlinkability of operations) Requires that users and/or subjects are unable to determine whether the same user caused cer- tain specific operations in the system, or whether operations are related in some other manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' In the context of V2X communication in C-ITS with many users, the cryptographically signed mes- sages broadcasted by the ITS-SUs (controlled by these users) should have the property of definition 1 (Hicks and Garcia, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' ETSI TS 102 941 V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='1, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Nonetheless, such interpretation has certain disadvantages, major of which is inflexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Indeed, ‘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='unable to determine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='..’ statement can either be false or true, meaning that the unlinkability of the whole C-ITS (with many cars and observable during many hours) is expressed using a binary value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' This issue has been recognized by practitioners and researchers alike, which is reflected in comments and best prac- tice recommendationsto ITS engineers and managers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' For example, (ISO/SAE 21434, 2021) contains the ta- ble ‘Example privacy impact rating criteria’: it in- cludes Impact rating Criteria interpreting the mean- ing for the severity degrees (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=', Negligible, Moder- ate, Major, Severe) for privacy impact rating indica- tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Importantly enough, interpretations in this table evolve around two aspects: a) the level of sensitivity of the information about road user;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' b) how easily it can be linked to a PII (Personally Identifiable Infor- mation) principal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Such emphasis on the easiness of linking motivates us to modify definition 1 in the fol- lowing manner: Definition 2 (Unlinkability of operations*) Is the degree of inability to determine (by users and/or sub- jects) whether the same user caused certain specific operations in the system, or whether operations are related in some other manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' To provide convenience in comparing unlinkabil- ity in C-ITS under different conditions, we use Shan- non entropy: it is an integral criterion of uncertainty in a system that fully captures the ‘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='degree of inabil- ity to determine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='..’ (Wagner and Eckhoff, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Henceforth, the main aim of our paper is to de- velop a methodology providing high level of assur- ance that entropy for CAMs’ origins is high in C-ITS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='2 Settings for the study In fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' 1, we introduce a general setup for our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' In fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' 1(a), two cars (ITS-SUs) are driven by Alice and Bob, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Both ITS-SUs transmit CAMs with the same frequency, and the roadside unit (RSU) receives them without losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' The role of the attacker is played by the RSU, who tries to separate CAMs of Alice from CAMs of Bob: this allows the attacker to link CAMs belonging to the same entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Although CAMs are signed, in our study we assume that a sig- nature scheme providing unlinkability is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' There- fore, the separation is done based on the content of the CAMs (since the are transmitted unencrpyted) and the order of arrival of CAMs within each time in- terval – see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' We consider the ordering of CAMs’ arrivals to be non-uniform in general: for example, in one extreme case, the CAM from Alice arrives first, and Bob’s CAM arrives second at any time interval i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' If an attacker knows about such a unique property, he can link CAMs without consider- ing their payload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' However, these cases are unlikely, meaning that an attacker should also be able to in- fer the source (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=', ‘from Alice’ or ‘from Bob’) of a CAM based on its content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' The requirements for the content of CAMs can be found in (DS/ETSI EN 302 637-2 V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='1, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' In particular, we consider that geo-position, velocity and acceleration are essen- tial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' On the one hand, these parameters are mandatory for the HighFrequency container in CAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' On the other hand, numerous techniques using these parame- ters have been developed for the domain of Multiple- Target Tracking (MTT): corresponding estimators can be of great use for reasoning about privacy in C-ITS (Blackman, 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Karim Emara, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Figure 1: Setting for our study of unlinkability of CAMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' In this study, CAMs’ content unlinkability is the core of our attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' We exclude from further con- sideration the following CAM payload: 1) cryp- tographically produced proofs of authenticity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=', signatures);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' 2) categorical data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=', vehicle role).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' These exclusions are due to substantial attention to issue ‘1)’ among the members of the crypto- graphic community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' For example, pseudonym un- linking solutions were proposed in (Camenisch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Hicks and Garcia, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Neverthe- less, there is a need to complement these efforts by our study: the absence of encryption (due to safety reasons) in CAMs makes pseudonym unlink- ing necessary but not sufficient for CAMs’ unlink- ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' This is because other data, such as geo- graphic positions, in CAMs may be used for link- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Issue ‘2)’ can be omitted since categorical data is a part of basicVehicleContainerLowFrequency and is OPTIONAL in CAMs (DS/ETSI EN 302 637-2 V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='1, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' We also exclude VehicleLength and VehicleWidth (compulsory for the HighFrequency container), which otherwise are likely to be of great use in discriminating different vehicles (Escher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' For such an exclusion we find justification in (ETSI TS 102 894-2 V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='1, 2018) which allows usage of the codes 1023 and 62 for the length and width, respectively, if the corresponding information is un- available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Because of the details described above, we will model CAM as a vector in Rz where z ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Such a step is beneficial: we can apply commonly used dis- tortion measures such as, for example, Squared Error (SE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' This is a clear and straightforward way to re- fer to the quality degradation of essential location ser- vices (Shokri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' We, nevertheless, refrain from further discussions about the chosen distortion measure in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='3 Privacy assumptions and threats Here we provide a high-level intuition for the system and the threat of linkability, while the details will be introduced in the subsequent sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Alice and Bob coordinate their efforts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' They distribute the total al- lowed distortion among N − 1 time steps: as a result, they know the distortion limit for every time step i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' At the beginning of every time interval i, Alice and Bob know the true measurements (including position, speed, acceleration, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=') of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' To obfus- cate data in their CAMs they randomly agree on the order of their arrival at RSU at every i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' For every i they define a joint distribution according to which they change (obfuscate) their actual measurements: in expectation, they remain within the distortion limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' An attacker who fully controls RSU statistically infers the source of every pair of CAMs which he ob- serves during time i: this statistical inference is used to calculate entropy and aligns with definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' For this, the attacker refers to the joint distribution used by Alice and Bob during the obfuscation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' He also knows other information, such as the original geo-positions of the players at every i, and the probabilities for the order of CAMs’ arrivals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' The resulting unlinkability in the system depends on: i) statistics for the order of arrival of CAMs from the players;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' ii) the level of allowed distortion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' iii) how far apart actual measure- ments of Alice and Bob are at every i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' 3 MATHEMATICAL MODEL We explain our mathematical model in the follow- ing sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' To easy the reading, table 1 contains an overview about our notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='RSU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='Alice ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='Bob ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='CAM(B) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='CAM(A) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='CAM(B) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='CAM(A) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='N-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='N-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='b)Table 1: Notations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='Notation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='Description ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='ITS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='Intelligent Transport Systems ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='C-ITS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='Cooperative Intelligent Transport Systems ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='ITS-SU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='ITS Station Unit (including installed in vehicles) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='V2X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='Vehicle-to-Everything ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='CAM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='Cooperative Awareness Message ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='RSU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='Roadside Unit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='HMM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='Hidden Markov Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='Du ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='Set of user-related data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='Ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='Set of information system-related data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='U ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='Set of information system’s users ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='P ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='Set of data processing procedures at the information system ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='P ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='Set of players including Alice and Bob ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='xA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='k ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' 1 ≤ k ≤ µ A hidden state for Alice XA = {xA k } Set of hidden states for Alice xB j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' 1 ≤ j ≤ ω A hidden state for Bob XB = {xB j } Set of hidden states for Bob X(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='B) Set of joint hidden states for � Alice,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Bob � X(B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='A) Set of joint hidden states for � Bob,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Alice � R Index (label) for rose nodes XR = X(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='B) Set of all rose nodes B Index (label) for blue nodes XB = X(B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='A) Set of all blue nodes L = {R ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='B} Set of labels encoding |P|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' combinations Y Set of joint observable states for Alice and Bob i ∈ {1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=',N − 1} Time-step in discrete HMM XA i Variable on XA at i XB i Variable on XB at i Xi Variable for joint hidden state on step i ℓi Variable on L at i Yi Variable on Y on step i Pr(Xi+1 | Xi) Probability of transition between hidden states Pr(Yi | Xi) Conditional probability for observable states ϕ Order mixing (label permuting) probability ρi Distribution over hidden states on step i ρi+1|ρi Conditional distribution over hidden states on step i+ 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='1 Model setup We consider the generic case of information systems with the passive attacker (Eve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' An information sys- tem processes data set Du for certain users, who form set U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' For this, information system executes a set of data processing algorithms P: data Ds, which de- scribes configuration and parameters of these pro- cessing algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Any data processing algorithm ∀p ∈ P is triggered by either a user or information system’s state described by its configuration and pa- rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' At the data processing algorithm p, spe- cific user data is taken for input that results in cre- ation of new or altering of existing user’s data as well as change of information system’s configuration and parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' In certain cases data processing al- gorithms within system can alter even user set U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Therefore, p is a mapping p : U× D → U× D, where D = Du ∪ Ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Thus, information system is described as a tuple {U,D,P,Eve}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' For uι ∈ U user’s identification information sys- tem performs special data processing algorithms identifys(·) ∈ P, so that identifys(duι) = uι ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' To protect it’s data the information system may apply obfuscation algorithm ob fuscate(·) ∈ P to achieve such data alteration d∗ uι = ob fuscate(duι) that identifys(d∗ uι) ̸= uι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Let’s consider Eve is watching over the data pro- cessing flow with a certain ability level that allows her to get access to the data of the information sys- tem – Eve sees d∗ u,d∗ s , where d∗ u ⊆ Du and d∗ s ⊆ Ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' For the unlinkability property of user’s data within the considered system it is necessary that ∀d∗ uι ⊆ Du so that Eve sees d∗ uι, it is infeasible for Eve to find such a transformation identifye(duι) that identifye(d∗ uι) = uι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' The following interpretations are possible in the context of unlinkability for C-ITS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' du = {registrationNumber,manu facturer,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=',PATHS}, where ∀pathι ∈ PATHS, pathι = {⟨xj,yj⟩}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Some ways of user linking are: search(registrationNumber) = uι: is usually performed by law enforcement agencies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' recon(manu facturer,color) = uι: can be per- formed by private detectives, for instance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' drivingModelling(pathsι) = uι: can be per- formed by gathering of data from roadside units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' The research focuses on the latter way of user link- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Therefore we considering the states of users ve- hicles at different moments in time (communicated in CAMs) and assessing unlinkability in C-ITS through Eve’s ability to infer information using, for instance, drivingModelling(·) as identifye(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='2 Markov model for unlinkability To study unlinkability in V2X we use Hidden Markov Model which graphical representation is given on fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' The following sets are needed to de- scribe the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' The set of all players is P = {Alice, Bob,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' For each player, there exists a set of hidden states for his vehicle, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=', for Alice there is XA = � xA 1,xA 2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=',xA k ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=',xA µ � and for Bob there is XB = � xB 1,xB 2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=',xB j ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=',xB ω � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Each state, for example, xA 1 can be a vector including specific position, veloc- ity, acceleration and other characteristics applicable to Alice’s vehicle at certain time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Throughout the paper we assume that XA ∩XB is in general non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' The system of |P| players is characterized by hid- den and observable joint states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Transition happens between hidden states Xi and Xi+1 when time step i proceeds to i + 1, where joint state Xi = � XA i ,XB i � is the composition (concatenation) of variables XA i ∈ XA and XB i ∈ XB .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' As such, ∀k, j(xA k ,xB j ) ∈ X(A,B), where |X(A,B)| = |XA| × |XB| (for simplicity of representa- tion we further assume |P| = 2, |XA| = µ = 2, |XB| = ω = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Possible transitions from Xi to Xi+1 are denoted using indices 1 − 16 (see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' 2): these transitions are governed by corresponding probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' For ex- ample, the transition from Xi = � XA i = xA 1,XB i = xB 1 � to Xi+1 = � XA i+1 = xA 2,XB i+1 = xB 2 � is denoted by index 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' The probability of such a transition is Pr � XA i+1 = xA 2,XB i+1 = xB 2 | XA i = xA 1,XB i = xB 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' In practice, these probabilities can be obtained based on the well-studied physical models for vehicles (Black- man, 1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' For each Xi of the hidden joint states there are |P|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' possible permutations for its concatenated com- ponents originating from the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' These permu- tations are the major cause of uncertainty when an attacker attempts to label combined CAMs of Alice and Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' In practice, this is caused by the unpre- dictable arrangement of CAMs within each scan (or session) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Hence, a permutation should be selected by randomly following one of the possible transi- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' For example, while the system is in a joint state � XA i = xA 1,XB i = xB 1 � permutation � xA 1,xB 1 � (rose colored node) should be considered if transition with index 17 takes place, and � xB 1,xA 1 � (blue coloured node) should be considered if transition 18 happens (see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' We will use notations Xi,R and Xi,B for rose and blue nodes, respectively, where Xi,R ∈ XR , Xi,B ∈ XB, and XR = X(A,B), XB = X(B,A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Fur- ther in the text, we will refer to the states repre- sented by the coloured nodes as ‘labelled states’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' For the sake of simplicity and without loss of generality, for all realizations of hidden states Xi, we consider Pr � Xi,R |Xi � = ϕ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='5, and Pr � Xi,B|Xi � = 1 − ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' 1 2 3 4 17 18 6 5 7 8 10 11 9 12 14 15 16 13 19 20 21 23 22 24 25 27 26 28 � XA = xA 1 , XB = xB 1 � � xA 1 , xB 1 � (ˆy1, ˇy1) � XA = xA 1 , XB = xB 2 � � xB 1 , xA 1 � (ˇy1, ˆy1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' � XA = xA 2 , XB = xB 1 � � xA 2 , xB 2 � (ˆy2, ˇy2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' � XA = xA 2 , XB = xB 2 � � xB 2 , xA 2 � (ˇy2, ˆy2) Figure 2: Hidden Markov Model for 2 players sending CAMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' To denote the totality of hidden permuted joint states we use set X{R ,B} = XR ∪ XB, where |XR | ≤ |X{R ,B}| ≤ 2|XR |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' For every Xi,R and Xi,B there are transitions to observable joint states Yi ∈ Y, Y = � (ˆy1, ˇy1),(ˇy1, ˆy1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=',(ˆyq, ˇyq),(ˇyq, ˆyq) ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=', (ˆyξ, ˇyξ),(ˇyξ, ˆyξ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Some of these transitions to observ- able states are denoted with indices 21 − 28 on fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Until proven otherwise, the cardinality of Y is consid- ered independent on |X{R ,B}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Measuring uncertainty about label ℓ ∈ L, L = {R ,B}, is of our main interest: this is done based on observable states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' 4 MODEL PROPERTIES To formally express unlinkability following definition 2 we will use conditional entropy H (ℓ1,ℓ2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='|Y1,Y2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=') for the sequence of la- bels ℓ1,ℓ2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=',ℓN−1 given that an attacker observes Y1,Y2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=',YN−1 (Wagner and Eckhoff, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='1 General expression for unlinkability For the described HMM, probability of any hidden state at any time step can be specified using multivari- ate discrete distribution ρρρ : X(A,B)×{0,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=',N−1} → [0,1]N|X(A,B)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' We will further use ρi slices of ρρρ such that ρρρ = N−1 � i=0 ρi, where each slice represents a distri- bution over hidden states at step i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Slice ρ0 defines distribution over the hidden states before the start of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Because HMM has been previously de- fined (see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' 2) using transitional probabilities that remain unchanged for all time steps, each slice can be fully determined in a conditioned sequential man- ner: ρi+1|ρi means that ρi+1 is trivially derived if ρi is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Since an attacker observes Y1,Y2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=',YN−1 and knows ρρρ analysis of H (ℓ1,ℓ2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' | Y1,Y2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=',ρρρ) is cen- tral to our reasoning about unlinkability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' We state the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Lemma 1 Unlinkability in V2X system (as per fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' 2) is expressed as: H � ℓ1,ℓ2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' ��Y1,Y2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=',ρρρ � = N−2 ∑ i=0 H � ℓi+1 ��Yi+1,{ρi+1|ρi} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' (1) Proof: For simplicity, we consider N = 3 only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' First, it should be noted that H � ℓ1,ℓ2 ��Y1,Y2,ρρρ � = H � ℓ1,ℓ2,Y1,Y2 ��ρρρ � − H � Y1,Y2 ��ρρρ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' (2) We then ponder at the right-hand side of the eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Each of these terms can be expressed as: H � ℓ1,ℓ2,Y1,Y2 ��ρρρ � = H � ℓ2,Y2 ��ℓ1,Y1,ρρρ � + H � ℓ1,Y1 ��ρρρ � , (3) and H � Y1,Y2 ��ρρρ � = H � Y2 ��Y1,ρρρ � + H � Y1 ��ρρρ � , (4) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' We point out that H � ℓ2,Y2 ��ℓ1,Y1,ρρρ � = H � ℓ2,Y2 ��ρρρ � and H � Y2 ��Y1,ρρρ � = H � Y2 ��ρρρ � in eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' (2) and (3), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' This follows from the fact that realizations of ℓi,Yi do not affect ℓi+1,Yi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' We finally stress that ρρρ is redundant for determining ℓi+1,Yi+1 since only ρi+1|ρi has rele- vance: H � Y1 ��ρρρ � = H � Y1 ��{ρ1|ρ0} � , H � ℓ1,Y1 ��ρρρ � = H � ℓ1,Y1 ��{ρ1|ρ0} � , H � Y2 ��ρρρ � = H � Y2 ��{ρ2|ρ1} � , H � ℓ2,Y2 ��ρρρ � = H � ℓ2,Y2 ��{ρ2|ρ1} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Hence, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' (2) can be rewritten as H � ℓ1,ℓ2 ��Y1,Y2,ρρρ � = H � ℓ1,Y1 ��{ρ1|ρ0} � + H � ℓ2,Y2 ��{ρ2|ρ1} � − � H � Y1 ��{ρ1|ρ0} � + H � Y2 ��{ρ2|ρ1} �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' (5) The latter eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' (5) can be regrouped H � ℓ1,ℓ2 ��Y1,Y2,ρρρ � = � H � ℓ1,Y1 ��{ρ1|ρ0} � − H � Y1 ��{ρ1|ρ0} �� + � H � ℓ2,Y2 ��{ρ2|ρ1} � − H � Y2 ��{ρ2|ρ1} �� = H � ℓ1 ��Y1,{ρ1|ρ0} � + H � ℓ2 ��Y2,{ρ2|ρ1} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' (6) □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='2 Worst-case unlinkability We aim to obtain a computationally feasible estima- tion of unlinkability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Direct utilization of the re- sults of lemma 1 presupposes computing {ρi+1|ρi} which has several disadvantages: a) transition prob- abilities for hidden states need to be specified (which usually requires studying physical models of move- ment for the users);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' b) total computational com- plexity for defining distributions over the hidden states is therefore O(Nµ2ω2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' To avoid these complications, we develop our unlinkability assur- ance based on a rational lower bound Hr for H � ℓ1,ℓ2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' ��Y1,Y2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=',ρρρ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' The concept of the ratio- nal lower bound is explained through the following assumptions (Sniedovich, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Assumption 1 (Worst-case unlinkability) Requires that an attacker knows sets for the hidden, labelled and observable states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' He knows all the transitions and the order mixing probability ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' For each ob- servable state at time i he then defines the worst possible hidden state(s) which does not contradict his knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' We nevertheless stress that despite assumption 1 might be viewed as excessive, the attacker does not know the labelled state ℓi (and can not force its selec- tion) at time i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Assumption 2 (Rational lower bound Hr) Requires that users are rational and maximize worst-case unlinkability: observable states are obtained through rational obfuscation of the worst labelled states considered by the attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' There are several aspects affecting the task of cal- culating such Hr: 1) probabilities for transitions be- tween hidden states (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=', the probabilities defining ρρρ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' 2) probabilities for transitions from the hidden states to the labelled states (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=', ϕ, 1 − ϕ), and from the labelled states to the observable states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Further, we consider a situation where the worst case ρρρ (minimiz- ing entropy) is defined for 1) while the most optimal probabilities (maximizing entropy) are then specified for 2) under constraint ˜D on the total distortion over N − 1 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' We use the results of lemma 1 to require the fol- lowing: Hr = min ρρρ � H � ℓ1,ℓ2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' ��Y1,Y2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=',ρρρ �� = N−2 ∑ i=0 min {ρi+1|ρi} � H � ℓi+1 ��Yi+1,{ρi+1|ρi} �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' (7) To obfuscate hidden states in the way maximizing Hr we need to determine properties of ρmin,i+1 = arg min {ρi+1|ρi} � H � ℓi+1 ��Yi+1,{ρi+1|ρi} �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' (8) Probabilities Pr � ℓi+1 = R ,Yi+1 ��{ρi+1|ρi} � , Pr � ℓi+1 = B,Yi+1 ��{ρi+1|ρi} � will be used in our further derivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' To simplify notations we will use Pr(ℓi+1 = R ,Yi+1), Pr(ℓi+1 = B,Yi+1), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' The probabilities are defined as: Pr(ℓi+1 = R ,Yi+1) = ∑ Xi+1,R ∈XR Pr � Yi+1 | Xi+1,R � Pr � Xi+1,R � = ∑ Xi+1,R ∈XR Pr � Yi+1 | Xi+1,R � ϕPr(Xi+1) , (9) Pr(ℓi+1 = B,Yi+1) = ∑ Xi+1,B∈XB Pr � Yi+1 | Xi+1,B � Pr � Xi+1,B � = ∑ Xi+1,B∈XB Pr � Yi+1 | Xi+1,B � (1 − ϕ)Pr(Xi+1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' (10) We then point out that Pr(ℓi+1 = R | Yi+1) = Pr(ℓi+1 = R ,Yi+1) Pr(Yi+1) , (11) Pr(ℓi+1 = B | Yi+1) = Pr(ℓi+1 = B,Yi+1) Pr(Yi+1) , (12) where Pr(Yi+1) = Pr(ℓi+1 = R ,Yi+1)+ Pr(ℓi+1 = B,Yi+1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' (13) The following result establishes an important property of ρmin,i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Lemma 2 For all i ∈ [1,N − 1] distribution ρmin,i is degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Proof: We presume that ρmin,i is non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' For simplicity and without loss of generality we consider two-point distribution ρ∗ min,i : {x1,x2} → [0,1]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' For instance, realizations x1 = (xA 1,xB 1), x2 = (xA 2,xB 2) can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Here Pr(Xi = x1) = ξ, and Pr(Xi = x2) = 1 − ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Minimization of conditional entropy in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' (8) is equivalent to the minimization of pℓi,min where pℓi,min = min � Pr(ℓi = R | Yi),Pr(ℓi = B | Yi) � , (14) and without loss of generality, we assume that pℓi,min = Pr(ℓi = R | Yi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' To express pℓi,min we then use eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' (9) to (11) and (13) with the following substitutions (simplifying ex- pressions): α1 = ϕPr � Yi | Xi,R = (xA 1,xB 1) � , β1 = ϕPr � Yi | Xi,R = (xA 1,xB 1) � + (1 − ϕ)Pr � Yi | Xi,B = (xB 1,xA 1) � , α2 = ϕPr � Yi | Xi,R = (xA 2,xB 2) � , β2 = ϕPr � Yi | Xi,R = (xA 2,xB 2) � + (1 − ϕ)Pr � Yi | Xi,B = (xB 2,xA 2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' The minimization task is then min ξ pℓi,min = min ξ ξα1 + (1 − ξ)α2 ξβ1 + (1 − ξ)β2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' (15) By analyzing ∂ ∂ξ pℓi,min, we conclude that there are no local extrema for ξ ∈ (0,1) and, hence, minimum is obtained in one of the end points, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=', ξ ∈ {0,1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' □ Based on the result of lemma 2, for every Yi there is one and only worst-case hidden state ˜Xi (because Pr � ˜Xi | ρmin,i � = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' It implies the following: Corollary 1 Design of HMM where for every state (realization) in Y there is one and only transition from X(A,B) explicitly satisfies assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Therefore, we will further adhere to such design principle and use ˜Xi to denote hidden states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Next, we will elaborate on: a) what is the optimal number of different observable states Yi for every ˜Xi?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' b) how should we define optimal observable states?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' c) what are the probabilities of transition (from the labelled states to the observable states)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='3 Requirements for the observable states Here we provide our analysis from the standpoints of the system that obfuscates hidden states (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=', the sys- tem produces observable states) on behalf of Alice and Bob, and hence ˜Xi is assumed to be known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' The pos- sibilities of transitions ˜Xi,R → Yi and ˜Xi,B → Yi im- ply that a non-zero distortion E[Di] takes place: E[Di] = ∑ y(i) j ∈Y(i) Di,jPr � Yi = y(i) j | ˜Xi � , (16) where Di,j = Pr � ℓi = R | Yi = y(i) j � d � ˜Xi,R ,y(i) j � + Pr � ℓi = B | Yi = y(i) j � d � ˜Xi,B,y(i) j � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' (17) Here Y(i) is the set of all observable states to which transitions exist from the realizations of ˜Xi,R and ˜Xi,B at time i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' y(i) j is an element in Y(i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' d (·,·) is some distortion measure (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=', SE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' The optimization effort is two-fold: i) how shall we obtain observable states Y(i) in a way that Hr,i is maximized under constraint ˜Di ≥ E[Di]?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' ii) how shall we define ˜Di for every time step i such that Hr is maximized and the total distortion constraint ˜D ≥ ∑i E[Di] is satisfied?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' We start with answering question i), which will assist us in answering question ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' For the obfuscation, we utilize the following principles: every element y(i) j in Y(i) can be fully specified by the realizations of ˜Xi,R , ˜Xi,B, and pa- rameter λ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Probabilities Pr � ℓi = R | Yi = y(i) j � , Pr � ℓi = B | Yi = y(i) j � then affect Hr,i,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' All these pa- rameters affect Di,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' The diagram explaining relations between all the mentioned parameters is provided on fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' In this example, labelled states are ˜Xi,R = � xA,xB� , ˜Xi,B = � xB,xA� ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' set Y(i) contains only two elements y(i) 1 = � ˆy(i), ˇy(i)� and y(i) 2 = � ˇy(i), ˆy(i)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' For example, to specify y(i) 1 we only need λ1 in addi- tion to the labelled states (∆ is the distance between them).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' To obtain y(i) 2 we should apply a similar pro- cedure where λ2 is known (in our particular example λ2 = 1 − λ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Probability Pr � ℓi = R | Yi = y(i) 1 � is denoted as p1: its value affects attacker’s uncertainty Hr,i,j as well as the distortion Di,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Figure 3: Scheme for obfuscation principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' To maximize Hr,i under ˜Di ≥ E[Di] we consider realizations of Yi and optimal adjustment of λ: such adjustment then allows us to increase p1 and 1 − p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' We note that Yi shall belong to a line segment (in a multidimensional space) connecting ˜Xi,R and ˜Xi,B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' This property is trivial (goes without proof) and can be best understood if triangle △ ˜Xi,R Yi ˜Xi,B is consid- ered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' As a result: ∀Yi � Yi ∈ Y(i) =⇒ � ∃λ ∈ [0,1] � ∧ �⃗Yi = −−→ ˜Xi,R + λ −−−−−→ ˜Xi,R ˜Xi,B �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' (18) We then establish the following: Lemma 3 To minimize Di,j it is required that λ j = 1 − Pr � ℓi = R | Yi = y(i) j � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Proof: From eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' (17) and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' (18) we derive that Di,j = p j∆2 i λ2 j + (1 − p j)∆2 i (1 − λ j)2 , where p j = Pr � ℓi = R | Yi = y(i) j � ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='5, and ∆2 i = d � ˜Xi,R , ˜Xi,B � = ��� −−−−−→ ˜Xi,R ˜Xi,B ��� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' We next analyse ∂ ∂λj Di,j and find that λ j = 1 − p j is the extremum (minimum) of Di,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' □ Corollary 2 Minimal distortion is Di,j = ∆2 i p j(1 − p j) ≤ ∆2 i 4 , where p j = Pr � ℓi = R | Yi = y(i) j � , ∆2 i = d � ˜Xi,R , ˜Xi,B � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Corollary 3 For every i, the highest lower bound (maxmin entropy) is: Hr,i = −νi log2 νi − (1 − νi)log2 (1 − νi) , (19) where νi = min � ϕ, ∆i−√ ∆2 i −4E[Di] 2∆i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Proof: It is required: 1) to determine Y(i) and prob- ability distribution over it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' 2) to determine Pr(ℓi | Yi) for every element in Y(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' For this, we demonstrate that maximum entropy under distortion constraint on E[Di] is achieved for ���Y(i)��� ≤ 2: we analyze the case for Y(i) = � y(i) 1 ,y(i) 2 � where Pr � ℓi = R | Yi = y(i) 1 � = Pr � ℓi = B | Yi = y(i) 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' (20) To prove the optimality of such settings, we con- sider several alternative cases where E[Di] = ˜Di is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Let us first consider an alternative case where ���Y(i)��� = 2 but Pr � ℓi = R | Yi = y(i) 1 � ̸= Pr � ℓi = R | Yi = y(i) 2 � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Pr � ℓi = R | Yi = y(i) 1 � ̸= Pr � ℓi = B | Yi = y(i) 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' (21) For simplicity, we use the following notations: Pr � Yi = y(i) 1 | ˜Xi � = α, and Pr � Yi = y(i) 2 | ˜Xi � = 1 − α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Pr � ℓi = R | Yi = y(i) 1 � = p1 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='5, and Pr � ℓi = R | Yi = y(i) 2 � = p2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Taking into ac- count the expression for conditional entropy, we then require: � max �Hr,i � = max � αH1 + (1 − α)H2 � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' ˜Di = αDi,1+ (1 − α)Di,2 , (22) where H1 = H � ℓi | Yi = y(i) 1 � , H2 = H � ℓi | Yi = y(i) 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Based on eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' (21) Di,1 ̸= Di,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' We now show that H1 and H2 are functions of Di,1 and Di,2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' For this, we only point out that p1 (similar results can be obtained for p2) is a mono- tonically increasing function of Di,1: it follows from corollary 2 that p1 = ∆i−√ ∆2 i −4Di,1 2∆i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' To demonstrate the fallacy of attaining both eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' (21) and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' (22) it is sufficient to show the following (concavity): αF(x)+ (1 − α)F � ˜Di − αx 1 − α � ≤ F( ˜Di) , (23) where x = Di,1, and F(x) = −p1(x)log � p1(x) � − � 1− p1(x) � log � 1 − p1(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' The validity of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' (23) fol- lows from ∂ ∂xF(x) = 1 ∆iθ log � ∆i+θ ∆i−θ � ≥ 0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' ∂2 ∂x2 F(x) = − 2 ∆iθ2 � 1 ∆i+θ + 1 ∆i−θ − 1 θ log � ∆i+θ ∆i−θ �� ≤ 0 , where θ = � ∆2 i − 4x, and x ∈ � 0, ∆2 i 4 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Next, we point out a different case where ���Y(i)��� > 2 and demonstrate that it is non-optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' For this we consider ���Y(i)��� = 3 while the conclusions for ���Y(i)��� > 3 can be derived inductively then.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Similarly to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' (22) we demand � max �Hr,i � = max � αH1+ βH2 + (1 − α− β)H3 � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' ˜Di = αDi,1 + βDi,2+ (1 − α− β)Di,3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' The task is then to show that there is y(i) 4 for which Di,4 = αDi,1+βDi,2 α+β , and maxH4 ≥ max � α α+βH1 + β α+βH2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' We hence- forth maintain that ���Y(i)��� ≤ 2 represents optimal settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' To obtain max � αH1 + (1 − α)H2 � in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' (22) it is sufficient that H1 = H2 and Di,1 = Di,2 = ˜Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' The lat- ter requires that either λ1 = λ2 or λ1 = (1 − λ2): the first condition implies p1 = p2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='5 and leads to a trivial situation where y(i) 1 = y(i) 2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='5 � ˜Xi,R + ˜Xi,B � meaning that ���Y(i)��� = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' The second condition implies p1 = 1 − p2 and leads to y(i) 1 ̸= y(i) 2 if ˜Di < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='25∆2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Requirement α ∈ [0,1] must be consistent with the order mixing probability ϕ: αp1 + (1 − α)p2 = ϕ , (24) from which we derive α = ϕ+p1−1 2p1−1 demanding ϕ ≥ p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Alternatively, this demand can be understood based on the fact H (ℓi) ≥ H (ℓi | Yi): setting p1 > ϕ results in a greater distortion, but this does not increase en- tropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' □ There are several important takeaways from the proof of corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' First, for every hidden state ˜Xi there are two observable states that are obtained ac- cording to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' (18) where λ(i) 1 = 1 − νi is used to de- fine realisation y(i) 1 , and λ(i) 2 = 1−λ(i) 1 is used for y(i) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Second, maximum allowed distortion should be used at step i meaning that E[Di] = ˜Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Third, probabili- ties for transitions from labelled states to observable states are Pr � Yi = y(i) 1 | ℓi = R � = νi ϕ ϕ+νi−1 2νi−1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Pr � Yi = y(i) 2 | ℓi = R � = 1 − Pr � Yi = y(i) 1 | ℓi = R � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Pr � Yi = y(i) 1 | ℓi = B � = 1−νi 1−ϕ ϕ+νi−1 2νi−1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Pr � Yi = y(i) 2 | ℓi = B � = 1 − Pr � Yi = y(i) 1 | ℓi = B � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' (25) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='4 Optimal obfuscation for N −1 time steps For every i we now define ˜Di such that Hr = ∑iHr,i is maximized under the total distortion constraint ˜D ≥ ∑i ˜Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' For this reason, we obtain optimal observ- able states and corresponding transition probabilities (from the labelled states) for all the time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' From the proof of the corollary 3 we use that ∂ ∂ ˜Di Hr,i ≥ 0 and ∂2 ∂ ˜D2 i Hr,i ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' To maximize Hr we therefore require \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 ∀i ∂ ∂ ˜Di Hr,i = 1 ∆2 i √1−κi log � 1+√1−κi 1−√1−κi � = C ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' ˜D = N−1 ∑ i=1 ˜Di = 1 4 N−1 ∑ i=1 κi∆2 i , (26) where C is some constant, κi = 4 ˜Di ∆2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' We then solve the system eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' (26) for all κi, i ∈ [1,N − 1], and according to corollary 3 obtain νi = min � ϕ, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='5 − √0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='25 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='25κi � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' 5 OBFUSCATION ALGORITHM Here we represent our aforementioned findings in the form of obfuscation algorithm (see algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' It is practical and can be implemented in real settings: its complexity (excluding the complexity of solve pro- cedure) is only O(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' For input, the algorithm ac- cepts arrays (of size N) XA, XB, and scalars ˜D, ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' El- ements of these arrays are scalar/vector realizations for XA i and XB i characterizing geo-positions of Alice and Bob, respectively, at time i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' In practice, these ar- rays may contain extrapolations based on historical data and repetitive patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' For example, Alice and Bob may commute to work using the same routes and roughly at the same time every day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Procedure solve provides a solution to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' (26): array κκκ contains ele- ments κi needed to define realizations for obfuscated state Yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' It is also needed to calculate the unlink- ability criterion (entropy) Hr,i dependent on the ob- fuscation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Procedure send RSU encapsulates data obfuscated at time i in accordance with one of the V2X communication formats and sends it to the near- est RSU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' The output of the algorithm is, therefore, an array Y containing all the obfuscated records and the indicator of the total unlinkability in the system over N − 1 steps, Hr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Algorithm 1: Obfuscation algorithm input : XA, XB, ˜D, ϕ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' output: Y, Hr ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' begin Hr ← 0, Y ← ∅, κκκ ← solve � ˜D,XA,XB� ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' for i ← 1 to N −1 do νi ← min � ϕ, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='5−√0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='25−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='25κi � , α ← (ϕ+νi −1)/(2νi −1), Hr,i ← −νi log(νi)−(1−νi)log(1−νi), Hr ← Hr +Hr,i, P1,R ← νiα/ϕ, P1,B ← (1−νi)α/(1−ϕ), r1 ← UniRand � [0,1] � , r2 ← UniRand � [0,1] � , ΛR ← 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='5+(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='5−νi) sign± � P1,R −r2 � , ΛB ← 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='5+(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='5−νi) sign± � P1,B −r2 � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' if r1 ≤ ϕ then ˆy(i) ← X A i +ΛR � X B i −X A i � , ˇy(i) ← X B i +ΛR � X A i −X B i � , Yi = concat(ˆy(i), ˇy(i));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' else ˆy(i) ← X A i +ΛB � X B i −X A i � , ˇy(i) ← X B i +ΛB � X A i −X B i � , Yi = concat(ˇy(i), ˆy(i));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' send RSU(Yi), Y = concat(Y,Yi) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' 6 DISCUSSION Here we discuss how well the main aim (see sec- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='1) – “to develop a methodology providing a high level of assurance that entropy for CAMs’ ori- gins is high in C-ITS” – was achieved by our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' For this, we provide characteristics about the major results, their advantages, limitations, and plans for further work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='1 Results and their characteristics In this paper, we combine: (i) the classical definition of unlinkability and (ii) assumptions about a strong at- tacker to (iii) measure and improve unlinkability in C- ITS by developing the optimal joint obfuscation tech- nique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' The academic novelty is due to the combina- tion of points (i-iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Next, we discuss the importance of each point in greater detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' First, we lean towards the classical definition of unlinkability demanded by the standards governing the domain of C-ITS applications (ETSI TS 102 941 V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='1, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' ISO 24102-1, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' The concept of this study is, therefore, closer to some early works on location privacy, such as (Shokri, 2012) relying on Bayesian inference in HMM and contrasts with many later works reliant on k-anonymity, differential privacy, and geo-indistinguishability (Andr´es et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Bordenabe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Corser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' In this work, the definition of unlinkability is cap- tured through Bayesian inference and is further re- flected by entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' We also stress on advantages of such an approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Based on definition 2, the at- tacker’s reasoning about the operations plays the cen- tral role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Such reasoning may go beyond properties observable within the object (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=', cryptographically signed CAM message): it may additionally rely on meta-information collected, for example, on a sys- tem level (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=', the order of CAMs arrivals at RSU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' This feature accords with many concepts in science and philosophy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Among others, Leibniz stated that indiscernible objects have identities (Hacking, 1975): preferences about these identities can be expressed statistically (and, hence, used in Bayesian inference).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' For instance, in statistics such reasoning may be as- sisted by means of additional indexing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' In contrast, geo-indistinguishability does not support further rea- soning about indiscernible pieces of geo-data, mak- ing this privacy concept less demanding (and, hence, inferior) compared to unlinkability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' To witness the differences between these concepts, one should ob- serve order mixing (and corresponding probability ϕ) in our HMM (see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' 2): even if XA i = XB i , the model sets (XA i ,XB i ) apart from (XB i ,XA i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' For that reason, we agree with the authors of (Montazeri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Takbiri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=', 2017, 2020), stating that both obfusca- tion and permutation (order mixing) are required for strong privacy assurance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Second, our focus on unlinkability (and not on the location protection) provides consistency even under the assumption that an adversary is strong: the at- tacker knows the actual locations of Alice and Bob at every moment i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' He also knows the probabilistic obfuscation and order mixing algorithm used by the players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' However, he does not know the outputs of this probabilistic algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' His goal is then to infer the origin of the digitally signed obfuscated messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' As a result of applying assumption 1, reasoning about statistical inference made by the attacker is very much simplified compared to (Shokri, 2012): information about HMM’s hidden states (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=', actual locations, ve- locities, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=') and transition probabilities are not re- quired for such reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' The latter detail is benefi- cial for the privacy assurance since establishing prob- abilities for transitions in HMM is a laborious and of- ten imprecise procedure relying on Kalman-like esti- mators (Blackman, 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Lehmann and Pieczynski, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Third, strong (and simplifying) assumptions as- sist us in specifying the lower bound of unlinkabil- ity in C-ITS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' For every time step i unlinkability is expressed through entropy Hr,i: the worst-case in- ference is made by an attacker meaning that Hr,i is the lower bound (see assumption 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Components Hr,i are then summed over N − 1 steps to obtain Hr (see lemma 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Such summation is a simple and in- tuitive step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' It is, nevertheless, justified because for any i ∈ {1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=',N − 1}, inference about the source (origin) of arrived CAM is independent from such inference at i − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' An analogy can be established between entropy (unlinkability) values Hr,i and Hr, and the concepts of microscopic and macroscopic pri- vacy, respectively, in (Shokri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Better pro- tection of macroscopic privacy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=', trajectories) re- quires higher uncertainty about labels ℓi for the lo- cations reported on the microscopic level (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=', geo- graphic points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Higher uncertainty about ℓi can only be provided at the cost of higher expected distortion E[Di] of the players’ CAMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' To maximize uncer- tainty Hr,i about labels under the constraint on distor- tions we propose a new simple way to define a joint distribution that must be followed during the obfus- cation (to obtain CAMs) conducted cooperatively by Alice and Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Optimality of (multivariate) noise pa- rameters under various constraints on distortions has been studied by many authors in the past (Andr´es et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Geng and Viswanath, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Takbiri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Our approach, however, differs: to improve microscopic privacy at i we insist on joint probabilis- tic obfuscation of the samples from different play- ers (see proof of corollary 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Because of the lat- ter feature, our obfuscation approach is clearly data- dependent (Croft et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Guan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' We then analyse how to optimally distribute distortions over N − 1 time steps if the corresponding distortion cap is specified for the whole duration of C-ITS ob- servation (see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' (26)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' All the findings of this paper are incorporated in algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Procedure solve is one of the major factors contributing to the time com- plexity of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' This, nevertheless, can be addressed if the obfuscation optimality is slightly sac- rificed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' For example, solve can be pre-computed for several cases only: each case would produce a dis- tinct kind of distribution for a random variable ∆2 i˜D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Then, the actual input data should be approximated by the best-matching distribution, and the corresponding pre-computed outputs of solve should be used for the obfuscation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Such workaround can also turn our al- gorithm into a ‘real-time algorithm’: if Alice and Bob believe that their future data will align well with one of the pre-computed distributions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=', because of ha- bitual daily commutes) they can obfuscate it ‘on-the- fly’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Hence, the pre-computed cases for solve can be treated as profiles that pairs of players agree to use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='2 Limitations and future work This paper has certain limitations which we plan to address in our further studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Only one particular composition of entropy (to measure unlinkability) and squared error (to measure distortion) is considered in our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Besides en- tropy, other uncertainty measures may be useful for expressing unlinkability in C-ITS (Wagner and Eck- hoff, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Also, distortion measures other than SE have been analysed and recommended in the past by some authors studying Multiple-Target Tracking and its applications (Gorji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Only high assurance about minimally achievable unlinkability (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=', rational lower bound, see assump- tions 1 and 2) is studied here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' However, to further improve the practicality of our methodology, indica- tors obtained for the worst-case scenario (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=', a very strong attacker) may be complemented by other in- dicators relying on less pessimistic scenarios (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=', a weaker attacker).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Only 2 players are considered in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' This substantially limits the number of permutations for CAMs: as a result, for every hidden state there are only 2 labelled states (see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' Because of that, the methodology defining observable states is also simple (see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' We plan to increase the number of players in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' However, for larger numbers of players, defining optimal observable states and corresponding probabilities for transitions is a non-trivial task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' This is because more complex structures and transforma- tions in Rz, z ≥ 1, need to be analyzed to optimize instant and joint obfuscation of CAMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' 7 ACKNOWLEDGMENT This research is co-financed by public funding of the state of Saxony, Germany.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' 2010 Seventh International Conference on Wireless On-demand Net- work Systems and Services (WONS), 176– 183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content=' u1 (1 - X1)△ V2 () () y 2 B r j= V2 3=2 (1 - >1)△ ()) V2 [] B p1 V2ScileP TeSS ScienceandTechnologyPublicationsThis figure "orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='png" is available in "png"� format from: http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='org/ps/2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} +page_content='04130v1' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfzQjl/content/2301.04130v1.pdf'} diff --git a/nNE3T4oBgHgl3EQf6wum/content/tmp_files/2301.04793v1.pdf.txt b/nNE3T4oBgHgl3EQf6wum/content/tmp_files/2301.04793v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0f749941d14e9414cfbbfafcfd9cbe401cb2f03d --- /dev/null +++ b/nNE3T4oBgHgl3EQf6wum/content/tmp_files/2301.04793v1.pdf.txt @@ -0,0 +1,587 @@ +PINN for Dynamical Partial Differential +Equations is Not Training Deeper Networks +Rather Learning Advection and Time +Variance +Siddharth Rout +Department of Mechanical Engineering. +The University of British Columbia, Vancouver, BC, Canada. +Contributing authors: sidrout@mail.ubc.ca; +Abstract +The concepts and techniques of physics-informed neural networks +(PINNs) is studied and limitations are identified to make it effi- +cient to approximate dynamical equations. Potential working research +domains are explored for increasing the robustness of this technique +for the solvability of partial differential equations. It is identified that +PINNs potentially fails to stronger advection and longer time duration. +Also, optimization function and constraint posing needs to be smarter. +Even a shallow network is good for a lot of problems while power- +ful deeper network fails. Reservoir computing based recurrent neural +network architecture is recommended to solve dynamical problems. +Keywords: Dynamical Systems, Chaos, Physics Informed Neural Networks, +PINN, Kuramoto–Sivashinsky Equation, ODE solution, PDE solution, +Differential Equations, Recurrent Neural Network, Reservoir Computing +1 Introduction +The solution of differential equations have been an important topic for the +almost every field of the world, say it be finance, mechanics, meteorology etc. +Starting from the days of Newton and Leibniz solving differential equations +have been core to developments in this world. Not all differential equations are +1 +arXiv:2301.04793v1 [physics.comp-ph] 12 Jan 2023 + +– +2 +PINN for Dynamical PDEs +solvable by hand and initiate limitations, especially when multiple independent +variables come into the equation or build system of equations. However, solving +these equations are need of the hour and hence we shifted our focus from exact +solutions to approximate solutions as function operations and transformations +are limiting in concepts. There arose the generation for solving equations by +using the basic definitions or first principles of limits, discretization and numer- +ical analysis[1]. Popular such techniques are finite element methods, finite +volume methods, finite difference approximations etc. These methods are gen- +eralizable and that is the benefit. Using these techniques we can solve almost +any equation for any geometry. But with increased complexity, there are a +couple of problems accompanied like how good the approximation is and com- +putation expense. Discretization gives us a long list of simplified approximate +equations to solve. Though we know how to solve however solving them will +require us to use computers to do those hectic mathematical calculations. For +a stable and accurate solution, a lot of time and energy are consumed in the +process[1]. Though we can calculate a lot of things in this world but we have +lack of time, resources, money and problems could be endless. A type of differ- +ential equation called dynamical systems is tough to solve after a certain range +in time, there are reasons to it. Dynamical systems are tough to solve as the +solution my bifurcate and that shall make the system chaotic. In a chaotic sys- +tem, a minute change in the initial condition or equation coefficients will have +drastically different outcomes. This is sometimes referred by ’The Butterfly +Effect’. The aim of this project is to develop a function approximation method +that can potentially replace computationally expensive solvers for dynamical +systems. The one dimensional Kuramoto-Sivashinsky equation is solved for +trials and research[2, 3]. +Function approximations or analytical solutions are better known for their +light-weight[4]. These techniques can get rid of the three primary types of errors +that are evident in full order discretized approximation, namely instability, +inaccuracy and shift[5]. A system of dynamical systems is mainly sensitive +inaccuracy due to insanely evident sensitivity to initial conditions. A benefit +of function approximation is the ability of correction and reproduction. Also, +an added benefit is extrapolability. Among the function approximators, neural +networks have been excellent candidate as universal approximators[6, 7]. In +the past decade, deep neural networks which are basically multiple layers of +neural networks have been used for various complex regression problems due to +its ability to capture high dimensional strong non-linearity. These models are +fitted to the data directly as input to output mapping[7]. Neural networks can +also be trained to differential equations by optimizing the residuals after fitting +to random points in the input domain. Such models are called physics-informed +neural networks or popularly called PINNs[4, 5, 8, 9]. +Solving partial differential equations using PINNs are universally accepted +by the scientific community. There are a plenty of advantages of these meth- +ods over conventional methods. The major once are ability to solve wide + +– +PINN for Dynamical PDEs +3 +category of problems that were tough to solve otherwise. Moreover, these meth- +ods do not require meshing and discretization which is sometimes a tough +task. Another advantage unlike other analytical models does not require data +from full order solutions to set the parameters. However, being newly devel- +oped these techniques are not robust enough for solving complex equations +like hyperbolic equations, strongly non-linear equations, strongly advective +equations[5], chaotic dynamical systems, coupled system of equations, shock +wave equations etc. +2 Dynamical Partial Differential Equations +Activities in the world is mostly the four dimensions, three in space and one +in time. Each new dimension adds a layer of complexity. Dynamical systems +generally mean functions that describe the dependence of state of a system +with time. Henri Poincare was the first one to identify the special behaviour of +dynamical systems. The theory of these dynamical systems is highly relevant +in studying behaviour of complex dynamics, usually in the form of differential +equations, which makes it continuous dynamical systems. The major points of +focus in this domain are the attractors, chaos, fractals and bifurcations that +explains the long term behaviour of states qualitatively. This helps under- +standing evolution of dynamical events like turbulence, storm, mixing fluids, +environment change, economic changes, planetary motions and many more. +The necessary applications of dynamical systems theory are to find struc- +tural stability, Lyapunov time, bifurcation points, position tracking and +quantitative approximations which one way or the other determines the pre- +dictability of the state at a particular time. Predictability of dynamical systems +is a tough job. Before the advent of computing machines prediction required +sophisticated mathematical techniques that were specific to specific classes +of dynamical systems. These are sometimes among the toughest differential +equations to solve. Also considering other factors mentioned above, accurate +prediction is a great deal for these kinds of systems. +3 Case Selection +The cases below point out two major difficulties in solving differential equations +clearly. The concepts are explained with reference to the terms and frame- +work of the equation mentioned. The two equations shall be good examples to +analyse the theory of PINNs. +3.1 1D Steady Advection-Diffusion Equation +The differential equation below is the governing equation for steady one +dimensional flow of combined advection and diffusion phenomena. +αux = uxx + +– +4 +PINN for Dynamical PDEs +Fig. 1 Solution of steady state advection-diffusion +Fig. 2 Solution of One dimensional Kuramoto-Sivashinsky equation +If we notice, α is the weight for the advection term in the equation. That means +larger is α, more dominant is the advection effect, which introduces directional +characters and hence discrete approximation becomes tougher. Figure 1 shows +the difference in the solution with advection dominance. Higher is the Peclet +number, more dominant is the advection. The figure compares the solution for +Pe 1 and 50. Hence the numerical integration sees rapidly growing error that +makes the solution unstable and inaccurate[10]. Hence, a major class of higher +order methods are developed to tackle this particular issue. +3.2 1D Kuramoto–Sivashinsky Equation +The equation below is the one dimensional Kuramoto-Sivashinsky equation +ut + αuux + βuxx + γuxxxx = 0 +The linear form of it is as below: +ut + αux + βuxx + γuxxxx = 0 + +1.0 +Pe=1 +Pe=50 +0.8 +0.6 +0.4 +0.2 +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0Kuramoto-Sivashinskydynamics +100 +80 +60 +40 +20 +0 +0 +20 +40 +60 +80 +100 +120 +x– +PINN for Dynamical PDEs +5 +This equation has the advection, diffusion and dissipation effect. It is one of the +equations where the solution is extremely sensitive to the initial condition[2, 3]. +The higher order terms in the expansion of the difference equation are very +much relevant and hence sensitive for error propagation in time. Figure 2 shown +the solution of a case of one dimensional KS equation on Julia code developed +by Mahatab Lak et. al. from the University of New Hampshire. +4 Neural Networks as Universal Function +Approximator +George Cybenko was the one to prove arbitrary width case using neural net- +works with sigmoid activation in 1989[6]. Later in the same year, Hornik et. +al. proved multi-layer feed-forward networks are universal approximators[7]. +Multi-layer artificial neural networks are composites of weighted sum of inputs +passed through non-linear(activation) functions like tanh(), sigmoid(), etc. +This enables an extremely potent highly non-linear function with large num- +ber of trainable parameters(weights and biases). This makes it universal +approximation. +5 Physics-informed Neural Networks +Informing the physics to a neural network is a concept brought up by Lagaris +et. al[4]. in the late 1990s by using neural network as a trial function to solve dif- +ferential equations by reduction of the residuals using AutoGrad (an automatic +differentiation technique) at various points in the domain. The boundary con- +straints are forced into the neural network function by modifying it mannually. +In 2017, Raissi et al. proceed by using more accurate automatic differentia- +tion and deeper networks to approximate tougher problems[8, 9]. The novelty +in their work comes from the way they pose the loss function to reduce the +residual. They did not manually force the constraints by modifying the trial +function rather let the trial function fit to the boundary and initial constraints +by adding the mean square error from the data points satisfying the conditions +as summed constraints to mean squared residuals. This makes the technique +very much generalizable. Almost all the differential equations could be posed +to be solved using this technique, which they named PINNs. +5.1 Advancements in PINNs +The unknown solution u(t, x) is represented by a deep neural network +uθ(t, x), where θ denotes all tunable parameters of the network (e.g., weights +and biases). The physics-informed model can be trained by minimizing the +following loss function. +L(θ) = λicLic(θ) + λbcLbc(θ) + λrLr(θ), +where + +– +6 +PINN for Dynamical PDEs +Here +� +xi +ic +�Nic +i=1 , +� +ti +bc, xi +bc +�Nbc +i=1 and +� +ti +r, xi +r +�Nr +i=1 can be the vertices of a fixed +mesh or points that are randomly sampled at each iteration of a gradient +descent algorithm. The hyper-parameters {λic, λbc, λr} allow the flexibility of +assigning a different learning rate to each individual loss term in order to +balance their interplay during model training[12, 13]. These weights may be +user-specified or tuned automatically during training. +5.2 Advantages of PINNs over Other Neural Networks +The major advantage of this technique is that it does not require physical data +to train the analytical model. Moreover, the technique is generalizable in the +sense that with the exactly same concept various equations could be solved[4, +5, 8, 9, 12, 13]. Previous models required alteration of the learning function +depending on number of equations coupled, boundary conditions etc. in order +to force the constraints. Being a strong approximating function the three major +kinds of error, namely instability, inaccuracy and shifting errors, could be taken +care of simultaneously. These problems are taken care individually in finite +numerical techniques[5]. This kind of technique is an excellent candidate for +robust higher order methods. Neural network is not just an approximator but +rather a smart approximator. Hence, depending on the local physical property +it can act differently with switching kind of behavior. In particular for the +case of dynamical system, with time progression the integrated error increases +too rapidly. PINNs being an optimization technique for regression, training to +measured physical data points as additional loss terms or regularization can +be used for correcting the approximating function. +6 Experiments with the selected cases +The qualitative property in dynamical problems is strong translational vari- +ance. Hence, the two major causes for PINNs performing poorly are advection +dominance and time variance which are demonstrated below. +6.1 PINNs for 1-D Steady Advection-Diffusion +Peclet number is a good non-dimensional parameter to scale advective domi- +nance over diffusion characteristic of an equation. It is the ratio of advective +transport rate to diffusive transport rate. In our problem we can quantify that +by the ratio of coefficient of advective term times the length of domain space to +coefficient of diffusive term, so that is α in our particular case. PINN can solve +this problem but there is a limit set by advection characters in the differential +equations. No matter how deeper and how sophisticated we make the neural +network it is not possible to solve for problems with Peclet number more than +something close to 8. Figure 3 shows the results noted. For lower Peclet num- +ber problems, it is noticed that it is not necessary to have deeper and wider +layers. A good thing is in conventional numerical techniques fails for problems +set with this value more than 2. Hence, these schemes can be used for shape + +– +PINN for Dynamical PDEs +7 +Fig. 3 Optmimized architecture for solving one dimensional steady advection diffusion. +Fig. 4 Minimized losses with various activation functions while solving one dimensional +steady advection diffusion where C is coefficient of advective term. +Fig. 5 Loss trend while solving using various optimizer for one dimensional steady advection +diffusion +functions that let larger grids with similar accuracy. The work and figures in +this section has been sourced from by 2019 thesis titled ”Numerical Approxi- +mation in CFD Problems Using Physics Informed Machine Learning”[5]. +Parametric analysis is done to understand more about the performance and +effectiveness. The impact of change in the non-linearity function as well as loss +optimization algorithm is studied. Figure number 4 records the loss values in a +tabular form. Among various non-linearity functions tanh() and tan() performs +consistently as well as better than sigmoid(). However, tanh() wins this game +clearly. Figure number 5 shows the trend of loss value with iteration for vari- +ous optimizers. The neural network is trained using various optimizers however +L-BFGS-B and SL-SQP perfoms better than other specialised optimization +techniques. BFGS performs remarkably better than others. The prime reason +could be the fact that these optimizers are second order and perform better in +the case of optimizing multiobjective functions than other techniques. How- +ever, it can be noted that first order techniques like Adam performs decently + +Architecture +Valid for: +Layers +NeuronsperLayer +1 +2 +Pe<3.8 +2 +2 +Pe<4.8 +3 +2 +Pe<6.3 +3 +3 +Pe<6.5 +3 +10 +Pe<7.6 Models +Loss +C +Neuron +Optimize +Sigmoid +Tanh +SoftPlus +Log +ArcTan +s +r +Sigmoid +0.5 +2 +7.75615e-04 +1.47605e-05 +4.77046e-03 +7.31412e-03 +1.11537e-04 +Adam +1 +2 +1.55154e-03 +2.19862e-04 +4.46882e-03 +1.46074e-02 +4.01917e-04 +0.5 +2 +1.28998e-03 +1.27516e-06 +1.49661e-02 +1.79618e-02 +6.72869e-04 +1 +2 +1.70795e-02 +1.91109e-04 +1.93503e-02 +1.93258e-02 +1.08187e-03 +0.5 +4 +2.55921e-06 +1.59368e-05 +2.43783e-03 +2.39689e-03 +9.42070e-05 +1 +4 +L-BFGS +2.81592e-06 +3.33567e-04 +4.54190e-03 +2.79268e-03 +4.18945e-04 +0.5 +8 +B +9.95310e-05 +5.13360e-06 +1.79042e-02 +2.69044e-03 +2.04379e-05 +1 +8 +3.38175e-04 +1.34256e-05 +1.61106e-02 +9.27748e-03 +1.62906e-04 +1.5 +8 +1.92506e-03 +7.63881e-05 +1.70217e-02 +8.23745e-03 +7.14311e-04 +2 +8 +3.10389e-04 +2.35750e-04 +1.77785e-02 +7.77693e-03 +1.16140e-04CG +-BFGS-B +COBYLA +SLSQP +TNC +(sso) +Not an easy optimization! +-5 +Notall of themaresufficientlygood atfollowingthegradientpathologies +-6 +1000 +2000 +3000 +4000 +5000 +6000 +7000 +8000 +9000 +10000 +Iterations– +8 +PINN for Dynamical PDEs +even if they don’t fail. An important thing that can be observed that the col- +located PDE residuals and the fitting losses for constraints are clearly not in +similar scale which is not typical for regular data-driven neural network learn- +ing. Also the gradient pathology is not smooth and hence tough for other +optimizers. Guided optimization like hill climbing helps in few cases. +6.2 PINNs for 1-D Kuramoto-Sivashinsky +There are recent publications demonstrating how PINNs can solve one +dimensional Kuramoto-Sivashinsky equation which is among the standard +dynamical equation that can turn chaotic. However, there is a small intersect- +ing set of coefficients for advection, diffusion and dissipation terms for which +this works. As demonstrated in the previous case the dominance of advection +toughens the optimization of the PINN. Here, not only it is dynamical but +also non-linear. It has various orders of spatial derivatives making it difficult +especially when the system becomes chaotic. +CausalPINN by Wang et. al. is considered the state of art PINN[13]. They +have rightly identified the problem of multiobjctive optimization due to differ- +ence in scales of residuals and constraints stated by Rout et. al.[5] and devised +weights for each loss terms by normalizing with the each cumulative loss terms. +They are first people to be able to solve 1D KS Equation. We can validate their +model using their open source code provided. Figure 6 shows how CausalPINN +perfoms with time for the case provided in the figure. It can be noticed that +PINN can now solve complex dynamical problems. The initial sine smoothly +curls as expected while the constraints are obeyed. Like initial curve is a neat +sine function and the boundaries are continuously at base zero(0). However, +the typical equation where all the coefficients are considered 1 is taken for reg- +ular study of the equation. The state of art PINN fails to optimize even after +an effort equivalent to one-day’s run-time. The net loss is recorded to be 33.263 +where the constraint loss was 0.0016. This suggests the difficulty in fitting to +the PDE where as PINN could manage to obey the constraints. The residual +loss is noted to be 1808.506, where its weight in the loss function disappear +from the scale. This clearly shows the issue of multi-objective optimization. +7 Observations and Conclusion +Based on the experiments and analysis a few points could be commented. Sim- +ple addition or weighed by mean addition of squared loss terms of residuals +at collocation points and fitting points directly for constraints have different +scale and orders of magnitudes. It is one of the major issue as it can lead to +non-pareto optimal solution. It is also noticed that sometime the loss func- +tion gets stuck at local optima and gradient pathologies are tough and uneven +in the parameter hyperspace[5, 12]. Hence, stochastic first order optimizers +work otherwise higher order optimizers suitable for constrained optimizations +like SQP and BFGS works while other fails[5]. A better representation of loss + +– +PINN for Dynamical PDEs +9 +Fig. +6 Snapshots of velocity(U) along the x-axis overlapped through time for one +dimensional Kuramoto-Sivashinsky equation. +function like appropriate or adaptive weighted losses can help. Otherwise con- +straints could be forcefully enforced by modified architecture or trial function, +like explained by Isaac Lagaris et. al[4]. Specifically, in the context of prob- +lems in dynamical systems recurrent neural networks(RNN) could prove to be +better candidate over simple deep networks[14]. A concrete reasoning has been +provided by Eldad Haber et. al. where RNNs can be proved to be in the form +of differential equations and hence fit into the theory to learn the dynamical +differential equations better[15, 16]. Especially, for chaotic systems reservoir +computing have been proved to be performing better[11]. Reservoir comput- +ers are a class of RNNs where the intermediate nodes are randomly arranged +and connected[11, 14]. They have random recurrent connections. The interme- +diate nodes are jumbled and entangled however they are connected out to the +output layer linearly. The entangled architecture makes it tough to backprop- +agate and hence only the final layer of weights are trainable for convenience. +The trainable output layers makes the effective non-linear network linear with +respect to trainable parameters hence conserving strong non-linearity while +making it easy to train. Apparently, RNN can also be introduced with PINNs +kind of loss definition to solve chaotic problems for turbulence and extreme +event prediction[17]. +Ultimately, we can justify the errors and specify the right path to solve a +dynamical system of partial differential equations by identifying the two prime +cause of poor performance. The two causes are advection dominance and time +variance, which have been identified from the case studies. We can conclude +that there is not always a requirement for deep and bulky layers in the archi- +tecture. The criticality lies in the way it is posed for optimization and the +optimizability. Gradient pathology must be taken care of. Deeper layers give +the potential to capture extremely strong non-linearity in high dimensional + +Success +2 +1 +u +t=o +-1 +-2 +t= 1o +0 +100 +200 +300 +400 +500 +X +Overlapped snapshots for: +0 =xxxx n×s00'0 +xx n×s0'0+x n×n×s++n– +10 +PINN for Dynamical PDEs +and strongly coupled system of equations. Also, specifically for time variance +characteristic we should use recurrent neural networks, especially reservoir net- +works which are in fact light weight but performs better. ”Physics-Informed +Recurrent Neural Networks (PIRNN) is the right path for solving dynamical +and chaotic problems”. +References +[1] J. Strikwerda, “Front Matter,” Finite Difference Schemes and Par- +tial Differential Equations, Second Edition, pp. i–xii, Jan. 2004, doi: +10.1137/1.9780898717938.fm. +[2] Y. +Kuramoto, +“Diffusion-Induced +Chaos +in +Reaction +Systems,” +Progress of Theoretical Physics Supplement, pp. 346–367, 1978, doi: +10.1143/ptps.64.346. +[3] G. I. Sivashinsky, “Nonlinear analysis of hydrodynamic instability in lam- +inar flames—I. Derivation of basic equations,” Acta Astronautica, no. +11–12, pp. 1177–1206, Nov. 1977, doi: 10.1016/0094-5765(77)90096-0. +[4] I. E. Lagaris, A. Likas, and D. I. Fotiadis, “Artificial neural networks for +solving ordinary and partial differential equations,” IEEE Transactions on +Neural Networks, no. 5, pp. 987–1000, 1998, doi: 10.1109/72.712178. +[5] S. Rout, V. Dwivedi, and B. Srinivasan, “ Numerical Approximation +in CFD Problems Using Physics Informed Machine Learning,” ArXiv, +Nov. 2021, doi: 10.48550/arXiv.2111.02987. Master’s Thesis 2019, Indian +Institute of Technology Madras +[6] G. Cybenko, “Approximation by superpositions of a sigmoidal function,” +Mathematics of Control, Signals, and Systems, no. 4, pp. 303–314, Dec. +1989, doi: 10.1007/bf02551274. +[7] K. Hornik, M. Stinchcombe, and H. White, “Universal approximation +of an unknown mapping and its derivatives using multilayer feedfor- +ward networks,” Neural Networks, no. 5, pp. 551–560, Jan. 1990, doi: +10.1016/0893-6080(90)90005-6. +[8] M. Raissi, P. Perdikaris, and G. E. Karniadakis, “Physics-informed neural +networks: A deep learning framework for solving forward and inverse prob- +lems involving nonlinear partial differential equations,” Journal of Compu- +tational Physics, pp. 686–707, Feb. 2019, doi: 10.1016/j.jcp.2018.10.045. +[9] M. Raissi, P. Perdikaris, and G. Karniadsakis, “Physics Informed Deep +Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential +Equations,” ArXiv, Nov. 2017, doi: 10.48550/arXiv.1711.10561. + +– +PINN for Dynamical PDEs +11 +[10] S. Patankar, Numerical Heat Transfer and Fluid Flow (Computational +Methods in Mechanics Thermal Sciences). CRC Press, 1980. +[11] D. J. Gauthier, E. Bollt, A. Griffith, and W. A. S. Barbosa, “Next gen- +eration reservoir computing,” Nature Communications, no. 1, Sep. 2021, +doi: 10.1038/s41467-021-25801-2. +[12] S. Wang, Y. Teng, and P. Perdikaris, “Understanding and Mitigating +Gradient Flow Pathologies in Physics-Informed Neural Networks,” SIAM +Journal on Scientific Computing, no. 5, pp. A3055–A3081, Jan. 2021, doi: +10.1137/20m1318043. +[13] S. Wang, S. Sankaran, and P. Perdikaris, “Respecting causality is all you +need for training physics-informed neural networks,” arXiv.org, Mar. 2022, +doi: 10.48550/arXiv.2203.07404. +[14] A. Chattopadhyay, P. Hassanzadeh, and D. Subramanian, “Data-driven +predictions of a multiscale Lorenz 96 chaotic system using machine-learning +methods: reservoir computing, artificial neural network, and long short- +term memory network,” Nonlinear Processes in Geophysics, no. 3, pp. +373–389, Jul. 2020, doi: 10.5194/npg-27-373-2020. +[15] E. Haber and L. Ruthotto, “Stable architectures for deep neural net- +works,” Inverse Problems, no. 1, p. 014004, Dec. 2017, doi: 10.1088/1361- +6420/aa9a90. +[16] B. Chang, L. Meng, E. Haber, F. Tung, and D. Begert, “ Multi- +level Residual Networks from Dynamical Systems View,” arXiv.org, 2018. +https://arxiv.org/abs/1710.10348. +[17] N. A. K. Doan, W. Polifke, and L. Magri, “Physics-informed echo state +networks,” Journal of Computational Science, p. 101237, Nov. 2020, doi: +10.1016/j.jocs.2020.101237. + diff --git a/nNE3T4oBgHgl3EQf6wum/content/tmp_files/load_file.txt b/nNE3T4oBgHgl3EQf6wum/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..dea463d8d6bc597c74f9cd7acd1aa47448d23737 --- /dev/null +++ b/nNE3T4oBgHgl3EQf6wum/content/tmp_files/load_file.txt @@ -0,0 +1,420 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf,len=419 +page_content='PINN for Dynamical Partial Differential Equations is Not Training Deeper Networks Rather Learning Advection and Time Variance Siddharth Rout Department of Mechanical Engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' The University of British Columbia, Vancouver, BC, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Contributing authors: sidrout@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content='ubc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content='ca;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Abstract The concepts and techniques of physics-informed neural networks (PINNs) is studied and limitations are identified to make it effi- cient to approximate dynamical equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Potential working research domains are explored for increasing the robustness of this technique for the solvability of partial differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' It is identified that PINNs potentially fails to stronger advection and longer time duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Also, optimization function and constraint posing needs to be smarter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Even a shallow network is good for a lot of problems while power- ful deeper network fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Reservoir computing based recurrent neural network architecture is recommended to solve dynamical problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Keywords: Dynamical Systems, Chaos, Physics Informed Neural Networks, PINN, Kuramoto–Sivashinsky Equation, ODE solution, PDE solution, Differential Equations, Recurrent Neural Network, Reservoir Computing 1 Introduction The solution of differential equations have been an important topic for the almost every field of the world, say it be finance, mechanics, meteorology etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Starting from the days of Newton and Leibniz solving differential equations have been core to developments in this world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Not all differential equations are 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content='04793v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content='comp-ph] 12 Jan 2023 – 2 PINN for Dynamical PDEs solvable by hand and initiate limitations, especially when multiple independent variables come into the equation or build system of equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' However, solving these equations are need of the hour and hence we shifted our focus from exact solutions to approximate solutions as function operations and transformations are limiting in concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' There arose the generation for solving equations by using the basic definitions or first principles of limits, discretization and numer- ical analysis[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Popular such techniques are finite element methods, finite volume methods, finite difference approximations etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' These methods are gen- eralizable and that is the benefit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Using these techniques we can solve almost any equation for any geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' But with increased complexity, there are a couple of problems accompanied like how good the approximation is and com- putation expense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Discretization gives us a long list of simplified approximate equations to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Though we know how to solve however solving them will require us to use computers to do those hectic mathematical calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' For a stable and accurate solution, a lot of time and energy are consumed in the process[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Though we can calculate a lot of things in this world but we have lack of time, resources, money and problems could be endless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' A type of differ- ential equation called dynamical systems is tough to solve after a certain range in time, there are reasons to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Dynamical systems are tough to solve as the solution my bifurcate and that shall make the system chaotic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' In a chaotic sys- tem, a minute change in the initial condition or equation coefficients will have drastically different outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' This is sometimes referred by ’The Butterfly Effect’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' The aim of this project is to develop a function approximation method that can potentially replace computationally expensive solvers for dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' The one dimensional Kuramoto-Sivashinsky equation is solved for trials and research[2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Function approximations or analytical solutions are better known for their light-weight[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' These techniques can get rid of the three primary types of errors that are evident in full order discretized approximation, namely instability, inaccuracy and shift[5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' A system of dynamical systems is mainly sensitive inaccuracy due to insanely evident sensitivity to initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' A benefit of function approximation is the ability of correction and reproduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Also, an added benefit is extrapolability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Among the function approximators, neural networks have been excellent candidate as universal approximators[6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' In the past decade, deep neural networks which are basically multiple layers of neural networks have been used for various complex regression problems due to its ability to capture high dimensional strong non-linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' These models are fitted to the data directly as input to output mapping[7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Neural networks can also be trained to differential equations by optimizing the residuals after fitting to random points in the input domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Such models are called physics-informed neural networks or popularly called PINNs[4, 5, 8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Solving partial differential equations using PINNs are universally accepted by the scientific community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' There are a plenty of advantages of these meth- ods over conventional methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' The major once are ability to solve wide – PINN for Dynamical PDEs 3 category of problems that were tough to solve otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Moreover, these meth- ods do not require meshing and discretization which is sometimes a tough task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Another advantage unlike other analytical models does not require data from full order solutions to set the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' However, being newly devel- oped these techniques are not robust enough for solving complex equations like hyperbolic equations, strongly non-linear equations, strongly advective equations[5], chaotic dynamical systems, coupled system of equations, shock wave equations etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' 2 Dynamical Partial Differential Equations Activities in the world is mostly the four dimensions, three in space and one in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Each new dimension adds a layer of complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Dynamical systems generally mean functions that describe the dependence of state of a system with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Henri Poincare was the first one to identify the special behaviour of dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' The theory of these dynamical systems is highly relevant in studying behaviour of complex dynamics, usually in the form of differential equations, which makes it continuous dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' The major points of focus in this domain are the attractors, chaos, fractals and bifurcations that explains the long term behaviour of states qualitatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' This helps under- standing evolution of dynamical events like turbulence, storm, mixing fluids, environment change, economic changes, planetary motions and many more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' The necessary applications of dynamical systems theory are to find struc- tural stability, Lyapunov time, bifurcation points, position tracking and quantitative approximations which one way or the other determines the pre- dictability of the state at a particular time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Predictability of dynamical systems is a tough job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Before the advent of computing machines prediction required sophisticated mathematical techniques that were specific to specific classes of dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' These are sometimes among the toughest differential equations to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Also considering other factors mentioned above, accurate prediction is a great deal for these kinds of systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' 3 Case Selection The cases below point out two major difficulties in solving differential equations clearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' The concepts are explained with reference to the terms and frame- work of the equation mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' The two equations shall be good examples to analyse the theory of PINNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content='1 1D Steady Advection-Diffusion Equation The differential equation below is the governing equation for steady one dimensional flow of combined advection and diffusion phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' αux = uxx – 4 PINN for Dynamical PDEs Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' 1 Solution of steady state advection-diffusion Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' 2 Solution of One dimensional Kuramoto-Sivashinsky equation If we notice, α is the weight for the advection term in the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' That means larger is α, more dominant is the advection effect, which introduces directional characters and hence discrete approximation becomes tougher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Figure 1 shows the difference in the solution with advection dominance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Higher is the Peclet number, more dominant is the advection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' The figure compares the solution for Pe 1 and 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Hence the numerical integration sees rapidly growing error that makes the solution unstable and inaccurate[10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Hence, a major class of higher order methods are developed to tackle this particular issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content='2 1D Kuramoto–Sivashinsky Equation The equation below is the one dimensional Kuramoto-Sivashinsky equation ut + αuux + βuxx + γuxxxx = 0 The linear form of it is as below: ut + αux + βuxx + γuxxxx = 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content='0 Pe=1 Pe=50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content='0Kuramoto-Sivashinskydynamics 100 80 60 40 20 0 0 20 40 60 80 100 120 x– PINN for Dynamical PDEs 5 This equation has the advection, diffusion and dissipation effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' It is one of the equations where the solution is extremely sensitive to the initial condition[2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' The higher order terms in the expansion of the difference equation are very much relevant and hence sensitive for error propagation in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Figure 2 shown the solution of a case of one dimensional KS equation on Julia code developed by Mahatab Lak et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' from the University of New Hampshire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' 4 Neural Networks as Universal Function Approximator George Cybenko was the one to prove arbitrary width case using neural net- works with sigmoid activation in 1989[6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Later in the same year, Hornik et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' proved multi-layer feed-forward networks are universal approximators[7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Multi-layer artificial neural networks are composites of weighted sum of inputs passed through non-linear(activation) functions like tanh(), sigmoid(), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' This enables an extremely potent highly non-linear function with large num- ber of trainable parameters(weights and biases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' This makes it universal approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' 5 Physics-informed Neural Networks Informing the physics to a neural network is a concept brought up by Lagaris et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' al[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' in the late 1990s by using neural network as a trial function to solve dif- ferential equations by reduction of the residuals using AutoGrad (an automatic differentiation technique) at various points in the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' The boundary con- straints are forced into the neural network function by modifying it mannually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' In 2017, Raissi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' proceed by using more accurate automatic differentia- tion and deeper networks to approximate tougher problems[8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' The novelty in their work comes from the way they pose the loss function to reduce the residual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' They did not manually force the constraints by modifying the trial function rather let the trial function fit to the boundary and initial constraints by adding the mean square error from the data points satisfying the conditions as summed constraints to mean squared residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' This makes the technique very much generalizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Almost all the differential equations could be posed to be solved using this technique, which they named PINNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content='1 Advancements in PINNs The unknown solution u(t, x) is represented by a deep neural network uθ(t, x), where θ denotes all tunable parameters of the network (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=', weights and biases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' The physics-informed model can be trained by minimizing the following loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' L(θ) = λicLic(θ) + λbcLbc(θ) + λrLr(θ), where – 6 PINN for Dynamical PDEs Here � xi ic �Nic i=1 , � ti bc, xi bc �Nbc i=1 and � ti r, xi r �Nr i=1 can be the vertices of a fixed mesh or points that are randomly sampled at each iteration of a gradient descent algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' The hyper-parameters {λic, λbc, λr} allow the flexibility of assigning a different learning rate to each individual loss term in order to balance their interplay during model training[12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' These weights may be user-specified or tuned automatically during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content='2 Advantages of PINNs over Other Neural Networks The major advantage of this technique is that it does not require physical data to train the analytical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Moreover, the technique is generalizable in the sense that with the exactly same concept various equations could be solved[4, 5, 8, 9, 12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Previous models required alteration of the learning function depending on number of equations coupled, boundary conditions etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' in order to force the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Being a strong approximating function the three major kinds of error, namely instability, inaccuracy and shifting errors, could be taken care of simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' These problems are taken care individually in finite numerical techniques[5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' This kind of technique is an excellent candidate for robust higher order methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Neural network is not just an approximator but rather a smart approximator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Hence, depending on the local physical property it can act differently with switching kind of behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' In particular for the case of dynamical system, with time progression the integrated error increases too rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' PINNs being an optimization technique for regression, training to measured physical data points as additional loss terms or regularization can be used for correcting the approximating function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' 6 Experiments with the selected cases The qualitative property in dynamical problems is strong translational vari- ance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Hence, the two major causes for PINNs performing poorly are advection dominance and time variance which are demonstrated below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content='1 PINNs for 1-D Steady Advection-Diffusion Peclet number is a good non-dimensional parameter to scale advective domi- nance over diffusion characteristic of an equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' It is the ratio of advective transport rate to diffusive transport rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' In our problem we can quantify that by the ratio of coefficient of advective term times the length of domain space to coefficient of diffusive term, so that is α in our particular case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' PINN can solve this problem but there is a limit set by advection characters in the differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' No matter how deeper and how sophisticated we make the neural network it is not possible to solve for problems with Peclet number more than something close to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Figure 3 shows the results noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' For lower Peclet num- ber problems, it is noticed that it is not necessary to have deeper and wider layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' A good thing is in conventional numerical techniques fails for problems set with this value more than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Hence, these schemes can be used for shape – PINN for Dynamical PDEs 7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' 3 Optmimized architecture for solving one dimensional steady advection diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' 4 Minimized losses with various activation functions while solving one dimensional steady advection diffusion where C is coefficient of advective term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' 5 Loss trend while solving using various optimizer for one dimensional steady advection diffusion functions that let larger grids with similar accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' The work and figures in this section has been sourced from by 2019 thesis titled ”Numerical Approxi- mation in CFD Problems Using Physics Informed Machine Learning”[5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Parametric analysis is done to understand more about the performance and effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' The impact of change in the non-linearity function as well as loss optimization algorithm is studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Figure number 4 records the loss values in a tabular form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Among various non-linearity functions tanh() and tan() performs consistently as well as better than sigmoid().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' However, tanh() wins this game clearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Figure number 5 shows the trend of loss value with iteration for vari- ous optimizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' The neural network is trained using various optimizers however L-BFGS-B and SL-SQP perfoms better than other specialised optimization techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' BFGS performs remarkably better than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' The prime reason could be the fact that these optimizers are second order and perform better in the case of optimizing multiobjective functions than other techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' How- ever, it can be noted that first order techniques like Adam performs decently Architecture Valid for: Layers NeuronsperLayer 1 2 Pe<3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content='8 2 2 Pe<4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content='8 3 2 Pe<6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content='3 3 3 Pe<6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content='5 3 10 Pe<7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content='6 Models Loss C Neuron Optimize Sigmoid Tanh SoftPlus Log ArcTan s r Sigmoid 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content='5 2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content='75615e-04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content='47605e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content='77046e-03 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content='31412e-03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content='11537e-04 Adam 1 2 1.' metadata={'source': 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+page_content='77785e-02 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content='77693e-03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content='16140e-04CG BFGS-B COBYLA SLSQP TNC (sso) Not an easy optimization!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' 5 Notall of themaresufficientlygood atfollowingthegradientpathologies 6 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Iterations– 8 PINN for Dynamical PDEs even if they don’t fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' An important thing that can be observed that the col- located PDE residuals and the fitting losses for constraints are clearly not in similar scale which is not typical for regular data-driven neural network learn- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Also the gradient pathology is not smooth and hence tough for other optimizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Guided optimization like hill climbing helps in few cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content='2 PINNs for 1-D Kuramoto-Sivashinsky There are recent publications demonstrating how PINNs can solve one dimensional Kuramoto-Sivashinsky equation which is among the standard dynamical equation that can turn chaotic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' However, there is a small intersect- ing set of coefficients for advection, diffusion and dissipation terms for which this works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' As demonstrated in the previous case the dominance of advection toughens the optimization of the PINN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Here, not only it is dynamical but also non-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' It has various orders of spatial derivatives making it difficult especially when the system becomes chaotic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' CausalPINN by Wang et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' is considered the state of art PINN[13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' They have rightly identified the problem of multiobjctive optimization due to differ- ence in scales of residuals and constraints stated by Rout et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' [5] and devised weights for each loss terms by normalizing with the each cumulative loss terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' They are first people to be able to solve 1D KS Equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' We can validate their model using their open source code provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Figure 6 shows how CausalPINN perfoms with time for the case provided in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' It can be noticed that PINN can now solve complex dynamical problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' The initial sine smoothly curls as expected while the constraints are obeyed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Like initial curve is a neat sine function and the boundaries are continuously at base zero(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' However, the typical equation where all the coefficients are considered 1 is taken for reg- ular study of the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' The state of art PINN fails to optimize even after an effort equivalent to one-day’s run-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' The net loss is recorded to be 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content='263 where the constraint loss was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content='0016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' This suggests the difficulty in fitting to the PDE where as PINN could manage to obey the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' The residual loss is noted to be 1808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content='506, where its weight in the loss function disappear from the scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' This clearly shows the issue of multi-objective optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' 7 Observations and Conclusion Based on the experiments and analysis a few points could be commented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Sim- ple addition or weighed by mean addition of squared loss terms of residuals at collocation points and fitting points directly for constraints have different scale and orders of magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' It is one of the major issue as it can lead to non-pareto optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' It is also noticed that sometime the loss func- tion gets stuck at local optima and gradient pathologies are tough and uneven in the parameter hyperspace[5, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Hence, stochastic first order optimizers work otherwise higher order optimizers suitable for constrained optimizations like SQP and BFGS works while other fails[5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' A better representation of loss – PINN for Dynamical PDEs 9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' 6 Snapshots of velocity(U) along the x-axis overlapped through time for one dimensional Kuramoto-Sivashinsky equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' function like appropriate or adaptive weighted losses can help.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Otherwise con- straints could be forcefully enforced by modified architecture or trial function, like explained by Isaac Lagaris et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' al[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Specifically, in the context of prob- lems in dynamical systems recurrent neural networks(RNN) could prove to be better candidate over simple deep networks[14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' A concrete reasoning has been provided by Eldad Haber et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' where RNNs can be proved to be in the form of differential equations and hence fit into the theory to learn the dynamical differential equations better[15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Especially, for chaotic systems reservoir computing have been proved to be performing better[11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Reservoir comput- ers are a class of RNNs where the intermediate nodes are randomly arranged and connected[11, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' They have random recurrent connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' The interme- diate nodes are jumbled and entangled however they are connected out to the output layer linearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' The entangled architecture makes it tough to backprop- agate and hence only the final layer of weights are trainable for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' The trainable output layers makes the effective non-linear network linear with respect to trainable parameters hence conserving strong non-linearity while making it easy to train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Apparently, RNN can also be introduced with PINNs kind of loss definition to solve chaotic problems for turbulence and extreme event prediction[17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Ultimately, we can justify the errors and specify the right path to solve a dynamical system of partial differential equations by identifying the two prime cause of poor performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' The two causes are advection dominance and time variance, which have been identified from the case studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' We can conclude that there is not always a requirement for deep and bulky layers in the archi- tecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' The criticality lies in the way it is posed for optimization and the optimizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Gradient pathology must be taken care of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=" Deeper layers give the potential to capture extremely strong non-linearity in high dimensional Success 2 1 u t=o 1 2 t= 1o 0 100 200 300 400 500 X Overlapped snapshots for: 0 =xxxx n×s00'0 +xx n×s0'0+x n×n×s++n– 10 PINN for Dynamical PDEs and strongly coupled system of equations." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Also, specifically for time variance characteristic we should use recurrent neural networks, especially reservoir net- works which are in fact light weight but performs better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' ”Physics-Informed Recurrent Neural Networks (PIRNN) is the right path for solving dynamical and chaotic problems”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' References [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content=' Strikwerda, “Front Matter,” Finite Difference Schemes and Par- tial Differential Equations, Second Edition, 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content='jocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} +page_content='101237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQf6wum/content/2301.04793v1.pdf'} diff --git a/pdAyT4oBgHgl3EQfzfkv/content/tmp_files/2301.00701v1.pdf.txt b/pdAyT4oBgHgl3EQfzfkv/content/tmp_files/2301.00701v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..618b1955a43efa995f3327a3abd538d2a325c9a1 --- /dev/null +++ b/pdAyT4oBgHgl3EQfzfkv/content/tmp_files/2301.00701v1.pdf.txt @@ -0,0 +1,3108 @@ +arXiv:2301.00701v1 [math.OC] 2 Jan 2023 +Noname manuscript No. +(will be inserted by the editor) +Fast convex optimization via closed-loop time scaling of gradient dynamics +Hedy Attouch · Radu Ioan Bot¸ · Dang-Khoa Nguyen +January 3, 2023 +Abstract In a Hilbert setting, for convex differentiable optimization, we develop a general framework for +adaptive accelerated gradient methods. They are based on damped inertial dynamics where the coefficients +are designed in a closed-loop way. Specifically, the damping is a feedback control of the velocity, or of +the gradient of the objective function. For this, we develop a closed-loop version of the time scaling and +averaging technique introduced by the authors. We thus obtain autonomous inertial dynamics which involve +vanishing viscous damping and implicit Hessian driven damping. By simply using the convergence rates for +the continuous steepest descent and Jensen’s inequality, without the need for further Lyapunov analysis, +we show that the trajectories have several remarkable properties at once: they ensure fast convergence of +values, fast convergence of the gradients towards zero, and they converge to optimal solutions. Our approach +leads to parallel algorithmic results, that we study in the case of proximal algorithms. These are among +the very first general results of this type obtained using autonomous dynamics. +Keywords fast convex optimization; damped inertial dynamic; time scaling; averaging; closed-loop +control; Nesterov and Ravine algorithms; Hessian driven damping; proximal algorithms +AMS subject classification 37N40, 46N10, 49M30, 65B99, 65K05, 65K10, 90B50, 90C25 +1 Introduction +In a real Hilbert space H, we develop a dynamic approach to the rapid resolution of convex optimization +problems which relies on inertial dynamics whose damping is designed as a closed-loop control. We consider +the minimization problem +min {f(x) : x ∈ H} , +(1) +where, throughout the paper, we make the following assumptions on the function f to be minimized +(A) +� +f : H → R is a convex function of class C1; S = argminH f ̸= ∅; +∇f is Lipschitz continuous on the bounded sets of H. +(2) +Our study is part of the close links between dissipative dynamical systems and optimization algorithms, +the latter being obtained by temporal discretization of the continuous dynamics. Our study comes as a +natural extension of the authors’ previous work [4] where the technique of time scaling and averaging was +Hedy Attouch +IMAG, Univ. Montpellier, CNRS, Montpellier, France +E-mail: hedy.attouch@umontpellier.fr +Radu Ioan Bot¸ +Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria, +E-mail: radu.bot@univie.ac.at +Dang-Khoa Nguyen +Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria, +E-mail: dang-khoa.nguyen@univie.ac.at + +used in an open-loop way, giving rise to non-autonomous damped inertial dynamics with fast convergence +properties. In the present paper, we take advantage of the simplicity and flexibility of this technique to +develop it in a closed-loop way. This will give rise to autonomous damped inertial dynamics with fast +convergence properties. Recall that the low-resolution ODE obtained by Su, Boyd, and Cand`es [30] of the +accelerated gradient method of Nesterov, together with the corresponding high-resolution ODE [8], [28] +(which involves an additional Hessian driven damping term) are non-autonomous dynamics, the coefficient +of viscous friction being of the form α/t. Our study therefore opens a new path in the field of first-order +adaptive optimization methods. +1.1 Time scale and averaging: the open-loop approach +Let us first briefly explain the time scaling and averaging method in the open-loop case on a model example +(see [4] for more details). Then we will look at how to develop a corresponding closed-loop approach. As +the basic starting dynamic, we consider the continuous steepest descent +(SD) +˙z(s) + ∇f(z(s)) = 0, +(3) +for which we have the classical convergence result +f (z (s)) − inf +H f = o +�1 +s +� +as s → +∞. +Then, we make the change of time variable s = τ(t) in (SD), where τ(·) is an increasing function from R+ +to R+, continuously differentiable, and satisfying limt→+∞ τ(t) = +∞. Setting y(t) := z(τ(t)), we get +˙y(t) + ˙τ(t)∇f(y(t)) = 0. +(4) +The convergence rate becomes +f(y(t)) − inf +H f = o +� 1 +τ(t) +� +as t → +∞. +(5) +Taking τ(·) which grows faster than the identity, makes the solution trajectories unchanged but travelled +faster. The price to pay is that (4) is a non-autonomous dynamic in which the coefficient in front of the +gradient term tends to infinity as t → +∞. This prevents from using gradient methods to discretize it. +Recall that for gradient methods the step size has to be less than or equal to twice the inverse of the +Lipschitz constant of the gradient. To overcome this difficulty we come with the second step of our method +which is averaging. Let us attach to y(·) the new function x : [t0, +∞[→ H defined by +˙x(t) + +1 +˙τ(t)(x(t) − y(t)) = 0, +(6) +with x(t0) = x0 given in H. We shall explain further the averaging interpretation. Equivalently +y(t) = x(t) + ˙τ(t) ˙x(t). +(7) +By temporal derivation of (7) we get +˙y(t) = ˙x(t) + ¨τ(t) ˙x(t) + ˙τ(t)¨x(t). +(8) +Replacing y(t) and ˙y(t) as given by (7) and (8) in (4), we get +¨x(t) + 1 + ¨τ(t) +˙τ(t) +˙x(t) + ∇f +� +x(t) + ˙τ(t) ˙x(t) +� += 0. +(9) +In doing so, we passed from the first-order differential equation (4) to the second-order differential equation +(9), with the advantage that now the coefficient in front of the gradient is fixed. Let us now particularize +the time scale τ(·). Taking +τ(t) = +t2 +2(α − 1), +(10) +gives 1+¨τ(t) +˙τ(t) += α +t , and the corresponding dynamic with implicit Hessian driven damping +¨x(t) + α +t ˙x(t) + ∇f +� +x(t) + +t +α − 1 ˙x(t) +� += 0. +(11) +2 + +In this dynamic, the Hessian driven damping appears in an implicit form. This type of dynamic was initiated +in [1], see also [22] for a related autonomous system in the case of a strongly convex function f. The rationale +justifying the use of the term “implicit” comes from the observation that by a Taylor expansion (as t → +∞ +we have t ˙x(t) → 0 which justifies the use of Taylor expansion), we have +∇f +� +x(t) + +t +α − 1 ˙x(t) +� +≈ ∇f(x(t)) + +t +α − 1∇2f(x(t)) ˙x(t), +thus making the Hessian damping appear indirectly in (11). Because of its important role in attenuating +the oscillations, several recent studies have been devoted to inertial dynamics combining the asymptotic +vanishing damping with the geometric Hessian-driven damping (coined sometimes Newton-type inertial +dynamics); see e.g., [2,11,8,9,10,15,16,21,28]. In turn, the corresponding algorithms, among which IGAHD +enjoys several favorable properties, introduce a correction term in the Nesterov accelerated gradient method +(see [24,25]) which reduces the oscillatory aspects. +Note that in (11) the coefficient of the Hessian damping is proportional to the inverse of the viscosity +damping. Thus asymptotically when the viscous damping tends towards zero, and therefore can cause +many small oscillations to appear, the coefficient of the Hessian driven damping tends towards infinity, and +therefore has an effective effect on the attenuation of the oscillations. This is the situation considered by +Attouch-Bot¸-Nguyen [4], who obtained convergence rates comparable to those associated with the Nesterov +accelerated gradient method. A major advantage of this approach is that there is no need to do a Lyapunov +analysis, we only use the classical convergence rate for the continuous steepest descent. Moreover, the +convergence of the trajectories is a direct consequence of the known results for the steepest descent. +1.2 Closed-loop control +The idea is to exploit the time scaling and averaging method and the fact that (SD) provides several +quantities which are increasing and converge to +∞ as t → +∞, so which are eligible for time scaling. This +will enable us to perform time scaling and averaging in a closed-loop way. Indeed, in (SD), the velocity +and the norm of the gradient are monotonically decreasing to zero. So, the idea is to use their inverse for +defining the time scaling. Specifically, in a first result we are going to define the derivative of the time +scaling τ(·) as a function of the inverse of the speed. This means acceleration of the time scaling when the +speed decreases. Following this approach, we will obtain in Theorem 5 the following model result. +Theorem 1 Suppose that f : H → R satisfies (A). Let us choose the positive parameters according to q > 0, +p ≥ 1, and γ > 1. Let x: [t0, +∞[ → H be a solution trajectory of the following system + + + + + + + + + + + + + + + + + + + +¨x(t) + (1 + γ) ˙τ(t)2 − τ(t)¨τ(t) +τ(t) ˙τ(t) +˙x(t) + γ ˙τ(t)2 +τ(t) ∇f +� +x(t) + 1 +γ +τ(t) +˙τ(t) ˙x(t) +� += 0 +τ (t) − 1 +qq +� +t0 + +� t +t0 +[λ (r)] +1 +q dr +�q += 0 +[λ (t)]p |˙τ (t)|p−1 +����∇f +� +x (t) + 1 +γ +τ(t) +˙τ(t) ˙x (t) +����� +p−1 += 1. +(12) +Then we have the fast convergence of values: as t → +∞ +f(x(t)) − inf +H f = o +� +1 +t1+q− 1 +p +� +. +(13) +Moreover, the solution trajectory x(t) converges weakly as t → +∞, and its limit belongs to S = argminf. +As a special case, take p = 1, q = 2. Then, the last equation of (12) gives λ(t) ≡ 1. According to this, the +second equation of (12) gives τ(t) = t2 +4 , and we find a case with time scaling in an open-loop form. After +elementary calculation, the first equation of (12) is written as +¨x(t) + α +t ˙x(t) + α − 1 +2 +∇f +� +x(t) + +t +α − 1 ˙x(t) +� += 0, +with α = 2γ + 1 > 3, and the convergence rate of the values becomes +f(x(t)) − inf +H f = o +� 1 +t2 +� +. +(14) +3 + +We therefore recover the results obtained by the authors in the case of the open loop, giving the optimal +convergence rates for general convex differentiable optimization. This inertial formulation may seem at first +glance complicated. Indeed it is equivalent to the first-order system in time and space + + + + + + + + + + + + + + + + + + + +˙y (t) + ˙τ (t) ∇f (y (t)) += 0 +˙x(t) + γ ˙τ(t) +τ(t)x(t) − γ ˙τ(t) +τ(t)y(t) += 0 +τ (t) − 1 +qq +� +t0 + +� t +t0 +[λ (r)] +1 +q dr +�q += 0 +[λ (t)]p ∥ ˙y (t)∥p−1 += 1, +(15) +whose temporal discretization provides corresponding optimization algorithms, see Theorem 11. +1.3 Link with the existing literature +Contrary to the rich literature that has been devoted to non-autonomous damped inertial methods and +their links with the fast first-order optimization algorithms for general convex optimization (in particular +the Nesterov accelerated gradient method), only a small number of papers have been devoted to these +questions, based on autonomous methods. Indeed the heavy ball method of Polyak only provides the +asymptotic convergence rate 1/t for general convex functions. The idea is therefore to see if we can mimic +the fast convergence properties of the Su, Boyd, and Cand`es dynamic model (see [30]) of the Nesterov +accelerated gradient method, using autonomous dynamics. A natural idea is to design the damping term, +on which is based the optimization properties of the system, in a closed-loop way. In this direction, we can +mention the following contributions. +a) Our study has a natural link with works devoted to regularized Newton methods for solving monotone +inclusions (and (1) in particular). Given a general maximally monotone operator A : H ⇒ H, to overcome +the ill-posed character of the continuous Newton method, in line with [13], Attouch, Redont and Svaiter +have studied in [12] the following closed-loop dynamic version of the Levenberg-Marquardt method +� +v(t) ∈ A(x(t)) +∥v(t)∥γ ˙x(t) + β ˙v(t) + v(t) = 0. +When γ > 1, they showed the well-posedness of the above system, and analyzed its convergence properties. +When A = ∇f this system writes +∥∇f(x(t))∥γ ˙x(t) + β∇2f(x(t)) ˙x(t) + ∇f(x(t)) = 0. +Thus, its inertial version +¨x(t) + ∥∇f(x(t))∥γ ˙x(t) + β∇2f(x(t)) ˙x(t) + ∇f(x(t)) = 0 +falls within the framework of our study with the damping equal to a closed-loop control of the norm of +the gradient. The techniques developed in [12] are particularly useful for studying the well-posedness of +dynamics with implicit features. +b) Although significantly different, our approach has several points in common with the article by Lin +and Jordan [19]. In this article, the authors study the closed-loop dynamical system + + + + + + + + + + + + + + + +˙y (t) + ˙τ (t) ∇f (x (t)) += 0 +˙x (t) + ˙τ(t) +τ(t) (x (t) − y (t)) + [ ˙τ(t)]2 +τ(t) ∇f (x (t)) += 0 +τ (t) − 1 +4 +�� t +0 +� +λ (r)dr + c +�2 += 0 +[λ (t)]p ∥∇f (x (t))∥p−1 += θ, +(16) +where c > 0 and 0 < θ < 1. The corresponding second-order in time damped inertial system writes as +follows +¨x(t) + 2 [˙τ(t)]2 − τ(t)¨τ(t) +τ(t) ˙τ(t) +˙x(t) + [ ˙τ(t)]2 +τ(t) ∇2f (x(t)) ˙x(t) + ˙τ(t)( ˙τ(t) + ¨τ(t)) +τ(t) +∇f (x(t)) = 0. +(17) +4 + +In the above system, the Hessian driven damping comes in an explicit way because of the structure of +the first equation which differs from the structure of the continuous steepest descent. In contrast, in our +approach, the first equation is the rescaled continuous steepest descent, and the Hessian driven damping +comes implicitly. Let us highlight some advantages of our approach. +• Our system is introduced in a natural way by using the time scaling and averaging method. This +makes unnecessary to perform a Lyapunov analysis for the inertial system. It has already been done for +the continuous steepest descent. This results in a significantly simplified mathematical analysis. +• Our dynamic model contains an additional parameter q which, when q = 2, gives the setting of Lin +and Jordan, and which, when judiciously tuned, gives better convergence rates. +• Our approach provides the weak convergence of the trajectories to optimal solutions. +We shall return later to the precise comparison between the two systems. +c) In [3], Attouch, Bot¸ and Csetnek study the convergence properties of the Autonomous Damped +Inertial Gradient Equation +(ADIGE) +¨x(t) + G +� +˙x(t), ∇f(x(t)),∇2f(x(t)) +� ++ ∇f(x(t)) = 0, +where the damping term G +� +˙x(t), ∇f(x(t)),∇2f(x(t)) +� +acts as a closed-loop control. They pay particular +attention to the role played by the parameter r > 1 in the asymptotic convergence analysis of the dynamic +¨x(t) + ∥ ˙x(t)∥r−2 ˙x(t) + ∇f(x(t)) = 0. +They show that the case r = 2 separates the weak damping (r > 2) from the strong damping (r < 2), hence +the importance of this case. These questions have also been considered by Haraux and Jendoubi in [17]. +d) In [29], Song, Jiang, and Ma develop an interesting technique for accelerating high-order algorithms +under general H¨older continuity assumption. Their continuous-time framework reduces to an inertial system +without Hessian-driven damping in the first-order setting, which has been proven to be an inaccurate +surrogate. Although underlying their approach, the acceleration via time scaling, the averaging technique, +and the closed-loop tuning of the coefficients are not clearly identified. +1.4 Organization of the paper +After a general presentation of the article in the introduction, we provide in Section 2 a general estimate of +the time scaling for the continuous steepest descent when it is defined in a closed-loop way. This is crucial +for the rest of the paper. Then we specialize these results to situations of particular interest, and examine in +details the case of closed-loop systems induced respectively by velocities, and then by gradients. In Section +3, which is the main part of the paper, we develop the next important step in our approach, which is the +averaging operation. This provides accelerated damped inertial dynamics that are autonomous and with +fast convergence properties. Finally, in Section 4 we analyze the fast convergence properties of proximal +algorithms which come naturally from the temporal discretization of the continuous dynamics. +2 Closed-loop time scaling of the steepest descent +2.1 Formulation of the closed-loop time scaling +Given t0 ≥ 0, q > 0, and p ≥ 1, the time scale function τ : [t0, +∞[ → R++ is defined by + + + + + + + + + + + +˙y (t) + ˙τ (t) ∇f (y (t)) += 0 +τ (t) − 1 +qq +� +t0 + +� t +t0 +[λ (r)] +1 +q dr +�q += 0 +[λ (t)]p [G (y (t))]p−1 += 1, +(18) +where G(·) is a given positive, continuous function that depends on the information of the trajectory y(·). +This general formalism allows us to unify the various situations coming from different choices of the time +scaling as a feedback control of the state of the system. For example G may be a function of y, ˙y, f (y) , ∇f (y) +and/or any mixture combination of them. Then the function λ(·) is continuous and it links the coefficient +5 + +of ∇f, namely ˙τ(·), with the solution trajectory y(·). +As a useful result, note that for every t ≥ t0, it holds +˙τ (t) = +1 +qq−1 +�� t +t0 +[λ (r)] +1 +q dr + t0 +�q−1 +[λ (t)] +1 +q += [τ (t)] +q−1 +q +[λ (t)] +1 +q > 0. +(19) +Moreover, the relations (18) allow us to cover the open-loop case. In particular, when p = 1 it holds λ (t) = 1 +for every t ≥ t0. This yields for every q > 0 +τ (t) = +� +t +q +�q +. +Taking further q := 1, then τ (t) becomes the regular time in variable t, namely τ (t) = t for every t ≥ t0. +Let us specify the interpretation of (18) as a steepest descent dynamic which is rescaled in time in a +closed-loop way. +Proposition 1 Suppose that f : H → R satisfies (A). Let t0 ≥ 0, q > 0, p ≥ 1 and y: [t0, +∞[ → H be a +solution trajectory of the system (18). Suppose that +lim +s→+∞ τ (s) = +∞. +Then y(·) is a solution trajectory of a time rescaled continuous steepest descent (SD), as described below: +Let s0 = τ (t0) and z : [s0, +∞[ → H be a solution trajectory of the following system +˙z (s) + ∇f (z (s)) = 0. +(20) +Then we have +y (t) = z (τ (t)) +∀t ≥ t0, +and there exists a continuously differentiable function σ: [s0, +∞[ → R++ such that +z (s) = y (σ (s)) +∀s ≥ s0. +Proof We already interpreted how to go from a solution trajectory z(·) of (SD) to the closed-loop system +above via the time scaling function τ(·). Let us now show the reverse direction. Let y: [t0, +∞[ → H +be a solution trajectory of (18). We have that λ is continuous and positive on [t0, +∞[, therefore τ is a +monotonically increasing function, hence injective. On the other hand, we have t0 = τ (t0) = +� +t0 +q +�q +. Since +by assumption limt→+∞ τ (t) = +∞, this means τ is a continuous function whose image contains [s0, +∞[, +hence surjective. Combining these premises, we have shown that τ is a bijection, which means it is invertible. +Set σ ≡ τ −1 and make the change of time variable t := σ (s) in (18). Let us define +z (s) = y (σ (s)) = y +� +τ −1 (s) +� +. +Then by the chain rule, we have +˙z (s) = ˙y +� +τ −1 (s) +� +1 +˙τ (τ −1 (s)) = ˙y (σ (s)) +1 +˙τ (σ (s)). +This leads to +˙z (s) + ∇f (z (s)) = 0. +In other words, z : [s0, +∞[ → H is a solution trajectory of (SD). +⊓⊔ +The above assertion allows us to transfer the convergence results of (SD) to some closed-loop systems. +In particular, given a time scaling function τ(·) as described above, by making the change of time variable +s := τ (t), we obtain the following results from Theorem 12 in the appendix applied to the unperturbed +continuous steepest descent system. +� +∞ +t0 +τ (t) +˙τ (t) ∥ ˙y (t)∥2 dt < +∞, +(21) +� +∞ +t0 +τ (t) ˙τ (t) ∥∇f (y (t))∥2 dt < +∞, +(22) +f(y(t)) − inf +H f = o +� 1 +τ(t) +� +as t → +∞, +(23) +∥∇f (y (t))∥ = o +� 1 +τ(t) +� +as t → +∞. +(24) +6 + +2.2 Lower bound estimate of the time scaling τ(t) +As key ingredient of our approach, the next step is to establish a lower bound for τ(t) in terms of t. This +will reflect the acceleration of our dynamic via time scaling and allow us to achieve fast convergence rates. +For this, we will need the following technical lemma, which can be seen as a nonlinear Gronwall result. +Lemma 1 Suppose that there exists C0 > 0 and b > a ≥ 0 such that +� t +t0 +[τ (r)]a [λ (r)]−b dr ≤ C0 < +∞ +∀t ≥ t0. +(25) +Then there exists C1 > 0 such that +τ (t) ≥ C1 (t − t0) +qb+1 +b−a +∀t ≥ t0. +(26) +Proof Let t ≥ t0 be fixed. By applying the H¨older inequality, we get +� t +t0 +[τ (r)] +a +qb+1 dr ≤ +�� t +t0 +[τ (r)]a [λ (r)]−b dr +� +1 +qb+1 �� t +t0 +[λ (r)] +1 +q dr +� +qb +qb+1 +≤ C +1 +qb+1 +0 +� +t0 + +� t +t0 +[λ (r)] +1 +q dr +� +qb +qb+1 += +� +C0qqb� +1 +qb+1 [τ (t)] +b +qb+1 . +(27) +If a = 0 then (26) follows immediately. From now on suppose that a > 0, so that the inequality (27) can be +rewritten as +� t +t0 +[τ (r)] +a +qb+1 dr ≤ +� +C0qqb� +1 +qb+1 � +[τ (t)] +a +qb+1 +� b +a +(28) +The arguments are now adapted from [19], which is inspired by the proof of Bihari-LaSalle inequality. Let +Cq,b := +� +C0qqb� +1 +qb+1 > 0 +and +A (t) := +� t +t0 +[τ (r)] +a +qb+1 dr +∀t ≥ t0, +so that (28) becomes +A (t) ≤ Cq,b +� ˙A (t) +� b +a +∀t ≥ t0 +or, equivalently, +C− a +b +q,b ≤ [A (t)]− a +b ˙A (t) +∀t ≥ t0. +Integrating from t0 to t, we obtain +C− a +b +q,b (t − t0) ≤ +� +1 − a +b +� � +[A (t)]1− a +b − [A (t0)]1− 1 +b +� +≤ [A (t)]1− a +b ≤ +� +Cq,b [τ (t)] +b +qb+1 +�1− a +b += C +b−a +b +q,b +[τ (t)] +b−a +qb+1 , +where the last inequality comes from (27). Since b > a, the conclusion follows. +⊓⊔ +Let us now particularize our results to some model situations. +2.3 Closed-loop control of (SD) via the velocity +Theorem 2 Suppose that f : H → R satisfies (A). Let q > 0, p ≥ 1 and y: [t0, +∞[ → H be a solution trajectory +of the following system + + + + + + + + + + + +˙y (t) + ˙τ (t) ∇f (y (t)) += 0 +τ (t) − 1 +qq +� +t0 + +� t +t0 +[λ (r)] +1 +q dr +�q += 0 +[λ (t)]p ∥ ˙y (t)∥p−1 += 1. +(29) +Then the following statements are satisfied: +(i) (convergence of values) +f (y (t)) − infH f = o +� +t−(1+q− 1 +p)� +as t → +∞. +(ii) (convergence of gradients towards zero) +∥∇f (y (t))∥ = o +� +t−(1+q− 1 +p)� +as t → +∞. +7 + +(iii) (integral estimate of the velocities) +� +∞ +t0 +t(1+ 1 +q − 1 +pq) ∥ ˙y (t)∥2+ p−1 +pq dt < +∞. +(iv) The solution trajectory y(t) converges weakly as t → +∞, and its limit belongs to S = argminf. +Proof When p = 1, we recover the open loop case with the time scaling function τ(t) = +� +t +q +�q +. The result is +a direct consequence of Theorem 12. Therefore, from now on we only consider the case p > 1. Recall that +from (21) we have +� +∞ +t0 +τ (t) +˙τ (t) ∥ ˙y (t)∥2 dt < +∞. +(30) +By using successively the definition of λ, and relation (19), we obtain +τ (t) +˙τ (t) ∥ ˙y (t)∥2 = τ (t) +˙τ (t) [λ (t)]− +2p +p−1 = [τ (t)] +1 +q [λ (t)]− 1 +q − +2p +p−1 +∀t ≥ t0. +According to the two above results we get +� +∞ +t0 +[τ (r)] +1 +q [λ (r)]− 1 +q − +2p +p−1 dr < +∞. +We are now in position to apply Lemma 1 with p > 1, a := 1 +q and b := 1 +q + +2p +p−1. We have +qb + 1 +b − a = +2 + 2pq +p−1 +2p +p−1 += p − 1 + pq +p += 1 + q − 1 +p, +and therefore there exists some constant C1 > 0 such that +τ (t) ≥ C1 (t − t0)1+q− 1 +p +∀t ≥ t0. +(31) +This leads to limt→+∞ τ (t) = +∞. Therefore, according to Proposition 1, we can extract the results from +Theorem 12 and the corresponding formulas (21), (22), (23). Specifically, we obtain +(i) for the values +f (y (t)) − inf +H f = o +� 1 +τ(t) +� += o +� +1 +t1+q− 1 +p +� +, +(ii) for the gradients +∥∇f (y (t))∥ = o +� 1 +τ(t) +� += o +� +1 +t1+q− 1 +p +� +. +(iii) for the velocities: we start from (30), i.e. � +∞ +t0 +τ (t) +˙τ (t) ∥ ˙y (t)∥2 dt < +∞, that we evaluate as follows: +τ (t) +˙τ (t) ∥ ˙y (t)∥2 = +τ(t) +τ(t) +q−1 +q λ(t) +1 +q +∥ ˙y (t)∥2 += τ(t) +1 +q +λ(t) +1 +q +∥ ˙y (t)∥2 += τ(t) +1 +q ∥ ˙y (t)∥2+ p−1 +pq +∀t ≥ t0. +According to (31) we deduce that +� +∞ +t0 +t(1+ 1 +q − 1 +pq) ∥ ˙y (t)∥2+ p−1 +pq dt < +∞. +(iv) Let us finally examine the convergence of the solution trajectories. We know that the solution +trajectory of the continuous steepest descent converges weakly when t → +∞, and its limit belong to +S = argminH f ̸= ∅; see Theorem 12 in appendix. With our notation we therefore have that z(s) converges +weakly when s → +∞. Since τ(t) → +∞ as t → +∞, we immediately deduce that y(t) = z(τ(t) converges +weakly as t → +∞, and its limit belong to S = argminH f ̸= ∅. This completes the proof. +⊓⊔ +8 + +2.4 Closed-loop control of (SD) via the norm of gradient +We develop an analysis parallel to that of the previous section, replacing speed control with gradient control. +Theorem 3 Suppose that f : H → R satisfies (A). Let q ≥ +1 +2, p ≥ 1 and y: [t0, +∞[ → H be a solution +trajectory of the following system + + + + + + + + + + + +˙y (t) + ˙τ (t) ∇f (y (t)) += 0 +τ (t) − 1 +qq +� +t0 + +� t +t0 +[λ (r)] +1 +q dr +�q += 0 +[λ (t)]p ∥∇f (y (t))∥p−1 += 1. +(32) +Then the following statements are satisfied: +(i) (convergence of values) +f (y (t)) − infH f = o � +t−pq� +as t → +∞. +(ii) (convergence of gradients towards zero) +∥∇f (y (t))∥ = o � +t−pq� +as t → +∞. +(iii) (integral estimate of the gradients) +� +∞ +t0 +tpq(2− 1 +q) ∥∇f (y (t))∥2+ p−1 +pq dt < +∞. +(iv) The solution trajectory y(t) converges weakly as t → +∞, and its limit belongs to S = argminf. +Proof Again, we only consider the case p > 1. We know from (22) that +� +∞ +t0 +τ (t) ˙τ (t) ∥∇f (y (t))∥2 dt < +∞. +By using successively the definition of λ, and the relation (19), we obtain +τ (t) ˙τ (t) ∥∇f (y (t))∥2 = τ (t) ˙τ (t) [λ (t)]− +2p +p−1 = [τ (t)]2− 1 +q [λ (t)] +1 +q − +2p +p−1 +∀t ≥ t0. +Therefore +� +∞ +t0 +[τ (t)]2− 1 +q [λ (t)] +1 +q − +2p +p−1 dt < +∞. +Let us apply Lemma 1 with a := 2 − 1 +q and b = +2p +p−1 − 1 +q. We have b > a, a ≥ 0 for q ≥ 1 +2, and +qb + 1 +b − a = +2pq +p−1 +2p +p−1 − 2 = pq. +Therefore +τ (t) ≥ C1 (t − t0)pq +∀t ≥ t0. +(33) +This gives limt→+∞ τ (t) = +∞. According to Proposition 1, we can extract the results from Theorem 12 +and the corresponding formulas (21), (22), (23). Specifically, we obtain +(i) for the values +f (y (t)) − inf +H f = o +� 1 +τ(t) +� += o +� 1 +tpq +� +; +(ii) for the gradients +∥∇f (y (t))∥ = o +� 1 +τ(t) +� += o +� +t−pq� +; +(iii) for the integral estimate of the gradients: we start from (30) +� +∞ +t0 +τ (t) ˙τ (t) ∥∇f (y (t))∥2 dt < +∞, +that we evaluate as follows: +τ (t) ˙τ (t) ∥∇f (y (t))∥2 = τ (t) [τ (t)] +q−1 +q +[λ (t)] +1 +q ∥∇f (y (t))∥2 += τ(t)2− 1 +q λ(t) +1 +q ∥∇f (y (t))∥2 += τ(t)2− 1 +q ∥∇f(y(t))∥2− p−1 +pq +∀t ≥ t0. +According to (33) we deduce that� +∞ +t0 +tpq(2− 1 +q) ∥∇f (y (t))∥2+ p−1 +pq dt < +∞. +(iv) The convergence of the solution trajectory follows from an argument similar to that of the previous +section. This completes the proof. +⊓⊔ +9 + +Remark 1 a) We thus achieved our first goal which was to accelerate the convergence properties of the +continuous steepest descent using closed-loop time scaling. For example, concerning the convergence rate +of the values, we passed from the convergence rate 1/t for the steepest descent to 1/t(1+q− 1 +p) when the +closed-loop control acts on the velocity, and 1/tpq in the case of the gradient. Clearly, by playing with +the parameters p and q we can get arbitrary fast convergence results. The same observation holds for the +convergence of the gradients towards zero. +b) By introducing a time scale function τ(·) which grows faster than the identity (i.e. τ(t) ≥ t) either +in open-loop or closed-loop, we have thus accelerated the continuous steepest descent dynamic. The price +to pay is that we no longer have an autonomous dynamic in (4), with as major drawback the fact that +the coefficient in front of the gradient term tends towards infinity as t → +∞. This prevents from using +gradient methods to discretize it. Recall that for gradient methods, the step size has to be less than or +equal to twice the inverse of the Lipschitz constant of the gradient. To overcome this, we come with the +second step of our method which is averaging. +3 Accelerated gradient systems with closed-loop control of the damping +3.1 General results concerning time scale and averaging +We will prove the following general result which puts forward a damped inertial dynamics which comes by +time scale and averaging of the continuous steepest descent. Then we will specialize it and consider time +scale obtained in a closed-loop way, and thus cover the two model situations. +Theorem 4 Suppose that f : H → R satisfies (A). Let γ > 1, and let τ : [t0, +∞[→ R++ be an increasing func- +tion, continuously differentiable, such that limt→+∞ τ(t) = +∞. Let x: [t0, +∞[ → H be a solution trajectory +of the following second-order differential equation +¨x(t) + (1 + γ) ˙τ(t)2 − τ(t)¨τ(t) +τ(t) ˙τ(t) +˙x(t) + γ ˙τ(t)2 +τ(t) ∇f +� +x(t) + 1 +γ +τ(t) +˙τ(t) ˙x(t) +� += 0. +(34) +Then we have the convergence rate of the values: as t → +∞ +f (x(t)) − inf +H f = o +� 1 +τ (t) +� +, +(35) +and x(t) converges weakly as t → +∞, and its limit belongs to S = argminf. +Proof a) We first interpret x as coming from the time scale and averaging of the continuous steepest descent. +We start from y(·) solution of +˙y (t) + ˙τ (t) ∇f (y (t)) = 0. +(36) +According to the time scale analysis developed in (5) we have +f (y (t)) − inf +H f = o +� 1 +τ (t) +� +as t → +∞. +This means there exists a positive function ε which satisfies limt→+∞ ε (t) = 0 and +f (y (t)) − inf +H f = ε (t) +τ (t) +∀t ≥ t0. +(37) +Let us define the time averaging process as the transformation from y to x according to the formula +˙x(t) + γ ˙τ(t) +τ(t)x(t) = γ ˙τ(t) +τ(t)y(t), +(38) +where γ > 1. Equivalently +y(t) = x(t) + 1 +γ +τ(t) +˙τ(t) ˙x(t). +(39) +By derivating y(·) we get +˙y (t) = ˙x (t) + 1 +γ +τ(t) +˙τ(t) ¨x(t) + 1 +γ +˙τ(t)2 − τ(t)¨τ(t) +˙τ(t)2 +˙x(t). +(40) +10 + +Replacing ˙y (t) by this expression in the constitutive rescaled steepest descent equation (36), we get +˙x (t) + 1 +γ +τ(t) +˙τ(t) ¨x(t) + 1 +γ +˙τ(t)2 − τ(t)¨τ(t) +˙τ(t)2 +˙x(t) + ˙τ (t) ∇f +� +x(t) + 1 +γ +τ(t) +˙τ(t) ˙x(t) +� += 0. +Equivalently +1 +γ +τ(t) +˙τ(t) ¨x(t) + (1 + γ) ˙τ(t)2 − τ(t)¨τ(t) +γ ˙τ(t)2 +˙x(t) + ˙τ (t) ∇f +� +x(t) + 1 +γ +τ(t) +˙τ(t) ˙x(t) +� += 0. +After multiplication by γ ˙τ(t) +τ(t) we get +¨x(t) + (1 + γ) ˙τ(t)2 − τ(t)¨τ(t) +τ(t) ˙τ(t) +˙x(t) + γ ˙τ(t)2 +τ(t) ∇f +� +x(t) + 1 +γ +τ(t) +˙τ(t) ˙x(t) +� += 0. +(41) +b) Let us now come to the corresponding estimate of the convergence rates with x(t) instead of y(t). The +idea is to express x as an average of y, and then conclude thanks to Jensen’s inequality. Set +b(t) = ˙τ(t) +τ(t) ≥ 0 +(42) +B(t) = +� t +t0 +b(u)du = +� t +t0 +˙τ(u) +τ(u)du = ln +� +τ(t) +τ(t0) +� +. +(43) +Therefore +eB(t) = τ(t) +τ(t0). +(44) +In order to express x in terms of y, we need to integrate the first-order linear differential equation (38) +which is written equivalently as follows +˙x(t) + γb(t)x(t) = γb(t)y(t). +After multiplying by eγB(t), we get equivalently +eγB(t) ˙x(t) + γb(t)eγB(t)x(t) = γb(t)eγB(t)y(t), +that is, +d +dt +� +eγB(t)x(t) +� += γb(t)eγB(t)y(t). +After integration we get +eγB(t)x(t) = eγB(t0)x(t0) + γ +� t +t0 +b(u)eγB(u)y(u)du. +According to eγB(t0) = e0 = 1 we get +x(t) = e−γB(t)x(t0) + γe−γB(t) +� t +t0 +b(u)eγB(u)y(u))du += e−γB(t)y(t0) + γe−γB(t) +� t +t0 +b(u)eγB(u)y(u)du, +(45) +where the last equality follows from the choice of the Cauchy data y(t0) = x(t0). Then, observe that x(t) +can be simply written as follows +x(t) = +� t +t0 +y(u) dµt(u), +(46) +where µt is the positive Radon measure on [t0, t] defined by +µt = e−γB(t)δt0 + γb(u)eγ(B(u)−B(t))du. +(47) +Precisely, in (47), δt0 is the Dirac measure at t0, and b(u)eB(u)−B(t)du is the measure with density +b(u)eB(u)−B(t) with respect to the Lebesgue measure on [t0, t]. According to +γe−γB(t) +� t +t0 +b(u)eγB(u)du = 1 − e−γB(t), +we have that µt is a positive Radon measure on [t0, t] whose total mass is equal to 1. It is therefore a +probability measure, and x(t) is obtained by averaging the trajectory y(·) on [t0, t] with respect to µt. +11 + +From there, let us show how to deduce fast convergence properties for the so defined trajectory x(·). +According to the convexity of f, and Jensen’s inequality, we deduce that +f +�� t +t0 +y(u) dµt(u) +� +− inf +H f = (f − inf +H f) +�� t +t0 +y(u)dµt(u) +� +≤ +� t +t0 +� +f(y(u)) − inf +H f +� +dµt(u) += +� t +t0 +ε (u) +τ (u)dµt(u), +where the last inequality above comes from (37). According to the definition of µt (see (47)) and the +formulation of x(t) (see (46)), we deduce that +f (x(t)) − inf +H f ≤ ε (t0) +τ (t0)e−γB(t) + γe−γB(t) +� t +t0 +ε (u) +τ (u)b(u)eγB(u)du. +Equivalently, +τ (t) +� +f (x(t)) − inf +H f +� +≤ ε (t0) +� τ (t) +τ (t0) +�1−γ ++ γτ (t) e−γB(t) +� t +t0 +ε (u) +τ (u)b(u)eγB(u)du. +(48) +Since γ > 1 and limt→+∞ τ(t) = +∞, it holds +lim sup +t→+∞ +τ (t) +� +f (x(t)) − inf +H f +� +≤ γ lim sup +t→+∞ +τ (t) e−γB(t) +� t +t0 +ε (u) +τ (u)b(u)eγB(u)du. +It is therefore enough to show that +lim sup +t→+∞ +� +γτ (t) e−γB(t) +� t +t0 +ε (u) +τ (u)b(u)eγB(u)du +� +≤ 0. +In order to prepare for integration by parts, note that +γb(u)eγB(u) = d +du +� +eγB(u)� +and +˙τ (u) +[τ (u)]2−γ = d +du +� +1 +γ − 1 +1 +[τ (t)]1−γ +� +. +Given an arbitrary η > 0 we consider Tη > t0 such that ε (u) ≤ η for every u ≥ Tη. Therefore, for every +t ≥ Tη, by integration by parts and by taking into consideration the relations (42)-(44), we get +γτ (t) e−γB(t) +� t +t0 +ε (u) +τ (u)b(u)eγB(u)du += γτ (t) e−γB(t) +�� Tη +t0 +ε (u) +τ (u)b(u)eγB(u)du + +� t +Tη +ε (u) +τ (u)b(u)eγB(u)du +� +≤ τ (t) e−γB(t) +� +γ +� Tη +t0 +ε (u) +τ (u)b(u)eγB(u)du + ηγ +� t +Tη +1 +τ (u)b(u)eγB(u)du +� += τ (t) e−γB(t) +� +γ +� Tη +t0 +ε (u) +τ (u)b(u)eγB(u)du + +η +τ (t)eγB(t) − +η +τ (Tη)eγB(Tη) + η +� t +Tη +˙τ (u) +[τ (u)]2 eγB(u)du +� += τ (t) e−γB(t) +� +γ +� Tη +t0 +ε (u) +τ (u)b(u)eγB(u)du + +η +τ (t)eγB(t) − +η +τ (Tη)eγB(Tη) + +η +[τ (t0)]γ +� t +Tη +˙τ (u) +[τ (u)]2−γ du +� += τ (t) e−γB(t) +� +γ +� Tη +t0 +ε (u) +τ (u)b(u)eγB(u)du + +η +τ (t)eγB(t) − +η +τ (Tη)eγB(Tη) + η [τ (t0)]−γ +γ − 1 +� +1 +[τ (t)]1−γ − +1 +[τ (Tη)]1−γ +�� +≤ +� +γ +� Tη +t0 +ε (u) +τ (u)b(u)eγB(u)du +� +τ (t) e−γB(t) + η + +η +γ − 1 +≤ C [τ (t)]1−γ + +ηγ +γ − 1. +Since limt→+∞ τ(t) = +∞, and γ > 1, we obtain +lim sup +t→+∞ +� +γτ (t) e−γB(t) +� t +t0 +ε (u) +τ (u)b(u)eγB(u)du +� +≤ +ηγ +γ − 1. +(49) +This being true for every η > 0, we infer +f (x(t)) − inf +H f = o +� 1 +τ (t) +� +. +(50) +12 + +c) For trajectories convergence, we take advantage of the fact that the solution trajectory z (·) of the +continuous steepest descent converges weakly towards a solution x∗ ∈ S. Since limt→+∞ τ(t) = +∞, this +immediately implies that y(t) = z(τ(t)) converges weakly to x∗ as s → +∞. In other words, for each v ∈ H +⟨y (t) , v⟩ → ⟨x∗, v⟩ as t → +∞. +To pass from the convergence of y to that of x, we use the interpretation of x as an average of y. The +convergence then results from the general property which says that convergence entails ergodic convergence. +Let us make this precise. Using again that limt→+∞ τ(t) = +∞, we have +x(t) ∼ γe−γB(t) +� t +t0 +b (u) eγB(u)y (u) du = +γ +[τ(t)]γ +� t +t0 +˙τ (u) [τ(u)]γ−1 y(u)du. +After elementary calculus, we just need to prove that if a(·) is a positive real-valued function which verifies +limu→+∞ a(u) = 0, then limt→+∞ A(t) = 0, where +A(t) = +γ +[τ(t)]γ +� t +t0 +˙τ (u) [τ(u)]γ−1 a(u)du. +Given an arbitrary η > 0, let us take Tη such that t0 < Tη and a(u) ≤ η for u ≥ Tη. For t > Tη, we have +A(t) = +γ +[τ(t)]γ +� Tη +t0 +˙τ (u) [τ(u)]γ−1 a(u)du + +γ +[τ(t)]γ +� t +Tη +˙τ (u) [τ(u)]γ−1 a(u)du +≤ +γ +[τ(t)]γ +� Tη +t0 +˙τ (u) [τ(u)]γ−1 a(u)du + ητ(t0). +Letting t converge to +∞ we get +lim sup +t→+∞ +A(t) ≤ ητ(t0). +This being true for any η > 0, we infer that limt→+∞ A(t) = 0, which completes the proof. +⊓⊔ +Remark 2 By taking γ := α−1 +2 +and τ (t) := +t2 +2(α−1), equation (41) becomes (see [4]) +¨x(t) + α +t ˙x(t) + ∇f +� +x(t) + +t +α − 1 ˙x(t) +� += 0. +We have γ > 1 ⇐⇒ α > 3, which is in accordance with the convergence results attached to Nesterov method. +3.2 Damped inertial system via closed-loop control of the velocity +Let us now examine the model situation where the time scaling is defined in a closed-loop way as a feedback +control of the velocity. Completing this construction with the averaging process, as described as above, we +get that (x, y): [t0, +∞[ → H × H is a solution trajectory of the following algebraic-differential system + + + + + + + + + + + + + + + + + + + + + + + +˙y (t) + ˙τ (t) ∇f (y (t)) += 0 +˙x(t) + γ ˙τ(t) +τ(t)x(t) − γ ˙τ(t) +τ(t)y(t) += 0 +τ (t) − 1 +qq +� +t0 + +� t +t0 +[λ (r)] +1 +q dr +�q += 0 +[λ (t)]p ∥ ˙y (t)∥p−1 += 1. +(51) +By specializing Theorem 4 to this situation we get the following result. +Theorem 5 Suppose that f : H → R satisfies (A). Let q > 0, p ≥ 1, γ > 1 and x: [t0, +∞[ → H be a solution +trajectory of the following system + + + + + + + + + + + + + + + + + + + +¨x(t) + (1 + γ) ˙τ(t)2 − τ(t)¨τ(t) +τ(t) ˙τ(t) +˙x(t) + γ ˙τ(t)2 +τ(t) ∇f +� +x(t) + 1 +γ +τ(t) +˙τ(t) ˙x(t) +� += 0 +τ (t) − 1 +qq +� +t0 + +� t +t0 +[λ (r)] +1 +q dr +�q += 0 +[λ (t)]p |˙τ (t)|p−1 +����∇f +� +x (t) + 1 +γ +τ(t) +˙τ(t) ˙x (t) +����� +p−1 += 1. +(52) +13 + +Then we have the fast convergence of values: as t → +∞ +f(x(t)) − inf +H f = o +� +1 +t1+q− 1 +p +� +. +(53) +Moreover, the solution trajectory x(t) converges weakly as t → +∞, and its limit belongs to S = argminf. +Proof We showed in the proof of Theorem 4 how to pass from (51) to (52). Conversely, let x(·) be a solution +trajectory of the damped inertial dynamic (52). Let us show that by setting +y (t) = 1 +γ +τ (t) +˙τ (t) ˙x (t) + x (t) , +then (x, y): [t0, +∞[ → H × H is a solution trajectory of + + + + + + + + + + + + + + + + + + + + + + + +˙y (t) + ˙τ (t) ∇f (y (t)) += 0 +˙x(t) + γ ˙τ(t) +τ(t)x(t) − γ ˙τ(t) +τ(t)y(t) += 0 +τ (t) − 1 +qq +� +t0 + +� t +t0 +[λ (r)] +1 +q dr +�q += 0 +[λ (t)]p ∥ ˙y (t)∥p−1 += 1. +(54) +Indeed, by taking the time derivative of y(·), as given by the second equation of (54), we get +˙y (t) = 1 +γ +τ (t) +˙τ (t) ¨x (t) + 1 +γ +� +1 + γ − τ (t) ¨τ (t) +[˙τ (t)]2 +� +˙x (t) += 1 +γ +τ (t) +˙τ (t) +� +¨x (t) + (1 + γ) [ ˙τ(t)]2 − τ(t)¨τ(t) +τ(t) ˙τ(t) +˙x(t) +� += − ˙τ (t) ∇f +� +x (t) + 1 +γ +τ (t) +˙τ (t) ˙x (t) +� += − ˙τ (t) ∇f (y (t)) . +This gives the first equation in (54) and +[λ (t)]p ∥ ˙y (t)∥p−1 = [λ (t)]p | ˙τ (t)|p−1 +����∇f +� +x (t) + 1 +γ +τ (t) +˙τ (t) ˙x (t) +����� +p−1 += 1. +This shows the equivalence of the two systems. According to Theorem 2, and formula (31), there exists a +constant C1 > 0 such that +τ (t) ≥ C1 (t − t0)1+q− 1 +p . +(55) +Therefore limt→+∞ τ(t) = +∞. According to Theorem 4 we deduce +f (x(t)) − inf +H f = o +� +1 +t1+q− 1 +p +� +, +(56) +and the convergence of the trajectory. +⊓⊔ +3.3 Damped inertial system via closed-loop control of the gradient +We proceed in parallel to the previous section to obtain the following result. +Theorem 6 Suppose that f : H → R satisfies (A). Let q > 0, p ≥ 1, γ > 1, and x: [t0, +∞[ → H be a solution +trajectory of the following system + + + + + + + + + + + + + + + + + + + +¨x(t) + (1 + γ) ˙τ(t)2 − τ(t)¨τ(t) +τ(t) ˙τ(t) +˙x(t) + γ ˙τ(t)2 +τ(t) ∇f +� +x(t) + 1 +γ +τ(t) +˙τ(t) ˙x(t) +� += 0 +τ (t) − 1 +qq +� +t0 + +� t +t0 +[λ (r)] +1 +q dr +�q += 0 +[λ (t)]p +����∇f +� +x (t) + 1 +γ +τ (t) +˙τ (t) ˙x (t) +����� +p−1 += 1. +(57) +14 + +Then we have the fast convergence of values: as t → +∞ +f(x(t)) − inf +H f = o +� 1 +tpq +� +. +(58) +Moreover, the solution trajectory x(t) converges weakly as t → +∞, and its limit belongs to S = argminf. +Proof Let x(·) be a solution trajectory of the damped inertial dynamic (57). Let us show show that by +setting +y (t) = 1 +γ +τ (t) +˙τ (t) ˙x (t) + x (t) , +then (x, y): [t0, +∞[ → H × H is a solution trajectory of + + + + + + + + + + + + + + + + + + + + + + + +˙y (t) + ˙τ (t) ∇f (y (t)) += 0 +˙x(t) + γ ˙τ(t) +τ(t)x(t) − γ ˙τ(t) +τ(t)y(t) += 0 +τ (t) − 1 +qq +� +t0 + +� t +t0 +[λ (r)] +1 +q dr +�q += 0 +[λ (t)]p ∥∇f (y (t))∥p−1 += 1. +(59) +Indeed, by the same argument as for the velocity case, we get +˙y (t) = − ˙τ (t) ∇f (y (t)) . +This gives the first equation in (59) and +[λ (t)]p ∥∇f(y (t))∥p−1 = [λ (t)]p +����∇f +� +x (t) + 1 +γ +τ (t) +˙τ (t) ˙x (t) +����� +p−1 += 1. +This shows the equivalence of the two systems. According to Theorem 3, and formula (33), there exists a +constant C1 > 0 such that +τ (t) ≥ C1 (t − t0)pq . +(60) +Therefore from Theorem 4 we deduce, as t → +∞ +f (x(t)) − inf +H f = o +� 1 +tpq +� +, +(61) +and the convergence of the trajectory. +⊓⊔ +3.4 Comparison with the Lin-Jordan approach +In [19], the authors study the second-order closed-loop dynamical system + + + + + + + + + + + + + + + +¨x (t) + +�2 ˙τ (t) +τ (t) − ¨τ (t) +˙τ (t) +� +˙x (t) + [ ˙τ (t)]2 +τ (t) ∇2f (x (t)) ˙x (t) + ˙τ (t) [ ˙τ (t) + ¨τ (t)] +τ (t) +∇f (x (t)) += 0 +τ (t) − 1 +4 +�� t +0 +� +λ (t)dr + c +�2 += 0 +[λ (t)]p ∥∇f (x (t))∥p−1 += θ, +(62) +whose first-order reformulation reads + + + + + + + + + + + + + + + +˙y (t) + ˙τ (t) ∇f (x (t)) += 0 +˙x (t) + ˙τ(t) +τ(t) (x (t) − y (t)) + [ ˙τ(t)]2 +τ(t) ∇f (x (t)) += 0 +τ (t) − 1 +4 +�� t +0 +� +λ (t)dr + c +�2 += 0 +[λ (t)]p ∥∇f (x (t))∥p−1 += θ, +(63) +where c > 0 and 0 < θ < 1 are given parameters. See also [20] for some extensions to monotone inclusions. +a) In [19], the authors obtained the following convergence rate of function values +15 + +f (x (t)) − inf +H f = O +� +1 +t +3p+1 +2 +� +as t → +∞. +Note that the last two equations in (63) are nothing else than those in (32) with q := 2. +For comparison, in our approach the convergence rate of the values obtained in Theorem 6 when q = 2 is +f (x (t)) − inf +H f = o +� 1 +t2p +� +which is better for every p > 1. +b) Let us now compare the convergence estimates of the gradients. In [19], the authors obtain the +integral estimate +� +∞ +t0 +t +3p+1 +2 +∥∇f (x (t))∥ +p+1 +p +dt < +∞, +which leads to +inf +t0≤σ≤t ∥∇f (x (σ))∥ = O +� +t− 3p +2 +� +as t → +∞. +In our approach, the right variable to consider is y(t), instead of x(t). According to (22) we have +� +∞ +t0 +τ (t) ˙τ (t) ∥∇f (y (t))∥2 dt < +∞. +Since q = 2, according to (19) we have +˙τ (t) = [τ (t)] +1 +2 [λ (t)] +1 +2 . +Therefore +τ (t) ˙τ (t) ∥∇f (y (t))∥2 = τ (t) +3 +2 [λ (t)] +1 +2 ∥∇f (y (t))∥2 = τ (t) +3 +2 ∥∇f (y (t))∥2− p−1 +2p . +Since τ(t) ≥ Ct2p, we deduce that +� +∞ +t0 +t3p ∥∇f (y (t))∥ +3p+1 +2p +dt < +∞. +which leads to +inf +t0≤σ≤t ∥∇f (y (σ))∥ = O +� +t−2p� +as t → +∞. +Again, our approach gives a better convergence rate than [19]. Let us also specify that our analysis provides +the convergence of the trajectories, which is an open question for [19]. Moreover, since our approach is +consistent with the steepest continuous descent, it can naturally be extended to the non-smooth case, and +to the case of cocoercive operators, as it was done in the open-loop case in [4]. +3.5 The limiting case γ = 1 +Our previous results are valid under the assumption γ > 1. It is a natural question to examine the limiting +case γ = 1. Close examination of the proof of the theorem reveals a slight change in the integration procedure +and a logarithm factor appears. The corresponding result obtained is written as follows. +Theorem 7 Suppose that f : H → R satisfies (A). Let x: [t0, +∞[ → H be a solution trajectory of the following +second-order differential equation +¨x(t) + 2 [˙τ(t)]2 − τ(t)¨τ(t) +τ(t) ˙τ(t) +˙x(t) + [˙τ(t)]2 +τ(t) ∇f +� +x(t) + τ(t) +˙τ(t) ˙x(t) +� += 0 +(64) +where τ : [t0, +∞[ → R++ is an increasing function, continuously differentiable, and satisfying limt→+∞ τ(t) = ++∞. Then we have the convergence rate of the values: as t → +∞ +f (x(t)) − inf +H f = o +�ln (τ (t)) +τ (t) +� +, +(65) +16 + +and the solution trajectory x(t) converges weakly as t → +∞, and its limits belongs to S = argminf. +Suppose moreover that there exists some θ > 0 and C1 > 0 such that for t sufficiently large +(A)asymp +τ(t) ≥ C1 (t − t0)θ . +(66) +Then we have the fast convergence of values: as t → +∞ +f (x(t)) − inf +H f = o +�ln (t) +tθ +� +. +(67) +When specialized to the closed-loop control of the velocity, we obtain +f (x(t)) − inf +H f = o +� ln (t) +t1+q− 1 +p +� +, +(68) +and in the case of the closed-loop control of the gradient +f (x(t)) − inf +H f = o +�ln (t) +tpq +� +. +(69) +So, the convergence rates are a little less good because of the logarithm term. +4 Associated proximal algorithms +4.1 A proximal-explicit discretization +In the following, we present a numerical approach based on a proximal-explicit temporal discretization of +the closed-loop systems investigated in this paper. By proximal-explicit we mean that the function f is +evaluated using a proximal step while the step size sequence (λk)k≥0 and the time scaling sequence (τk)k≥0 +are computed explicitly. This makes our numerical scheme much easier implementable than the numerical +algorithm proposed in [19] as well as the large-step A HPE approach by Monteiro and Svaiter [23] which +are in fact approximations of a proximal-implicit discrete time method. We restrict ourselves to the case +q = 1, which gives ˙τ (t) = λ (t). In this case, the continuous time closed-loop dynamical system is written +as follows + + + +˙y (t) + λ (t) ∇f (y (t)) += 0 +[λ (t)]p [G (y (t))]p−1 += 1. +(70) +Let us describe the general structure of the algorithm which is obtained by a proximal-explicit discretization +of the continuous system (70). +Given yk, yk−1 in H, we first define λk by +[λk]p [G (yk, yk−1)]p−1 = 1. +and consider then an implicit finite difference scheme for the first equation of (70) +yk+1 − yk + λk∇f (yk+1) = 0. +(71) +This gives the following algorithm, called PEAS for Proximal Explicit Algorithm with Adaptive Step Size. +Algorithm 1: Proximal-explicit algorithm with adaptive step size (PEAS) +Input: y0 ̸= y−1 ∈ H +1 for k = 0, 1, · · · do +2 +λk := [G (yk, yk−1)]− p−1 +p +3 +yk+1 := proxλkf (yk) +4 end +Note that (λk)k≥0 is computed explicitly in terms of (yk)k≥0. In other words, the definition of the sequence +(λk)k≥0 is decoupled from the computation of (yk)k≥0. This is different from the method in [19], which +ultimately leads to the large-step A HPE approach by Monteiro and Svaiter in [23]. +17 + +Let us now specify the link between λk and τk. We start from the relation (recall that we take q = 1) +˙τ (t) = λ (t) . +(72) +Then, for every k ≥ 0 we discretize (72) as follows +τk+1 − τk = λk ⇐⇒ τk+1 = λk + τk +(73) +with the convention λ0 := t0 and τ0 := 0, which then yields τk = �k−1 +i=0 λi. +Drawing inspiration from continuous analysis, we will first show that the function value f (yk) − infH f +attains the o +� +1 +τk+1 +� +rate of convergence, and the sequence (yk)k≥0 converges weakly to a solution. Then, +as a crucial result, we will derive a lower bound of τk+1 in terms of k. +The following result emphasizes that the rate of convergence and summability results holds for (yk)k≥0 for +arbitrary step sizes λk that satisfy � +k≥0 λk = +∞. The proof is an adaptation of [4, Theorem 4.1]. +Theorem 8 Let y0 ∈ H, (λk)k≥0 be a given positive sequence satisfying +� +k≥0 +λk = +∞, τ0 = 0 and τk = +�k−1 +i=0 λi for every k ≥ 1. Then for any sequence (yk)k≥0 generated by the proximal algorithm +yk+1 := proxλkf (yk) +∀k ≥ 0, +(74) +the following properties are satisfied: +(i) (summability of function values) +� +k≥0 +λk +� +f (yk+1) − inf +H f +� +< +∞; +(ii) (summability of gradients) +� +k≥0 +τkλk ∥∇f (yk+1)∥2 < +∞; +(iii) (summability of velocities) +� +k≥0 +τk +λk +∥yk+1 − yk∥2 < +∞; +(iv) (convergence of function values) +f (yk+1) − inf +H f = o +� +1 +τk+1 +� +as k → +∞; +(v) (convergence of gradient) +∥∇f (yk+1)∥ = o + + +1 +��k +l=0 τlλl + + as k → +∞; +(vi) the sequence of iterates (yk)k≥0 converges weakly as k → +∞, and its limit belongs to S = argminH f. +Proof Let k ≥ 0 be fixed. Take z∗ ∈ S = argminf. According to (71) and the convexity of f, we deduce that +1 +2 ∥yk+1 − z∗∥2 = 1 +2 ∥yk − z∗∥2 + ⟨yk+1 − z∗, yk+1 − yk⟩ − 1 +2 ∥yk+1 − yk∥2 += 1 +2 ∥yk − z∗∥2 − λk ⟨yk+1 − z∗, ∇f (yk+1)⟩ − 1 +2λ2 +k ∥∇f (yk+1)∥2 +≤ 1 +2 ∥yk − z∗∥2 − λk +� +f (yk+1) − inf +H f +� +− 1 +2λ2 +k ∥∇f (yk+1)∥2 . +(75) +Statement (i) follows from [14, Lemma 5.31]. In addition, the limit limk→+∞ ∥yk − z∗∥ ∈ R exists, which +means that the first condition of the discrete Opial’s lemma is fulfilled. +On the other hand, the sequence (f (yk) − infH f)k≥0 is nonincreasing. Precisely, we have for every k ≥ 0 +� +f (yk) − inf +H f +� +− +� +f (yk+1) − inf +H f +� +≥ ⟨∇f (yk+1) , yk − yk+1⟩ = λk ∥∇f (yk+1)∥2 ≥ 0. +(76) +According to [6, Lemma 22] we get +f (yk+1) − inf +H f = o +� +1 +�k +i=0 λi +� +, +which proves (iv). +Let k ≥ 1. Multiplying both sides of (76) by τk = �k−1 +i=0 λi > 0, then adding the result into (75), we get +τk+1 +� +f (yk+1) − inf +H f +� ++ 1 +2 ∥yk+1 − z∗∥2 ≤ τk +� +f (yk) − inf +H f +� ++ 1 +2 ∥yk − z∗∥2 +− 1 +2λ2 +k ∥∇f (yk+1)∥2 − τkλk ∥∇f (yk+1)∥2 . +18 + +This implies +� +k≥0 +λ2 +k ∥∇f (yk+1)∥2 < +∞ +and +� +k≥1 +τkλk ∥∇f (yk+1)∥2 < +∞, +which yields (ii). From (71), we infer (iii). To deduce (v), it suffices to show that the sequence (∥∇f (yk) ∥)k≥0 +is nonincreasing. Indeed, it follows from (74) and the cocoercivity of ∇f that +1 +2 ∥∇f (yk+1)∥2 = 1 +2 ∥∇f (yk)∥2 + ⟨∇f (yk+1) , ∇f (yk+1) − ∇f (yk)⟩ − 1 +2 ∥∇f (yk+1) − ∇f (yk)∥2 += 1 +2 ∥∇f (yk)∥2 − 1 +λk +⟨yk+1 − yk, ∇f (yk+1) − ∇f (yk)⟩ − 1 +2 ∥∇f (yk+1) − ∇f (yk)∥2 +≤ 1 +2 ∥∇f (yk)∥2 . +Taking into account also (ii), we obtain (v). +Finally, according to the assumption � +k≥0 λk = +∞, and (iv), we have that limk→+∞ f (yk) = infH f. +Since f is convex and lower semicontinuous, the second condition of Opial’s lemma is also fulfilled. This +gives the weak convergence of the sequence (yk)k≥0 to an element in S = argminf. +⊓⊔ +Then, we give a statement which can be seen as a discrete counterpart of Lemma 1. The result is more +complex not only because we are in the discrete setting, but also because it allows an explicit choice of the +stepsize, as we will see later. +Lemma 2 Let (λk)k≥0 be a positive sequence and (τk)k≥0 such that τ0 = 0 and τk = �k−1 +i=0 λi for all k ≥ 1. +Suppose that there exist C2 > 0 and a, b, c ≥ 0 such that b + c > a and +� +k≥0 +τ a +k λ−b +k λ−c +k+1 ≤ C2 < +∞ +Then there exists C3 > 0 such that for every k ≥ 1 it holds +τk+1 ≥ C3k +b+c+1 +b+c−a . +(77) +Proof By applying the H¨older inequality twice we get for all k ≥ 0 +k +� +i=0 +τ +a +b+c+1 +i +≤ +� k +� +i=0 +τ a +i λ−b +i +λ−c +i+1 +� +1 +b+c+1 � k +� +i=0 +λi +� +b +b+c+1 � k +� +i=0 +λi+1 +� +c +b+c+1 +≤ C +1 +b+c+1 +2 +�k+1 +� +i=0 +λi +� +b+c +b+c+1 += C +1 +b+c+1 +2 +τ +b+c +b+c+1 +k+2 +. +(78) +If a = 0 then (77) follows immediately. From now on we suppose that a > 0. Inequality (78) becomes +k +� +i=0 +τ +a +b+c+1 +i +≤ C +1 +b+c+1 +2 +� +τ +a +b+c+1 +k+2 +� b+c +a +∀k ≥ 0. +(79) +Following the continuous counterpart, let us define +Cb+c := C +1 +b+c+1 +2 +> 0 +and +Ak := +k +� +i=0 +τ +a +b+c+1 +i +∀k ≥ 0 +so that (79) becomes +Ak ≤ Cb+c (Ak+2 − Ak+1) +b+c +a +∀k ≥ 0. +From here, +C +− +a +b+c +b+c +≤ A +− +a +b+c +k +(Ak+2 − Ak+1) +∀k ≥ 1. +(80) +For convenience, we define the following function ψ: R++ → R++ as ψ (r) := r− +a +b+c . It is clear that +d +dr +� +b + c +b + c − ar1− +a +b+c +� += ψ (r) +and +˙ψ (r) = − +a +b + cr− +a +b+c −1 < 0. +Since (Ak)k≥0 is increasing, this means ψ (Ak+2) ≤ ψ (r) ≤ ψ(Ak) for every Ak ≤ r ≤ Ak+2. +Let k ≥ 1 fixed. We consider two separate cases. +19 + +Case 1: ψ (Ak) ≤ 2ψ (Ak+2). Then (80) leads to +C +− +a +b+c +b+c +≤ A +− +a +b+c +k +(Ak+2 − Ak) = ψ (Ak) (Ak+2 − Ak) +≤ 2ψ (Ak+2) (Ak+2 − Ak) = 2ψ (Ak+2) +� Ak+2 +Ak +1dr +≤ 2 +� Ak+2 +Ak +ψ (r) dr = 2 +b + c +b + c − a +� +A +1− +a +b+c +k+2 +− A +1− +a +b+c +k +� +. +Case 2: ψ (Ak) > 2ψ (Ak+2). This is equivalent to Ak+2 > 2 +b+c +a Ak. Since b + c > a, we can deduce further +A +1− +a +b+c +k+2 +> 2 +b+c +a −1A +1− +a +b+c +k +. +Consequently, +A +1− +a +b+c +k+2 +− A +1− +a +b+c +k +> +� +2 +b+c +a −1 − 1 +� +A +1− +a +b+c +k +≥ +� +2 +b+c +a −1 − 1 +� +A +1− +a +b+c +1 +, +recall that the last inequality follows from the increasing property of (Ak)k≥1. +In conclusion, for every k ≥ 0 we have +A +1− +a +b+c +k+2 +− A +1− +a +b+c +k +≥ C4 := min +�1 +2 +� +1 − +a +b + c +� +C +− +a +b+c +b+c +, +� +2 +b+c +a −1 − 1 +� +A +1− +a +b+c +1 +, A +1− +a +b+c +2 +� +> 0. +Telescoping sum arguments combined with (79) imply for every k ≥ 1 +C4k ≤ A +1− +a +b+c +2k +− A +1− +a +b+c +0 +≤ A +1− +a +b+c +2k +≤ C +b+c−a +(b+c)(b+c+1) +2 +τ +b+c−a +b+c+1 +2k+2 . +This gives for every k ≥ 1 +τ2k+3 ≥ τ2k+2 ≥ �C3k +b+c+1 +b+c−a , +where �C3 > 0. We therefore deduce that there exists C3 > 0 such that +τk+1 ≥ C3k +b+c+1 +b+c−a +∀k ≥ 1, +which gives (77). +⊓⊔ +Following a plan identical to the continuous case, we successively consider the case where the control +by feedback is formulated in terms of speed, then of gradient. +4.2 Adaptive stepsize rules resulting from the discretization of the velocity based system +In this subsection we specialize the algorithm PEAS to the case where G(yk, yk−1) = ∥yk − yk−1∥. +Algorithm 2: Proximal algorithm with adaptive step size defined via velocity +Input: y0 ̸= y−1 ∈ H +1 for k = 0, 1, · · · do +2 +if ∇f (yk) = 0 then +3 +stop +4 +else +5 +λk := ∥yk − yk−1∥− p−1 +p +6 +yk+1 := proxλkf (yk) +7 +end +8 end +Theorem 9 Let (yk)k≥0 be the sequence generated by Algorithm 2. Then it holds +f (yk) − inf +H f = o +� +1 +k2− 1 +p +� +as k → +∞, +and the sequence of iterates (yk)k≥0 converges weakly as k → +∞, and its limit belongs to S = argminH f. +20 + +Proof By the choice of the step size, we have from Theorem 8 (iii) that +� +k≥0 +τk +λk +∥yk+1 − yk∥2 = +� +k≥0 +τkλ−1 +k λ +− +2p +p−1 +k+1 +< +∞, +where τ0 = 0 and τk := �k−1 +i=0 λi for every k ≥ 1. We are in position to apply Lemma 2 with (a, b, c) := +� +1, 1, +2p +p−1 +� +. We get +τk+1 ≥ C3k2− 1 +p +∀k ≥ 1. +(81) +Therefore � +k≥0 λk = limk→+∞ τk = +∞, and we can apply Theorem 8 to obtain the conclusion. +⊓⊔ +4.3 Adaptive stepsize resulting from the discretization of the gradient based system +Now let us specialize the algorithm PEAS to the case where G(yk, yk−1) = ∥∇f(yk)∥. +Algorithm 3: Proximal algorithm with adaptive step size defined via gradient +Input: y0 ∈ H +1 for k = 0, 1, · · · do +2 +if ∇f (yk) = 0 then +3 +stop +4 +else +5 +λk := ∥∇f (yk)∥− p−1 +p +6 +yk+1 := proxλkf (yk) +7 +end +8 end +Theorem 10 Let (yk)k≥0 be the sequence generated by Algorithm 3. Then it holds +f (yk+1) − inf +H f = o +� +1 +k2− 1 +p +� +as k → +∞ +and the sequence of iterates (yk)k≥0 converges weakly as k → +∞, and its limit belongs to S = argminH f. +Proof In this case we have from Theorem 8 (ii) +� +k≥0 +τkλk ∥∇f (yk+1)∥2 = +� +k≥0 +τkλkλ +− +2p +p−1 +k+1 +< +∞, +(82) +where τ0 = 0 and τk := �k−1 +i=0 λi for every k ≥ 1. +Let us establish an inequality of the type +∥∇f (yk)∥ ≤ Ck ∥∇f (yk+1)∥ , +for some sequence Ck > 0 which is to be precised. We have for all k ≥ 0 +∥∇f (yk)∥2 = ∥∇f (yk+1)∥2 − 2 ⟨∇f (yk+1) , ∇f (yk+1) − ∇f (yk)⟩ + ∥∇f (yk+1) − ∇f (yk)∥2 += ∥∇f (yk+1)∥2 + 2 +λk +⟨yk+1 − yk, ∇f (yk+1) − ∇f (yk)⟩ + ∥∇f (yk+1) − ∇f (yk)∥2 +≤ ∥∇f (yk+1)∥2 + +�2L +λk ++ L2 +� +∥yk+1 − yk∥2 = (1 + Lλk)2 ∥∇f (yk+1)∥2 +≤ (1 + Lλk+1)2 ∥∇f (yk+1)∥2 +(83) +where L > 0 denotes the Lipschitz constant of ∇f on a bounded set containing the sequence (yk)k≥0. +Combining (82) and (83), we get +� +k≥0 +τkλk +1 +(1 + Lλk+1)2 ∥∇f (yk)∥2 = +� +k≥0 +τkλk +1 +(1 + Lλk+1)2 λ +− +2p +p−1 +k +< +∞. +(84) +21 + +Let us now show that limk→+∞ λk = +∞. According to the decreasing property of the sequence +(f (yk) − infH f)k≥0, by summing inequalities (76) we get +� +k≥0 +λk ∥∇f (yk+1)∥2 < +∞. +(85) +From the closed-loop rule we deduce that +� +k≥0 +λkλ +− +2p +p−1 +k+1 +< +∞. +(86) +Therefore +lim +k→+∞ λkλ +− +2p +p−1 +k+1 += 0. +Since (λk)k≥0 is increasing, let us denote by l > 0 its limit. If l is finite then, by passing to the limit on the +above inequality we get l1− +2p +p−1 = 0, a clear contradiction with l > 0. Therefore +lim +k→+∞ λk = +∞. +In this case +1 +(1+Lλk+1)2 ∼ (Lλk+1)−2, which gives +� +k≥0 +τkλ +1− +2p +p−1 +k +λk+1 +−2 < +∞. +(87) +We are in position to apply Lemma 2 with (a, b, c) := +� +1, +2p +p−1 − 1,2 +� +. We get +τk+1 ≥ C3k2− 1 +p +∀k ≥ 1. +We have � +k≥0 λk = limk→+∞ τk = +∞, and we can apply Theorem 8 to obtain, as k → +∞ +f (yk+1) − inf +H f = o +� +1 +k2− 1 +p +� +. +This completes the proof. +⊓⊔ +Remark 3 Note that the closed-loop control of the velocity and the closed-loop control of the gradient give +the same convergence rate of the values. Clearly, we have obtained a faster convergence result. +5 Inertial proximal algorithms obtained by closed-loop damping +Let us now consider the convergence properties of the sequences (xk)k≥0 which are obtained by applying +the averaging process to the sequences generated by Algorithm 2. Indeed, we limit our investigation to the +closed loop control of the velocity, the case of the closed loop control of the gradient is very similar. Let us +discretize the continuous averaging relation +˙x(t) + ˙τ(t) +τ(t) (x(t) − y(t)) = 0 +as follows (recall that, because of the choice q = 1, we have ˙τ(t) = λ(t)) +xk+1 − xk + +λk +τk+1 +(xk − yk+1) = 0. +Equivalently +xk+1 = +� +1 − +λk +τk+1 +� +xk + +λk +τk+1 +yk+1. +This gives the following proximal inertial algorithm: +Theorem 11 Let (xk)k≥0 be the sequence generated by Algorithm 4. Then it holds +f (xk) − inf +H f = O +� +1 +k2− 1 +p +� +as k → +∞ +and the sequence of iterates (xk)k≥0 converges weakly as k → +∞, and its limit belongs to S = argminH f. +22 + +Algorithm 4: Proximal inertial algorithm with adaptive step size defined via velocity +Input: τ0 := 0 and x0, y0 ̸= y−1 ∈ H +1 for k = 0, 1, · · · do +2 +if ∇f (yk) = 0 then +3 +stop +4 +else +5 +λk := ∥yk − yk−1∥− p−1 +p +6 +yk+1 := proxλkf (yk) +7 +τk+1 := τk + λk +8 +xk+1 := +� +1 − +λk +τk+1 +� +xk + +λk +τk+1 +yk+1. +9 +end +10 end +Proof Let k ≥ 0. By definition of xk+1 we have +τk+1xk+1 = (τk+1 − λk)xk + λkyk+1 +which gives (recall that τk+1 = �k +i=0 λi) +τk+1xk+1 − τkxk = (τk+1 − λk)xk + λkyk+1 − τkxk = (τk+1 − τk − λk)xk + λkyk+1 = λkyk+1. +Therefore, by telescoping arguments we obtain +xk+1 = +�k +i=0 λiyi+1 +τk+1 +∀k ≥ 0. +(88) +By convexity of f we infer +f (xk+1) − inf +H f = +� +f − inf +H f +� +(xk+1) = +� +f − inf +H f +� ��k +i=0 λiyi+1 +τk+1 +� +≤ +1 +τk+1 +k +� +i=0 +λi +� +f − inf +H f +� +(yi+1) = +1 +τk+1 +k +� +i=0 +λi +� +f (yi+1) − inf +H f +� +, +By Theorem 8 (i), we have � +k≥0 λk (f (yk+1) − infH f) < +∞, and by (81) we have τk+1 ≥ k2− 1 +p , which +gives the claim. The weak convergence of (xk)k≥0 to an element in S = argminH f follows from the weak +convergence of (yk)k≥0 and the Stolz-Ces´aro Theorem. +⊓⊔ +5.1 Geometric interpretation of PEAS +First note that (PEAS) can be equivalently written as follows +xk+1 = +� +1 − +λk +τk+1 +� +xk + +λk +τk+1 +proxλkf +� +xk−1 + +τk +λk−1 +(xk − xk−1 +� +. +(89) +Since +τk +λk−1 > 1, the algorithm first involves an extrapolation step (this is the inertial aspect), then a +proximal step, and finally a relaxation step which balances the inertia effect and dampens the oscillations. +This is shown in the figure below. We set θk = +λk +τk+1 ∈]0,1[. +Despite some analogies, (PEAS) is different from the relaxed inertial proximal algorithm (RIPA) con- +sidered by Attouch and Cabot in [7], and which writes +� +yk += xk + αk(xk − xk−1) +xk+1 += (1 − ρk)yk + ρk proxλkf(yk) +(90) +As main difference, in (PEAS) relaxation is taken between xk and proxλkf(yk), while in (RIPA) it is +taken between yk and proxλkf(yk). Consequently (PEAS) involves a Hessian damping effect which is not +present in (RIPA). Note in (PEAS) the balance between the extrapolation (inertial, acceleration) effect and +the relaxation effect. Moreover, our construction provides coefficients which are generated automatically +in closed loop way, whereas in (RIPA) they require subtle adjustment. The importance of the relaxation +technique when combined with inertia has been put to the fore in [18]. According to (88), xk+1 can be +23 + +yk = xk−1 + +1 +θk−1 (xk − xk−1) +• +xk +• +xk−1 +• +• +xk+1 = (1 − θk) xk + θk proxλkf (yk) +proxλkf(yk) +S +� +� +�� +� +� +� +� +� +� +❛❛❛❛❛❛❛❛❛❛❛❛❛❛ +✚ +✚ +✚ +❂ +✄ +✄ +✄✄ +✪ +✪ +✪ +✪ +Fig.1 +A geometrical illustration of algorithm PEAS +interpreted as an average of the {yn : 0 ≤ n ≤ k + 1}, which makes our approach somewhat analogous to +the nonlinear averaging technique developed in [27], where it is assumed that there is a unique minimizer. +Averaging techniques have also been used in [26] in the context of hybrid systems. Indeed, adjusting the +damping in a closed-loop ad hoc manner bears some analogy to restarting methods. +6 Conclusion, perspective +Our study proposes new fast adaptive optimization methods for convex optimization. We have shown +that the time scaling and averaging technique, previously developed by the authors in the context of +non-autonomous systems, can be developed by taking closed-loop time parameterization, giving rise to au- +tonomous dynamics. The method turns out to be flexible, because it is based on elementary mathematical +tools, namely the dynamics of the steepest descent, and the operations of temporal parameterization and +averaging. It is therefore not necessary to redo a Lyapunov analysis, one relies on the classic results for the +steepest descent. The results obtained for the continuous dynamics pass quite naturally to the correspond- +ing proximal algorithms, where the iterates are expressed in a direct way according to the proximal terms. +This study is one of the very first to develop an algorithmic framework based on autonomous dynamics and +which, when specialized, provides the convergence rates of the dynamical surrogate of the Nesterov acceler- +ation gradient method. Another important aspect of our analysis is that it exhibits Hessian-driven damping, +which plays a key role in damping oscillations. Our work opens up many perspectives, our method natu- +rally extending to gradient algorithms, proximal-gradient algorithms for composite optimization, cocercive +monotone operators, and the study of the stochastic version, to name only a few. +7 Appendix +7.1 Classical facts concerning the continuous steepest descent +Consider the classical continuous steepest descent +(SD) +˙z(t) + ∇f(z(t)) = 0. +(91) +Under the standing assumption (A) on f, we know that, for any z0 ∈ H there exists a unique classical +global solution z ∈ C1([t0, +∞[: H) of (SD) satisfying z(t0) = z0, see [5, Theorem 17.1.1]. We fix t0 as the +origin of time. Recall classical facts concerning the continuous steepest descent. +Theorem 12 Suppose that f : H → R satisfies (A). Let z : [t0, +∞[ → H be a solution trajectory of +˙z(t) + ∇f(z(t)) = g(t) +(92) +where g: [t0, +∞[ → H is such that +� +∞ +t0 +∥g (t)∥ dt < +∞ and +� +∞ +t0 +t ∥g (t)∥2 dt < +∞. +(93) +Then the following statements are satisfied: +24 + +(i) (convergence of gradients towards zero) +∥∇f (z (t))∥ = o +� 1 +√ +t +� +as t → +∞. +(ii) (integral estimate of the velocities) +� +∞ +t0 +t ∥ ˙z (t)∥2 dt < +∞. +(iii) (integral estimate of the gradients) +� +∞ +t0 +t ∥∇f (z (t))∥2 dt < +∞. +(iv) (convergence of values) f (z (t)) − infH f = o +�1 +t +� +as t → +∞. +(v) (improved convergence rates of gradients) if g(t) ≡ 0, then ∥∇f (z (t))∥ = o +�1 +t +� +as t → +∞. +(vi) The solution trajectory z(t) converges weakly as t → +∞, and its limit belongs to S = argminf. +If g(t) ≡ 0 we have that t �→ ∥∇f (z (t))∥ is nonincreasing since in this case +d +dt ∥∇f (z (t))∥2 = 2 +� +∇f (z (t)) , d +dt∇f (z (t)) +� += −2 +� +˙z (t) , d +dt∇f (z (t)) +� +≤ 0 +∀t ≥ t0. +Therefore, from the integral estimate of the gradients we deduce that ∥∇f (z (t))∥ = o � 1 +t +�. +7.2 Auxiliary result +Opial’s Lemma is a basic ingredient of the convergence analysis. +Lemma 3 (Opial) Let S be a nonempty subset of H and let (xk)k≥0 be a sequence in H. Assume that +(i) for every z ∈ S, limk→+∞ ∥xk − z∥ exists; +(ii) every weak sequential limit point of (xk)k≥0, as k → +∞, belongs to S. +Then (xk)k≥0 converges weakly as k → +∞, and its limit belongs to S. +Funding +The research of RIB and DKN has been supported by FWF (Austrian Science Fund), projects W 1260 and +P 34922-N, respectively. +References +1. C.D. Alecsa, S. L´aszl´o, T. Pinta, An extension of the second order dynamical system that models Nesterov’s convex +gradient method, Applied Mathematics and Optimization, 84 (2021), 1687–1716. +2. H. Attouch, A. Balhag, Z. Chbani, H. Riahi, Fast convex optimization via inertial dynamics combining viscous and +Hessian-driven damping with time rescaling, Evolution Equations and Control Theory, 11 (2) (2022), 487–514. +3. H. Attouch, R.I. Bot¸, E.R. Csetnek, Fast optimization via inertial dynamics with closed-loop damping. J. Eur. Math. +Soc. (2022), DOI 10.4171/JEMS/1231 +4. H. Attouch, R.I. Bot¸, D.-K. 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Neural Information Processing Systems, 27 (2014), 2510–2518. +26 + diff --git a/pdAyT4oBgHgl3EQfzfkv/content/tmp_files/load_file.txt b/pdAyT4oBgHgl3EQfzfkv/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3911ebcee91869cf2395408e00f8d62ab75bb06c --- /dev/null +++ b/pdAyT4oBgHgl3EQfzfkv/content/tmp_files/load_file.txt @@ -0,0 +1,823 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf,len=822 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='00701v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='OC] 2 Jan 2023 Noname manuscript No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (will be inserted by the editor) Fast convex optimization via closed-loop time scaling of gradient dynamics Hedy Attouch · Radu Ioan Bot¸ · Dang-Khoa Nguyen January 3, 2023 Abstract In a Hilbert setting, for convex differentiable optimization, we develop a general framework for adaptive accelerated gradient methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' They are based on damped inertial dynamics where the coefficients are designed in a closed-loop way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Specifically, the damping is a feedback control of the velocity, or of the gradient of the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' For this, we develop a closed-loop version of the time scaling and averaging technique introduced by the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' We thus obtain autonomous inertial dynamics which involve vanishing viscous damping and implicit Hessian driven damping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' By simply using the convergence rates for the continuous steepest descent and Jensen’s inequality, without the need for further Lyapunov analysis, we show that the trajectories have several remarkable properties at once: they ensure fast convergence of values, fast convergence of the gradients towards zero, and they converge to optimal solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Our approach leads to parallel algorithmic results, that we study in the case of proximal algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' These are among the very first general results of this type obtained using autonomous dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Keywords fast convex optimization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' damped inertial dynamic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' time scaling;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' averaging;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' closed-loop control;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Nesterov and Ravine algorithms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Hessian driven damping;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' proximal algorithms AMS subject classification 37N40, 46N10, 49M30, 65B99, 65K05, 65K10, 90B50, 90C25 1 Introduction In a real Hilbert space H, we develop a dynamic approach to the rapid resolution of convex optimization problems which relies on inertial dynamics whose damping is designed as a closed-loop control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' We consider the minimization problem min {f(x) : x ∈ H} , (1) where, throughout the paper, we make the following assumptions on the function f to be minimized (A) � f : H → R is a convex function of class C1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' S = argminH f ̸= ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' ∇f is Lipschitz continuous on the bounded sets of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (2) Our study is part of the close links between dissipative dynamical systems and optimization algorithms, the latter being obtained by temporal discretization of the continuous dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Our study comes as a natural extension of the authors’ previous work [4] where the technique of time scaling and averaging was Hedy Attouch IMAG, Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Montpellier, CNRS, Montpellier, France E-mail: hedy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='attouch@umontpellier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='fr Radu Ioan Bot¸ Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria, E-mail: radu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='bot@univie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='at Dang-Khoa Nguyen Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria, E-mail: dang-khoa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='nguyen@univie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='at used in an open-loop way, giving rise to non-autonomous damped inertial dynamics with fast convergence properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' In the present paper, we take advantage of the simplicity and flexibility of this technique to develop it in a closed-loop way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' This will give rise to autonomous damped inertial dynamics with fast convergence properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Recall that the low-resolution ODE obtained by Su, Boyd, and Cand`es [30] of the accelerated gradient method of Nesterov, together with the corresponding high-resolution ODE [8], [28] (which involves an additional Hessian driven damping term) are non-autonomous dynamics, the coefficient of viscous friction being of the form α/t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Our study therefore opens a new path in the field of first-order adaptive optimization methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='1 Time scale and averaging: the open-loop approach Let us first briefly explain the time scaling and averaging method in the open-loop case on a model example (see [4] for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Then we will look at how to develop a corresponding closed-loop approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' As the basic starting dynamic, we consider the continuous steepest descent (SD) ˙z(s) + ∇f(z(s)) = 0, (3) for which we have the classical convergence result f (z (s)) − inf H f = o �1 s � as s → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Then, we make the change of time variable s = τ(t) in (SD), where τ(·) is an increasing function from R+ to R+, continuously differentiable, and satisfying limt→+∞ τ(t) = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Setting y(t) := z(τ(t)), we get ˙y(t) + ˙τ(t)∇f(y(t)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (4) The convergence rate becomes f(y(t)) − inf H f = o � 1 τ(t) � as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (5) Taking τ(·) which grows faster than the identity, makes the solution trajectories unchanged but travelled faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' The price to pay is that (4) is a non-autonomous dynamic in which the coefficient in front of the gradient term tends to infinity as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' This prevents from using gradient methods to discretize it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Recall that for gradient methods the step size has to be less than or equal to twice the inverse of the Lipschitz constant of the gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' To overcome this difficulty we come with the second step of our method which is averaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Let us attach to y(·) the new function x : [t0, +∞[→ H defined by ˙x(t) + 1 ˙τ(t)(x(t) − y(t)) = 0, (6) with x(t0) = x0 given in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' We shall explain further the averaging interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Equivalently y(t) = x(t) + ˙τ(t) ˙x(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (7) By temporal derivation of (7) we get ˙y(t) = ˙x(t) + ¨τ(t) ˙x(t) + ˙τ(t)¨x(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (8) Replacing y(t) and ˙y(t) as given by (7) and (8) in (4), we get ¨x(t) + 1 + ¨τ(t) ˙τ(t) ˙x(t) + ∇f � x(t) + ˙τ(t) ˙x(t) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (9) In doing so, we passed from the first-order differential equation (4) to the second-order differential equation (9), with the advantage that now the coefficient in front of the gradient is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Let us now particularize the time scale τ(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Taking τ(t) = t2 2(α − 1), (10) gives 1+¨τ(t) ˙τ(t) = α t , and the corresponding dynamic with implicit Hessian driven damping ¨x(t) + α t ˙x(t) + ∇f � x(t) + t α − 1 ˙x(t) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (11) 2 In this dynamic, the Hessian driven damping appears in an implicit form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' This type of dynamic was initiated in [1], see also [22] for a related autonomous system in the case of a strongly convex function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' The rationale justifying the use of the term “implicit” comes from the observation that by a Taylor expansion (as t → +∞ we have t ˙x(t) → 0 which justifies the use of Taylor expansion), we have ∇f � x(t) + t α − 1 ˙x(t) � ≈ ∇f(x(t)) + t α − 1∇2f(x(t)) ˙x(t), thus making the Hessian damping appear indirectly in (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Because of its important role in attenuating the oscillations, several recent studies have been devoted to inertial dynamics combining the asymptotic vanishing damping with the geometric Hessian-driven damping (coined sometimes Newton-type inertial dynamics);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=', [2,11,8,9,10,15,16,21,28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' In turn, the corresponding algorithms, among which IGAHD enjoys several favorable properties, introduce a correction term in the Nesterov accelerated gradient method (see [24,25]) which reduces the oscillatory aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Note that in (11) the coefficient of the Hessian damping is proportional to the inverse of the viscosity damping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Thus asymptotically when the viscous damping tends towards zero, and therefore can cause many small oscillations to appear, the coefficient of the Hessian driven damping tends towards infinity, and therefore has an effective effect on the attenuation of the oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' This is the situation considered by Attouch-Bot¸-Nguyen [4], who obtained convergence rates comparable to those associated with the Nesterov accelerated gradient method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' A major advantage of this approach is that there is no need to do a Lyapunov analysis, we only use the classical convergence rate for the continuous steepest descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Moreover, the convergence of the trajectories is a direct consequence of the known results for the steepest descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='2 Closed-loop control The idea is to exploit the time scaling and averaging method and the fact that (SD) provides several quantities which are increasing and converge to +∞ as t → +∞, so which are eligible for time scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' This will enable us to perform time scaling and averaging in a closed-loop way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Indeed, in (SD), the velocity and the norm of the gradient are monotonically decreasing to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' So, the idea is to use their inverse for defining the time scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Specifically, in a first result we are going to define the derivative of the time scaling τ(·) as a function of the inverse of the speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' This means acceleration of the time scaling when the speed decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Following this approach, we will obtain in Theorem 5 the following model result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Theorem 1 Suppose that f : H → R satisfies (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Let us choose the positive parameters according to q > 0, p ≥ 1, and γ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Let x: [t0, +∞[ → H be a solution trajectory of the following system \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ¨x(t) + (1 + γ) ˙τ(t)2 − τ(t)¨τ(t) τ(t) ˙τ(t) ˙x(t) + γ ˙τ(t)2 τ(t) ∇f � x(t) + 1 γ τ(t) ˙τ(t) ˙x(t) � = 0 τ (t) − 1 qq � t0 + � t t0 [λ (r)] 1 q dr �q = 0 [λ (t)]p |˙τ (t)|p−1 ����∇f � x (t) + 1 γ τ(t) ˙τ(t) ˙x (t) ����� p−1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (12) Then we have the fast convergence of values: as t → +∞ f(x(t)) − inf H f = o � 1 t1+q− 1 p � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (13) Moreover, the solution trajectory x(t) converges weakly as t → +∞, and its limit belongs to S = argminf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' As a special case, take p = 1, q = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Then, the last equation of (12) gives λ(t) ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' According to this, the second equation of (12) gives τ(t) = t2 4 , and we find a case with time scaling in an open-loop form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' After elementary calculation, the first equation of (12) is written as ¨x(t) + α t ˙x(t) + α − 1 2 ∇f � x(t) + t α − 1 ˙x(t) � = 0, with α = 2γ + 1 > 3, and the convergence rate of the values becomes f(x(t)) − inf H f = o � 1 t2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (14) 3 We therefore recover the results obtained by the authors in the case of the open loop, giving the optimal convergence rates for general convex differentiable optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' This inertial formulation may seem at first glance complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Indeed it is equivalent to the first-order system in time and space \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ˙y (t) + ˙τ (t) ∇f (y (t)) = 0 ˙x(t) + γ ˙τ(t) τ(t)x(t) − γ ˙τ(t) τ(t)y(t) = 0 τ (t) − 1 qq � t0 + � t t0 [λ (r)] 1 q dr �q = 0 [λ (t)]p ∥ ˙y (t)∥p−1 = 1, (15) whose temporal discretization provides corresponding optimization algorithms, see Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='3 Link with the existing literature Contrary to the rich literature that has been devoted to non-autonomous damped inertial methods and their links with the fast first-order optimization algorithms for general convex optimization (in particular the Nesterov accelerated gradient method), only a small number of papers have been devoted to these questions, based on autonomous methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Indeed the heavy ball method of Polyak only provides the asymptotic convergence rate 1/t for general convex functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' The idea is therefore to see if we can mimic the fast convergence properties of the Su, Boyd, and Cand`es dynamic model (see [30]) of the Nesterov accelerated gradient method, using autonomous dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' A natural idea is to design the damping term, on which is based the optimization properties of the system, in a closed-loop way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' In this direction, we can mention the following contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' a) Our study has a natural link with works devoted to regularized Newton methods for solving monotone inclusions (and (1) in particular).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Given a general maximally monotone operator A : H ⇒ H, to overcome the ill-posed character of the continuous Newton method, in line with [13], Attouch, Redont and Svaiter have studied in [12] the following closed-loop dynamic version of the Levenberg-Marquardt method � v(t) ∈ A(x(t)) ∥v(t)∥γ ˙x(t) + β ˙v(t) + v(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' When γ > 1, they showed the well-posedness of the above system, and analyzed its convergence properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' When A = ∇f this system writes ∥∇f(x(t))∥γ ˙x(t) + β∇2f(x(t)) ˙x(t) + ∇f(x(t)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Thus, its inertial version ¨x(t) + ∥∇f(x(t))∥γ ˙x(t) + β∇2f(x(t)) ˙x(t) + ∇f(x(t)) = 0 falls within the framework of our study with the damping equal to a closed-loop control of the norm of the gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' The techniques developed in [12] are particularly useful for studying the well-posedness of dynamics with implicit features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' b) Although significantly different, our approach has several points in common with the article by Lin and Jordan [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' In this article, the authors study the closed-loop dynamical system \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ˙y (t) + ˙τ (t) ∇f (x (t)) = 0 ˙x (t) + ˙τ(t) τ(t) (x (t) − y (t)) + [ ˙τ(t)]2 τ(t) ∇f (x (t)) = 0 τ (t) − 1 4 �� t 0 � λ (r)dr + c �2 = 0 [λ (t)]p ∥∇f (x (t))∥p−1 = θ, (16) where c > 0 and 0 < θ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' The corresponding second-order in time damped inertial system writes as follows ¨x(t) + 2 [˙τ(t)]2 − τ(t)¨τ(t) τ(t) ˙τ(t) ˙x(t) + [ ˙τ(t)]2 τ(t) ∇2f (x(t)) ˙x(t) + ˙τ(t)( ˙τ(t) + ¨τ(t)) τ(t) ∇f (x(t)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (17) 4 In the above system, the Hessian driven damping comes in an explicit way because of the structure of the first equation which differs from the structure of the continuous steepest descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' In contrast, in our approach, the first equation is the rescaled continuous steepest descent, and the Hessian driven damping comes implicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Let us highlight some advantages of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Our system is introduced in a natural way by using the time scaling and averaging method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' This makes unnecessary to perform a Lyapunov analysis for the inertial system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' It has already been done for the continuous steepest descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' This results in a significantly simplified mathematical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Our dynamic model contains an additional parameter q which, when q = 2, gives the setting of Lin and Jordan, and which, when judiciously tuned, gives better convergence rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Our approach provides the weak convergence of the trajectories to optimal solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' We shall return later to the precise comparison between the two systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' c) In [3], Attouch, Bot¸ and Csetnek study the convergence properties of the Autonomous Damped Inertial Gradient Equation (ADIGE) ¨x(t) + G � ˙x(t), ∇f(x(t)),∇2f(x(t)) � + ∇f(x(t)) = 0, where the damping term G � ˙x(t), ∇f(x(t)),∇2f(x(t)) � acts as a closed-loop control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' They pay particular attention to the role played by the parameter r > 1 in the asymptotic convergence analysis of the dynamic ¨x(t) + ∥ ˙x(t)∥r−2 ˙x(t) + ∇f(x(t)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' They show that the case r = 2 separates the weak damping (r > 2) from the strong damping (r < 2), hence the importance of this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' These questions have also been considered by Haraux and Jendoubi in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' d) In [29], Song, Jiang, and Ma develop an interesting technique for accelerating high-order algorithms under general H¨older continuity assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Their continuous-time framework reduces to an inertial system without Hessian-driven damping in the first-order setting, which has been proven to be an inaccurate surrogate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Although underlying their approach, the acceleration via time scaling, the averaging technique, and the closed-loop tuning of the coefficients are not clearly identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='4 Organization of the paper After a general presentation of the article in the introduction, we provide in Section 2 a general estimate of the time scaling for the continuous steepest descent when it is defined in a closed-loop way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' This is crucial for the rest of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Then we specialize these results to situations of particular interest, and examine in details the case of closed-loop systems induced respectively by velocities, and then by gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' In Section 3, which is the main part of the paper, we develop the next important step in our approach, which is the averaging operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' This provides accelerated damped inertial dynamics that are autonomous and with fast convergence properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Finally, in Section 4 we analyze the fast convergence properties of proximal algorithms which come naturally from the temporal discretization of the continuous dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' 2 Closed-loop time scaling of the steepest descent 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='1 Formulation of the closed-loop time scaling Given t0 ≥ 0, q > 0, and p ≥ 1, the time scale function τ : [t0, +∞[ → R++ is defined by \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ˙y (t) + ˙τ (t) ∇f (y (t)) = 0 τ (t) − 1 qq � t0 + � t t0 [λ (r)] 1 q dr �q = 0 [λ (t)]p [G (y (t))]p−1 = 1, (18) where G(·) is a given positive, continuous function that depends on the information of the trajectory y(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' This general formalism allows us to unify the various situations coming from different choices of the time scaling as a feedback control of the state of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' For example G may be a function of y, ˙y, f (y) , ∇f (y) and/or any mixture combination of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Then the function λ(·) is continuous and it links the coefficient 5 of ∇f, namely ˙τ(·), with the solution trajectory y(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' As a useful result, note that for every t ≥ t0, it holds ˙τ (t) = 1 qq−1 �� t t0 [λ (r)] 1 q dr + t0 �q−1 [λ (t)] 1 q = [τ (t)] q−1 q [λ (t)] 1 q > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (19) Moreover, the relations (18) allow us to cover the open-loop case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' In particular, when p = 1 it holds λ (t) = 1 for every t ≥ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' This yields for every q > 0 τ (t) = � t q �q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Taking further q := 1, then τ (t) becomes the regular time in variable t, namely τ (t) = t for every t ≥ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Let us specify the interpretation of (18) as a steepest descent dynamic which is rescaled in time in a closed-loop way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Proposition 1 Suppose that f : H → R satisfies (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Let t0 ≥ 0, q > 0, p ≥ 1 and y: [t0, +∞[ → H be a solution trajectory of the system (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Suppose that lim s→+∞ τ (s) = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Then y(·) is a solution trajectory of a time rescaled continuous steepest descent (SD), as described below: Let s0 = τ (t0) and z : [s0, +∞[ → H be a solution trajectory of the following system ˙z (s) + ∇f (z (s)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (20) Then we have y (t) = z (τ (t)) ∀t ≥ t0, and there exists a continuously differentiable function σ: [s0, +∞[ → R++ such that z (s) = y (σ (s)) ∀s ≥ s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Proof We already interpreted how to go from a solution trajectory z(·) of (SD) to the closed-loop system above via the time scaling function τ(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Let us now show the reverse direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Let y: [t0, +∞[ → H be a solution trajectory of (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' We have that λ is continuous and positive on [t0, +∞[, therefore τ is a monotonically increasing function, hence injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' On the other hand, we have t0 = τ (t0) = � t0 q �q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Since by assumption limt→+∞ τ (t) = +∞, this means τ is a continuous function whose image contains [s0, +∞[, hence surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Combining these premises, we have shown that τ is a bijection, which means it is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Set σ ≡ τ −1 and make the change of time variable t := σ (s) in (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Let us define z (s) = y (σ (s)) = y � τ −1 (s) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Then by the chain rule, we have ˙z (s) = ˙y � τ −1 (s) � 1 ˙τ (τ −1 (s)) = ˙y (σ (s)) 1 ˙τ (σ (s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' This leads to ˙z (s) + ∇f (z (s)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' In other words, z : [s0, +∞[ → H is a solution trajectory of (SD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' ⊓⊔ The above assertion allows us to transfer the convergence results of (SD) to some closed-loop systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' In particular, given a time scaling function τ(·) as described above, by making the change of time variable s := τ (t), we obtain the following results from Theorem 12 in the appendix applied to the unperturbed continuous steepest descent system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' � +∞ t0 τ (t) ˙τ (t) ∥ ˙y (t)∥2 dt < +∞, (21) � +∞ t0 τ (t) ˙τ (t) ∥∇f (y (t))∥2 dt < +∞, (22) f(y(t)) − inf H f = o � 1 τ(t) � as t → +∞, (23) ∥∇f (y (t))∥ = o � 1 τ(t) � as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (24) 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='2 Lower bound estimate of the time scaling τ(t) As key ingredient of our approach, the next step is to establish a lower bound for τ(t) in terms of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' This will reflect the acceleration of our dynamic via time scaling and allow us to achieve fast convergence rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' For this, we will need the following technical lemma, which can be seen as a nonlinear Gronwall result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Lemma 1 Suppose that there exists C0 > 0 and b > a ≥ 0 such that � t t0 [τ (r)]a [λ (r)]−b dr ≤ C0 < +∞ ∀t ≥ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (25) Then there exists C1 > 0 such that τ (t) ≥ C1 (t − t0) qb+1 b−a ∀t ≥ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (26) Proof Let t ≥ t0 be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' By applying the H¨older inequality, we get � t t0 [τ (r)] a qb+1 dr ≤ �� t t0 [τ (r)]a [λ (r)]−b dr � 1 qb+1 �� t t0 [λ (r)] 1 q dr � qb qb+1 ≤ C 1 qb+1 0 � t0 + � t t0 [λ (r)] 1 q dr � qb qb+1 = � C0qqb� 1 qb+1 [τ (t)] b qb+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (27) If a = 0 then (26) follows immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' From now on suppose that a > 0, so that the inequality (27) can be rewritten as � t t0 [τ (r)] a qb+1 dr ≤ � C0qqb� 1 qb+1 � [τ (t)] a qb+1 � b a (28) The arguments are now adapted from [19], which is inspired by the proof of Bihari-LaSalle inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Let Cq,b := � C0qqb� 1 qb+1 > 0 and A (t) := � t t0 [τ (r)] a qb+1 dr ∀t ≥ t0, so that (28) becomes A (t) ≤ Cq,b � ˙A (t) � b a ∀t ≥ t0 or, equivalently, C− a b q,b ≤ [A (t)]− a b ˙A (t) ∀t ≥ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Integrating from t0 to t, we obtain C− a b q,b (t − t0) ≤ � 1 − a b � � [A (t)]1− a b − [A (t0)]1− 1 b � ≤ [A (t)]1− a b ≤ � Cq,b [τ (t)] b qb+1 �1− a b = C b−a b q,b [τ (t)] b−a qb+1 , where the last inequality comes from (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Since b > a, the conclusion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' ⊓⊔ Let us now particularize our results to some model situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='3 Closed-loop control of (SD) via the velocity Theorem 2 Suppose that f : H → R satisfies (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Let q > 0, p ≥ 1 and y: [t0, +∞[ → H be a solution trajectory of the following system \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ˙y (t) + ˙τ (t) ∇f (y (t)) = 0 τ (t) − 1 qq � t0 + � t t0 [λ (r)] 1 q dr �q = 0 [λ (t)]p ∥ ˙y (t)∥p−1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (29) Then the following statements are satisfied: (i) (convergence of values) f (y (t)) − infH f = o � t−(1+q− 1 p)� as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (ii) (convergence of gradients towards zero) ∥∇f (y (t))∥ = o � t−(1+q− 1 p)� as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' 7 (iii) (integral estimate of the velocities) � +∞ t0 t(1+ 1 q − 1 pq) ∥ ˙y (t)∥2+ p−1 pq dt < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (iv) The solution trajectory y(t) converges weakly as t → +∞, and its limit belongs to S = argminf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Proof When p = 1, we recover the open loop case with the time scaling function τ(t) = � t q �q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' The result is a direct consequence of Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Therefore, from now on we only consider the case p > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Recall that from (21) we have � +∞ t0 τ (t) ˙τ (t) ∥ ˙y (t)∥2 dt < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (30) By using successively the definition of λ, and relation (19), we obtain τ (t) ˙τ (t) ∥ ˙y (t)∥2 = τ (t) ˙τ (t) [λ (t)]− 2p p−1 = [τ (t)] 1 q [λ (t)]− 1 q − 2p p−1 ∀t ≥ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' According to the two above results we get � +∞ t0 [τ (r)] 1 q [λ (r)]− 1 q − 2p p−1 dr < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' We are now in position to apply Lemma 1 with p > 1, a := 1 q and b := 1 q + 2p p−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' We have qb + 1 b − a = 2 + 2pq p−1 2p p−1 = p − 1 + pq p = 1 + q − 1 p, and therefore there exists some constant C1 > 0 such that τ (t) ≥ C1 (t − t0)1+q− 1 p ∀t ≥ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (31) This leads to limt→+∞ τ (t) = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Therefore, according to Proposition 1, we can extract the results from Theorem 12 and the corresponding formulas (21), (22), (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Specifically, we obtain (i) for the values f (y (t)) − inf H f = o � 1 τ(t) � = o � 1 t1+q− 1 p � , (ii) for the gradients ∥∇f (y (t))∥ = o � 1 τ(t) � = o � 1 t1+q− 1 p � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (iii) for the velocities: we start from (30), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' � +∞ t0 τ (t) ˙τ (t) ∥ ˙y (t)∥2 dt < +∞, that we evaluate as follows: τ (t) ˙τ (t) ∥ ˙y (t)∥2 = τ(t) τ(t) q−1 q λ(t) 1 q ∥ ˙y (t)∥2 = τ(t) 1 q λ(t) 1 q ∥ ˙y (t)∥2 = τ(t) 1 q ∥ ˙y (t)∥2+ p−1 pq ∀t ≥ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' According to (31) we deduce that � +∞ t0 t(1+ 1 q − 1 pq) ∥ ˙y (t)∥2+ p−1 pq dt < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (iv) Let us finally examine the convergence of the solution trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' We know that the solution trajectory of the continuous steepest descent converges weakly when t → +∞, and its limit belong to S = argminH f ̸= ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' see Theorem 12 in appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' With our notation we therefore have that z(s) converges weakly when s → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Since τ(t) → +∞ as t → +∞, we immediately deduce that y(t) = z(τ(t) converges weakly as t → +∞, and its limit belong to S = argminH f ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' ⊓⊔ 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='4 Closed-loop control of (SD) via the norm of gradient We develop an analysis parallel to that of the previous section, replacing speed control with gradient control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Theorem 3 Suppose that f : H → R satisfies (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Let q ≥ 1 2, p ≥ 1 and y: [t0, +∞[ → H be a solution trajectory of the following system \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ˙y (t) + ˙τ (t) ∇f (y (t)) = 0 τ (t) − 1 qq � t0 + � t t0 [λ (r)] 1 q dr �q = 0 [λ (t)]p ∥∇f (y (t))∥p−1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (32) Then the following statements are satisfied: (i) (convergence of values) f (y (t)) − infH f = o � t−pq� as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (ii) (convergence of gradients towards zero) ∥∇f (y (t))∥ = o � t−pq� as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (iii) (integral estimate of the gradients) � +∞ t0 tpq(2− 1 q) ∥∇f (y (t))∥2+ p−1 pq dt < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (iv) The solution trajectory y(t) converges weakly as t → +∞, and its limit belongs to S = argminf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Proof Again, we only consider the case p > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' We know from (22) that � +∞ t0 τ (t) ˙τ (t) ∥∇f (y (t))∥2 dt < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' By using successively the definition of λ, and the relation (19), we obtain τ (t) ˙τ (t) ∥∇f (y (t))∥2 = τ (t) ˙τ (t) [λ (t)]− 2p p−1 = [τ (t)]2− 1 q [λ (t)] 1 q − 2p p−1 ∀t ≥ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Therefore � +∞ t0 [τ (t)]2− 1 q [λ (t)] 1 q − 2p p−1 dt < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Let us apply Lemma 1 with a := 2 − 1 q and b = 2p p−1 − 1 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' We have b > a, a ≥ 0 for q ≥ 1 2, and qb + 1 b − a = 2pq p−1 2p p−1 − 2 = pq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Therefore τ (t) ≥ C1 (t − t0)pq ∀t ≥ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (33) This gives limt→+∞ τ (t) = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' According to Proposition 1, we can extract the results from Theorem 12 and the corresponding formulas (21), (22), (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Specifically, we obtain (i) for the values f (y (t)) − inf H f = o � 1 τ(t) � = o � 1 tpq � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (ii) for the gradients ∥∇f (y (t))∥ = o � 1 τ(t) � = o � t−pq� ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (iii) for the integral estimate of the gradients: we start from (30) � +∞ t0 τ (t) ˙τ (t) ∥∇f (y (t))∥2 dt < +∞, that we evaluate as follows: τ (t) ˙τ (t) ∥∇f (y (t))∥2 = τ (t) [τ (t)] q−1 q [λ (t)] 1 q ∥∇f (y (t))∥2 = τ(t)2− 1 q λ(t) 1 q ∥∇f (y (t))∥2 = τ(t)2− 1 q ∥∇f(y(t))∥2− p−1 pq ∀t ≥ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' According to (33) we deduce that� +∞ t0 tpq(2− 1 q) ∥∇f (y (t))∥2+ p−1 pq dt < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (iv) The convergence of the solution trajectory follows from an argument similar to that of the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' ⊓⊔ 9 Remark 1 a) We thus achieved our first goal which was to accelerate the convergence properties of the continuous steepest descent using closed-loop time scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' For example, concerning the convergence rate of the values, we passed from the convergence rate 1/t for the steepest descent to 1/t(1+q− 1 p) when the closed-loop control acts on the velocity, and 1/tpq in the case of the gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Clearly, by playing with the parameters p and q we can get arbitrary fast convergence results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' The same observation holds for the convergence of the gradients towards zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' b) By introducing a time scale function τ(·) which grows faster than the identity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' τ(t) ≥ t) either in open-loop or closed-loop, we have thus accelerated the continuous steepest descent dynamic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' The price to pay is that we no longer have an autonomous dynamic in (4), with as major drawback the fact that the coefficient in front of the gradient term tends towards infinity as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' This prevents from using gradient methods to discretize it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Recall that for gradient methods, the step size has to be less than or equal to twice the inverse of the Lipschitz constant of the gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' To overcome this, we come with the second step of our method which is averaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' 3 Accelerated gradient systems with closed-loop control of the damping 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='1 General results concerning time scale and averaging We will prove the following general result which puts forward a damped inertial dynamics which comes by time scale and averaging of the continuous steepest descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Then we will specialize it and consider time scale obtained in a closed-loop way, and thus cover the two model situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Theorem 4 Suppose that f : H → R satisfies (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Let γ > 1, and let τ : [t0, +∞[→ R++ be an increasing func- tion, continuously differentiable, such that limt→+∞ τ(t) = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Let x: [t0, +∞[ → H be a solution trajectory of the following second-order differential equation ¨x(t) + (1 + γ) ˙τ(t)2 − τ(t)¨τ(t) τ(t) ˙τ(t) ˙x(t) + γ ˙τ(t)2 τ(t) ∇f � x(t) + 1 γ τ(t) ˙τ(t) ˙x(t) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (34) Then we have the convergence rate of the values: as t → +∞ f (x(t)) − inf H f = o � 1 τ (t) � , (35) and x(t) converges weakly as t → +∞, and its limit belongs to S = argminf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Proof a) We first interpret x as coming from the time scale and averaging of the continuous steepest descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' We start from y(·) solution of ˙y (t) + ˙τ (t) ∇f (y (t)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (36) According to the time scale analysis developed in (5) we have f (y (t)) − inf H f = o � 1 τ (t) � as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' This means there exists a positive function ε which satisfies limt→+∞ ε (t) = 0 and f (y (t)) − inf H f = ε (t) τ (t) ∀t ≥ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (37) Let us define the time averaging process as the transformation from y to x according to the formula ˙x(t) + γ ˙τ(t) τ(t)x(t) = γ ˙τ(t) τ(t)y(t), (38) where γ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Equivalently y(t) = x(t) + 1 γ τ(t) ˙τ(t) ˙x(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (39) By derivating y(·) we get ˙y (t) = ˙x (t) + 1 γ τ(t) ˙τ(t) ¨x(t) + 1 γ ˙τ(t)2 − τ(t)¨τ(t) ˙τ(t)2 ˙x(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (40) 10 Replacing ˙y (t) by this expression in the constitutive rescaled steepest descent equation (36), we get ˙x (t) + 1 γ τ(t) ˙τ(t) ¨x(t) + 1 γ ˙τ(t)2 − τ(t)¨τ(t) ˙τ(t)2 ˙x(t) + ˙τ (t) ∇f � x(t) + 1 γ τ(t) ˙τ(t) ˙x(t) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Equivalently 1 γ τ(t) ˙τ(t) ¨x(t) + (1 + γ) ˙τ(t)2 − τ(t)¨τ(t) γ ˙τ(t)2 ˙x(t) + ˙τ (t) ∇f � x(t) + 1 γ τ(t) ˙τ(t) ˙x(t) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' After multiplication by γ ˙τ(t) τ(t) we get ¨x(t) + (1 + γ) ˙τ(t)2 − τ(t)¨τ(t) τ(t) ˙τ(t) ˙x(t) + γ ˙τ(t)2 τ(t) ∇f � x(t) + 1 γ τ(t) ˙τ(t) ˙x(t) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (41) b) Let us now come to the corresponding estimate of the convergence rates with x(t) instead of y(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' The idea is to express x as an average of y, and then conclude thanks to Jensen’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Set b(t) = ˙τ(t) τ(t) ≥ 0 (42) B(t) = � t t0 b(u)du = � t t0 ˙τ(u) τ(u)du = ln � τ(t) τ(t0) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (43) Therefore eB(t) = τ(t) τ(t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (44) In order to express x in terms of y, we need to integrate the first-order linear differential equation (38) which is written equivalently as follows ˙x(t) + γb(t)x(t) = γb(t)y(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' After multiplying by eγB(t), we get equivalently eγB(t) ˙x(t) + γb(t)eγB(t)x(t) = γb(t)eγB(t)y(t), that is, d dt � eγB(t)x(t) � = γb(t)eγB(t)y(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' After integration we get eγB(t)x(t) = eγB(t0)x(t0) + γ � t t0 b(u)eγB(u)y(u)du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' According to eγB(t0) = e0 = 1 we get x(t) = e−γB(t)x(t0) + γe−γB(t) � t t0 b(u)eγB(u)y(u))du = e−γB(t)y(t0) + γe−γB(t) � t t0 b(u)eγB(u)y(u)du, (45) where the last equality follows from the choice of the Cauchy data y(t0) = x(t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Then, observe that x(t) can be simply written as follows x(t) = � t t0 y(u) dµt(u), (46) where µt is the positive Radon measure on [t0, t] defined by µt = e−γB(t)δt0 + γb(u)eγ(B(u)−B(t))du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (47) Precisely, in (47), δt0 is the Dirac measure at t0, and b(u)eB(u)−B(t)du is the measure with density b(u)eB(u)−B(t) with respect to the Lebesgue measure on [t0, t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' According to γe−γB(t) � t t0 b(u)eγB(u)du = 1 − e−γB(t), we have that µt is a positive Radon measure on [t0, t] whose total mass is equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' It is therefore a probability measure, and x(t) is obtained by averaging the trajectory y(·) on [t0, t] with respect to µt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' 11 From there, let us show how to deduce fast convergence properties for the so defined trajectory x(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' According to the convexity of f, and Jensen’s inequality, we deduce that f �� t t0 y(u) dµt(u) � − inf H f = (f − inf H f) �� t t0 y(u)dµt(u) � ≤ � t t0 � f(y(u)) − inf H f � dµt(u) = � t t0 ε (u) τ (u)dµt(u), where the last inequality above comes from (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' According to the definition of µt (see (47)) and the formulation of x(t) (see (46)), we deduce that f (x(t)) − inf H f ≤ ε (t0) τ (t0)e−γB(t) + γe−γB(t) � t t0 ε (u) τ (u)b(u)eγB(u)du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Equivalently, τ (t) � f (x(t)) − inf H f � ≤ ε (t0) � τ (t) τ (t0) �1−γ + γτ (t) e−γB(t) � t t0 ε (u) τ (u)b(u)eγB(u)du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (48) Since γ > 1 and limt→+∞ τ(t) = +∞, it holds lim sup t→+∞ τ (t) � f (x(t)) − inf H f � ≤ γ lim sup t→+∞ τ (t) e−γB(t) � t t0 ε (u) τ (u)b(u)eγB(u)du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' It is therefore enough to show that lim sup t→+∞ � γτ (t) e−γB(t) � t t0 ε (u) τ (u)b(u)eγB(u)du � ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' In order to prepare for integration by parts, note that γb(u)eγB(u) = d du � eγB(u)� and ˙τ (u) [τ (u)]2−γ = d du � 1 γ − 1 1 [τ (t)]1−γ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Given an arbitrary η > 0 we consider Tη > t0 such that ε (u) ≤ η for every u ≥ Tη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' for every t ≥ Tη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' by integration by parts and by taking into consideration the relations (42)-(44),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' we get ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='γτ (t) e−γB(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='� t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='t0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='ε (u) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='τ (u)b(u)eγB(u)du ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='= γτ (t) e−γB(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='�� Tη ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='t0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='ε (u) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='τ (u)b(u)eγB(u)du + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='� t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='Tη ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='ε (u) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='τ (u)b(u)eγB(u)du ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='≤ τ (t) e−γB(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='� Tη ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='t0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='ε (u) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='τ (u)b(u)eγB(u)du + ηγ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='� t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='Tη ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='τ (u)b(u)eγB(u)du ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='= τ (t) e−γB(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='� Tη ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='t0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='ε (u) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='τ (u)b(u)eγB(u)du + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='η ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='τ (t)eγB(t) − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='η ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='τ (Tη)eγB(Tη) + η ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='� t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='Tη ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='˙τ (u) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='[τ (u)]2 eγB(u)du ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='= τ (t) e−γB(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='� Tη ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='t0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='ε (u) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='τ (u)b(u)eγB(u)du + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='η ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='τ (t)eγB(t) − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='η ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='τ (Tη)eγB(Tη) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='η ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='[τ (t0)]γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='� t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='Tη ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='˙τ (u) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='[τ (u)]2−γ du ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='= τ (t) e−γB(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='� Tη ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='t0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='ε (u) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='τ (u)b(u)eγB(u)du + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='η ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='τ (t)eγB(t) − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='η ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='τ (Tη)eγB(Tη) + η [τ (t0)]−γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='γ − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='[τ (t)]1−γ − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='[τ (Tη)]1−γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='≤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='� Tη ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='t0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='ε (u) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='τ (u)b(u)eγB(u)du ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='τ (t) e−γB(t) + η + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='η ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='γ − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='≤ C [τ (t)]1−γ + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='ηγ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='γ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Since limt→+∞ τ(t) = +∞, and γ > 1, we obtain lim sup t→+∞ � γτ (t) e−γB(t) � t t0 ε (u) τ (u)b(u)eγB(u)du � ≤ ηγ γ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (49) This being true for every η > 0, we infer f (x(t)) − inf H f = o � 1 τ (t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (50) 12 c) For trajectories convergence, we take advantage of the fact that the solution trajectory z (·) of the continuous steepest descent converges weakly towards a solution x∗ ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Since limt→+∞ τ(t) = +∞, this immediately implies that y(t) = z(τ(t)) converges weakly to x∗ as s → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' In other words, for each v ∈ H ⟨y (t) , v⟩ → ⟨x∗, v⟩ as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' To pass from the convergence of y to that of x, we use the interpretation of x as an average of y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' The convergence then results from the general property which says that convergence entails ergodic convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Let us make this precise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Using again that limt→+∞ τ(t) = +∞, we have x(t) ∼ γe−γB(t) � t t0 b (u) eγB(u)y (u) du = γ [τ(t)]γ � t t0 ˙τ (u) [τ(u)]γ−1 y(u)du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' After elementary calculus, we just need to prove that if a(·) is a positive real-valued function which verifies limu→+∞ a(u) = 0, then limt→+∞ A(t) = 0, where A(t) = γ [τ(t)]γ � t t0 ˙τ (u) [τ(u)]γ−1 a(u)du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Given an arbitrary η > 0, let us take Tη such that t0 < Tη and a(u) ≤ η for u ≥ Tη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' For t > Tη, we have A(t) = γ [τ(t)]γ � Tη t0 ˙τ (u) [τ(u)]γ−1 a(u)du + γ [τ(t)]γ � t Tη ˙τ (u) [τ(u)]γ−1 a(u)du ≤ γ [τ(t)]γ � Tη t0 ˙τ (u) [τ(u)]γ−1 a(u)du + ητ(t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Letting t converge to +∞ we get lim sup t→+∞ A(t) ≤ ητ(t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' This being true for any η > 0, we infer that limt→+∞ A(t) = 0, which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' ⊓⊔ Remark 2 By taking γ := α−1 2 and τ (t) := t2 2(α−1), equation (41) becomes (see [4]) ¨x(t) + α t ˙x(t) + ∇f � x(t) + t α − 1 ˙x(t) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' We have γ > 1 ⇐⇒ α > 3, which is in accordance with the convergence results attached to Nesterov method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='2 Damped inertial system via closed-loop control of the velocity Let us now examine the model situation where the time scaling is defined in a closed-loop way as a feedback control of the velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Completing this construction with the averaging process, as described as above, we get that (x, y): [t0, +∞[ → H × H is a solution trajectory of the following algebraic-differential system \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ˙y (t) + ˙τ (t) ∇f (y (t)) = 0 ˙x(t) + γ ˙τ(t) τ(t)x(t) − γ ˙τ(t) τ(t)y(t) = 0 τ (t) − 1 qq � t0 + � t t0 [λ (r)] 1 q dr �q = 0 [λ (t)]p ∥ ˙y (t)∥p−1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (51) By specializing Theorem 4 to this situation we get the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Theorem 5 Suppose that f : H → R satisfies (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Let q > 0, p ≥ 1, γ > 1 and x: [t0, +∞[ → H be a solution trajectory of the following system \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ¨x(t) + (1 + γ) ˙τ(t)2 − τ(t)¨τ(t) τ(t) ˙τ(t) ˙x(t) + γ ˙τ(t)2 τ(t) ∇f � x(t) + 1 γ τ(t) ˙τ(t) ˙x(t) � = 0 τ (t) − 1 qq � t0 + � t t0 [λ (r)] 1 q dr �q = 0 [λ (t)]p |˙τ (t)|p−1 ����∇f � x (t) + 1 γ τ(t) ˙τ(t) ˙x (t) ����� p−1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (52) 13 Then we have the fast convergence of values: as t → +∞ f(x(t)) − inf H f = o � 1 t1+q− 1 p � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (53) Moreover, the solution trajectory x(t) converges weakly as t → +∞, and its limit belongs to S = argminf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Proof We showed in the proof of Theorem 4 how to pass from (51) to (52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Conversely, let x(·) be a solution trajectory of the damped inertial dynamic (52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Let us show that by setting y (t) = 1 γ τ (t) ˙τ (t) ˙x (t) + x (t) , then (x, y): [t0, +∞[ → H × H is a solution trajectory of \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ˙y (t) + ˙τ (t) ∇f (y (t)) = 0 ˙x(t) + γ ˙τ(t) τ(t)x(t) − γ ˙τ(t) τ(t)y(t) = 0 τ (t) − 1 qq � t0 + � t t0 [λ (r)] 1 q dr �q = 0 [λ (t)]p ∥ ˙y (t)∥p−1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (54) Indeed, by taking the time derivative of y(·), as given by the second equation of (54), we get ˙y (t) = 1 γ τ (t) ˙τ (t) ¨x (t) + 1 γ � 1 + γ − τ (t) ¨τ (t) [˙τ (t)]2 � ˙x (t) = 1 γ τ (t) ˙τ (t) � ¨x (t) + (1 + γ) [ ˙τ(t)]2 − τ(t)¨τ(t) τ(t) ˙τ(t) ˙x(t) � = − ˙τ (t) ∇f � x (t) + 1 γ τ (t) ˙τ (t) ˙x (t) � = − ˙τ (t) ∇f (y (t)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' This gives the first equation in (54) and [λ (t)]p ∥ ˙y (t)∥p−1 = [λ (t)]p | ˙τ (t)|p−1 ����∇f � x (t) + 1 γ τ (t) ˙τ (t) ˙x (t) ����� p−1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' This shows the equivalence of the two systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' According to Theorem 2, and formula (31), there exists a constant C1 > 0 such that τ (t) ≥ C1 (t − t0)1+q− 1 p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (55) Therefore limt→+∞ τ(t) = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' According to Theorem 4 we deduce f (x(t)) − inf H f = o � 1 t1+q− 1 p � , (56) and the convergence of the trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' ⊓⊔ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='3 Damped inertial system via closed-loop control of the gradient We proceed in parallel to the previous section to obtain the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Theorem 6 Suppose that f : H → R satisfies (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Let q > 0, p ≥ 1, γ > 1, and x: [t0, +∞[ → H be a solution trajectory of the following system \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ¨x(t) + (1 + γ) ˙τ(t)2 − τ(t)¨τ(t) τ(t) ˙τ(t) ˙x(t) + γ ˙τ(t)2 τ(t) ∇f � x(t) + 1 γ τ(t) ˙τ(t) ˙x(t) � = 0 τ (t) − 1 qq � t0 + � t t0 [λ (r)] 1 q dr �q = 0 [λ (t)]p ����∇f � x (t) + 1 γ τ (t) ˙τ (t) ˙x (t) ����� p−1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (57) 14 Then we have the fast convergence of values: as t → +∞ f(x(t)) − inf H f = o � 1 tpq � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (58) Moreover, the solution trajectory x(t) converges weakly as t → +∞, and its limit belongs to S = argminf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Proof Let x(·) be a solution trajectory of the damped inertial dynamic (57).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Let us show show that by setting y (t) = 1 γ τ (t) ˙τ (t) ˙x (t) + x (t) , then (x, y): [t0, +∞[ → H × H is a solution trajectory of \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ˙y (t) + ˙τ (t) ∇f (y (t)) = 0 ˙x(t) + γ ˙τ(t) τ(t)x(t) − γ ˙τ(t) τ(t)y(t) = 0 τ (t) − 1 qq � t0 + � t t0 [λ (r)] 1 q dr �q = 0 [λ (t)]p ∥∇f (y (t))∥p−1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (59) Indeed, by the same argument as for the velocity case, we get ˙y (t) = − ˙τ (t) ∇f (y (t)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' This gives the first equation in (59) and [λ (t)]p ∥∇f(y (t))∥p−1 = [λ (t)]p ����∇f � x (t) + 1 γ τ (t) ˙τ (t) ˙x (t) ����� p−1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' This shows the equivalence of the two systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' According to Theorem 3, and formula (33), there exists a constant C1 > 0 such that τ (t) ≥ C1 (t − t0)pq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (60) Therefore from Theorem 4 we deduce, as t → +∞ f (x(t)) − inf H f = o � 1 tpq � , (61) and the convergence of the trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' ⊓⊔ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='4 Comparison with the Lin-Jordan approach In [19],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' the authors study the second-order closed-loop dynamical system \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ¨x (t) + �2 ˙τ (t) τ (t) − ¨τ (t) ˙τ (t) � ˙x (t) + [ ˙τ (t)]2 τ (t) ∇2f (x (t)) ˙x (t) + ˙τ (t) [ ˙τ (t) + ¨τ (t)] τ (t) ∇f (x (t)) = 0 τ (t) − 1 4 �� t 0 � λ (t)dr + c �2 = 0 [λ (t)]p ∥∇f (x (t))∥p−1 = θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (62) whose first-order reformulation reads \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ˙y (t) + ˙τ (t) ∇f (x (t)) = 0 ˙x (t) + ˙τ(t) τ(t) (x (t) − y (t)) + [ ˙τ(t)]2 τ(t) ∇f (x (t)) = 0 τ (t) − 1 4 �� t 0 � λ (t)dr + c �2 = 0 [λ (t)]p ∥∇f (x (t))∥p−1 = θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (63) where c > 0 and 0 < θ < 1 are given parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' See also [20] for some extensions to monotone inclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' a) In [19], the authors obtained the following convergence rate of function values 15 f (x (t)) − inf H f = O � 1 t 3p+1 2 � as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Note that the last two equations in (63) are nothing else than those in (32) with q := 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' For comparison, in our approach the convergence rate of the values obtained in Theorem 6 when q = 2 is f (x (t)) − inf H f = o � 1 t2p � which is better for every p > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' b) Let us now compare the convergence estimates of the gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' In [19], the authors obtain the integral estimate � +∞ t0 t 3p+1 2 ∥∇f (x (t))∥ p+1 p dt < +∞, which leads to inf t0≤σ≤t ∥∇f (x (σ))∥ = O � t− 3p 2 � as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' In our approach, the right variable to consider is y(t), instead of x(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' According to (22) we have � +∞ t0 τ (t) ˙τ (t) ∥∇f (y (t))∥2 dt < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Since q = 2, according to (19) we have ˙τ (t) = [τ (t)] 1 2 [λ (t)] 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Therefore τ (t) ˙τ (t) ∥∇f (y (t))∥2 = τ (t) 3 2 [λ (t)] 1 2 ∥∇f (y (t))∥2 = τ (t) 3 2 ∥∇f (y (t))∥2− p−1 2p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Since τ(t) ≥ Ct2p, we deduce that � +∞ t0 t3p ∥∇f (y (t))∥ 3p+1 2p dt < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' which leads to inf t0≤σ≤t ∥∇f (y (σ))∥ = O � t−2p� as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Again, our approach gives a better convergence rate than [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Let us also specify that our analysis provides the convergence of the trajectories, which is an open question for [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Moreover, since our approach is consistent with the steepest continuous descent, it can naturally be extended to the non-smooth case, and to the case of cocoercive operators, as it was done in the open-loop case in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='5 The limiting case γ = 1 Our previous results are valid under the assumption γ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' It is a natural question to examine the limiting case γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Close examination of the proof of the theorem reveals a slight change in the integration procedure and a logarithm factor appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' The corresponding result obtained is written as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Theorem 7 Suppose that f : H → R satisfies (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Let x: [t0, +∞[ → H be a solution trajectory of the following second-order differential equation ¨x(t) + 2 [˙τ(t)]2 − τ(t)¨τ(t) τ(t) ˙τ(t) ˙x(t) + [˙τ(t)]2 τ(t) ∇f � x(t) + τ(t) ˙τ(t) ˙x(t) � = 0 (64) where τ : [t0, +∞[ → R++ is an increasing function, continuously differentiable, and satisfying limt→+∞ τ(t) = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Then we have the convergence rate of the values: as t → +∞ f (x(t)) − inf H f = o �ln (τ (t)) τ (t) � , (65) 16 and the solution trajectory x(t) converges weakly as t → +∞, and its limits belongs to S = argminf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Suppose moreover that there exists some θ > 0 and C1 > 0 such that for t sufficiently large (A)asymp τ(t) ≥ C1 (t − t0)θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (66) Then we have the fast convergence of values: as t → +∞ f (x(t)) − inf H f = o �ln (t) tθ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (67) When specialized to the closed-loop control of the velocity, we obtain f (x(t)) − inf H f = o � ln (t) t1+q− 1 p � , (68) and in the case of the closed-loop control of the gradient f (x(t)) − inf H f = o �ln (t) tpq � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (69) So, the convergence rates are a little less good because of the logarithm term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' 4 Associated proximal algorithms 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='1 A proximal-explicit discretization In the following, we present a numerical approach based on a proximal-explicit temporal discretization of the closed-loop systems investigated in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' By proximal-explicit we mean that the function f is evaluated using a proximal step while the step size sequence (λk)k≥0 and the time scaling sequence (τk)k≥0 are computed explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' This makes our numerical scheme much easier implementable than the numerical algorithm proposed in [19] as well as the large-step A HPE approach by Monteiro and Svaiter [23] which are in fact approximations of a proximal-implicit discrete time method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' We restrict ourselves to the case q = 1, which gives ˙τ (t) = λ (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' In this case, the continuous time closed-loop dynamical system is written as follows \uf8f1 \uf8f2 \uf8f3 ˙y (t) + λ (t) ∇f (y (t)) = 0 [λ (t)]p [G (y (t))]p−1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (70) Let us describe the general structure of the algorithm which is obtained by a proximal-explicit discretization of the continuous system (70).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Given yk, yk−1 in H, we first define λk by [λk]p [G (yk, yk−1)]p−1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' and consider then an implicit finite difference scheme for the first equation of (70) yk+1 − yk + λk∇f (yk+1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (71) This gives the following algorithm, called PEAS for Proximal Explicit Algorithm with Adaptive Step Size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Algorithm 1: Proximal-explicit algorithm with adaptive step size (PEAS) Input: y0 ̸= y−1 ∈ H 1 for k = 0, 1, · · · do 2 λk := [G (yk, yk−1)]− p−1 p 3 yk+1 := proxλkf (yk) 4 end Note that (λk)k≥0 is computed explicitly in terms of (yk)k≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' In other words, the definition of the sequence (λk)k≥0 is decoupled from the computation of (yk)k≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' This is different from the method in [19], which ultimately leads to the large-step A HPE approach by Monteiro and Svaiter in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' 17 Let us now specify the link between λk and τk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' We start from the relation (recall that we take q = 1) ˙τ (t) = λ (t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (72) Then, for every k ≥ 0 we discretize (72) as follows τk+1 − τk = λk ⇐⇒ τk+1 = λk + τk (73) with the convention λ0 := t0 and τ0 := 0, which then yields τk = �k−1 i=0 λi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Drawing inspiration from continuous analysis, we will first show that the function value f (yk) − infH f attains the o � 1 τk+1 � rate of convergence, and the sequence (yk)k≥0 converges weakly to a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Then, as a crucial result, we will derive a lower bound of τk+1 in terms of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' The following result emphasizes that the rate of convergence and summability results holds for (yk)k≥0 for arbitrary step sizes λk that satisfy � k≥0 λk = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' The proof is an adaptation of [4, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Theorem 8 Let y0 ∈ H, (λk)k≥0 be a given positive sequence satisfying � k≥0 λk = +∞, τ0 = 0 and τk = �k−1 i=0 λi for every k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Then for any sequence (yk)k≥0 generated by the proximal algorithm yk+1 := proxλkf (yk) ∀k ≥ 0, (74) the following properties are satisfied: (i) (summability of function values) � k≥0 λk � f (yk+1) − inf H f � < +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (ii) (summability of gradients) � k≥0 τkλk ∥∇f (yk+1)∥2 < +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (iii) (summability of velocities) � k≥0 τk λk ∥yk+1 − yk∥2 < +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (iv) (convergence of function values) f (yk+1) − inf H f = o � 1 τk+1 � as k → +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (v) (convergence of gradient) ∥∇f (yk+1)∥ = o \uf8eb \uf8ed 1 ��k l=0 τlλl \uf8f6 \uf8f8 as k → +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (vi) the sequence of iterates (yk)k≥0 converges weakly as k → +∞, and its limit belongs to S = argminH f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Proof Let k ≥ 0 be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Take z∗ ∈ S = argminf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' According to (71) and the convexity of f, we deduce that 1 2 ∥yk+1 − z∗∥2 = 1 2 ∥yk − z∗∥2 + ⟨yk+1 − z∗, yk+1 − yk⟩ − 1 2 ∥yk+1 − yk∥2 = 1 2 ∥yk − z∗∥2 − λk ⟨yk+1 − z∗, ∇f (yk+1)⟩ − 1 2λ2 k ∥∇f (yk+1)∥2 ≤ 1 2 ∥yk − z∗∥2 − λk � f (yk+1) − inf H f � − 1 2λ2 k ∥∇f (yk+1)∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (75) Statement (i) follows from [14, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' In addition, the limit limk→+∞ ∥yk − z∗∥ ∈ R exists, which means that the first condition of the discrete Opial’s lemma is fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' On the other hand, the sequence (f (yk) − infH f)k≥0 is nonincreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Precisely, we have for every k ≥ 0 � f (yk) − inf H f � − � f (yk+1) − inf H f � ≥ ⟨∇f (yk+1) , yk − yk+1⟩ = λk ∥∇f (yk+1)∥2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (76) According to [6, Lemma 22] we get f (yk+1) − inf H f = o � 1 �k i=0 λi � , which proves (iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Let k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Multiplying both sides of (76) by τk = �k−1 i=0 λi > 0, then adding the result into (75), we get τk+1 � f (yk+1) − inf H f � + 1 2 ∥yk+1 − z∗∥2 ≤ τk � f (yk) − inf H f � + 1 2 ∥yk − z∗∥2 − 1 2λ2 k ∥∇f (yk+1)∥2 − τkλk ∥∇f (yk+1)∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' 18 This implies � k≥0 λ2 k ∥∇f (yk+1)∥2 < +∞ and � k≥1 τkλk ∥∇f (yk+1)∥2 < +∞, which yields (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' From (71), we infer (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' To deduce (v), it suffices to show that the sequence (∥∇f (yk) ∥)k≥0 is nonincreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Indeed, it follows from (74) and the cocoercivity of ∇f that 1 2 ∥∇f (yk+1)∥2 = 1 2 ∥∇f (yk)∥2 + ⟨∇f (yk+1) , ∇f (yk+1) − ∇f (yk)⟩ − 1 2 ∥∇f (yk+1) − ∇f (yk)∥2 = 1 2 ∥∇f (yk)∥2 − 1 λk ⟨yk+1 − yk, ∇f (yk+1) − ∇f (yk)⟩ − 1 2 ∥∇f (yk+1) − ∇f (yk)∥2 ≤ 1 2 ∥∇f (yk)∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Taking into account also (ii), we obtain (v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Finally, according to the assumption � k≥0 λk = +∞, and (iv), we have that limk→+∞ f (yk) = infH f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Since f is convex and lower semicontinuous, the second condition of Opial’s lemma is also fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' This gives the weak convergence of the sequence (yk)k≥0 to an element in S = argminf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' ⊓⊔ Then, we give a statement which can be seen as a discrete counterpart of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' The result is more complex not only because we are in the discrete setting, but also because it allows an explicit choice of the stepsize, as we will see later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Lemma 2 Let (λk)k≥0 be a positive sequence and (τk)k≥0 such that τ0 = 0 and τk = �k−1 i=0 λi for all k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Suppose that there exist C2 > 0 and a, b, c ≥ 0 such that b + c > a and � k≥0 τ a k λ−b k λ−c k+1 ≤ C2 < +∞ Then there exists C3 > 0 such that for every k ≥ 1 it holds τk+1 ≥ C3k b+c+1 b+c−a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (77) Proof By applying the H¨older inequality twice we get for all k ≥ 0 k � i=0 τ a b+c+1 i ≤ � k � i=0 τ a i λ−b i λ−c i+1 � 1 b+c+1 � k � i=0 λi � b b+c+1 � k � i=0 λi+1 � c b+c+1 ≤ C 1 b+c+1 2 �k+1 � i=0 λi � b+c b+c+1 = C 1 b+c+1 2 τ b+c b+c+1 k+2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (78) If a = 0 then (77) follows immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' From now on we suppose that a > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Inequality (78) becomes k � i=0 τ a b+c+1 i ≤ C 1 b+c+1 2 � τ a b+c+1 k+2 � b+c a ∀k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (79) Following the continuous counterpart, let us define Cb+c := C 1 b+c+1 2 > 0 and Ak := k � i=0 τ a b+c+1 i ∀k ≥ 0 so that (79) becomes Ak ≤ Cb+c (Ak+2 − Ak+1) b+c a ∀k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' From here, C − a b+c b+c ≤ A − a b+c k (Ak+2 − Ak+1) ∀k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (80) For convenience, we define the following function ψ: R++ → R++ as ψ (r) := r− a b+c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' It is clear that d dr � b + c b + c − ar1− a b+c � = ψ (r) and ˙ψ (r) = − a b + cr− a b+c −1 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Since (Ak)k≥0 is increasing, this means ψ (Ak+2) ≤ ψ (r) ≤ ψ(Ak) for every Ak ≤ r ≤ Ak+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Let k ≥ 1 fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' We consider two separate cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' 19 Case 1: ψ (Ak) ≤ 2ψ (Ak+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Then (80) leads to C − a b+c b+c ≤ A − a b+c k (Ak+2 − Ak) = ψ (Ak) (Ak+2 − Ak) ≤ 2ψ (Ak+2) (Ak+2 − Ak) = 2ψ (Ak+2) � Ak+2 Ak 1dr ≤ 2 � Ak+2 Ak ψ (r) dr = 2 b + c b + c − a � A 1− a b+c k+2 − A 1− a b+c k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Case 2: ψ (Ak) > 2ψ (Ak+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' This is equivalent to Ak+2 > 2 b+c a Ak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Since b + c > a, we can deduce further A 1− a b+c k+2 > 2 b+c a −1A 1− a b+c k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Consequently, A 1− a b+c k+2 − A 1− a b+c k > � 2 b+c a −1 − 1 � A 1− a b+c k ≥ � 2 b+c a −1 − 1 � A 1− a b+c 1 , recall that the last inequality follows from the increasing property of (Ak)k≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' In conclusion, for every k ≥ 0 we have A 1− a b+c k+2 − A 1− a b+c k ≥ C4 := min �1 2 � 1 − a b + c � C − a b+c b+c , � 2 b+c a −1 − 1 � A 1− a b+c 1 , A 1− a b+c 2 � > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Telescoping sum arguments combined with (79) imply for every k ≥ 1 C4k ≤ A 1− a b+c 2k − A 1− a b+c 0 ≤ A 1− a b+c 2k ≤ C b+c−a (b+c)(b+c+1) 2 τ b+c−a b+c+1 2k+2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' This gives for every k ≥ 1 τ2k+3 ≥ τ2k+2 ≥ �C3k b+c+1 b+c−a , where �C3 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' We therefore deduce that there exists C3 > 0 such that τk+1 ≥ C3k b+c+1 b+c−a ∀k ≥ 1, which gives (77).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' ⊓⊔ Following a plan identical to the continuous case, we successively consider the case where the control by feedback is formulated in terms of speed, then of gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='2 Adaptive stepsize rules resulting from the discretization of the velocity based system In this subsection we specialize the algorithm PEAS to the case where G(yk, yk−1) = ∥yk − yk−1∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Algorithm 2: Proximal algorithm with adaptive step size defined via velocity Input: y0 ̸= y−1 ∈ H 1 for k = 0, 1, · · · do 2 if ∇f (yk) = 0 then 3 stop 4 else 5 λk := ∥yk − yk−1∥− p−1 p 6 yk+1 := proxλkf (yk) 7 end 8 end Theorem 9 Let (yk)k≥0 be the sequence generated by Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Then it holds f (yk) − inf H f = o � 1 k2− 1 p � as k → +∞, and the sequence of iterates (yk)k≥0 converges weakly as k → +∞, and its limit belongs to S = argminH f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' 20 Proof By the choice of the step size, we have from Theorem 8 (iii) that � k≥0 τk λk ∥yk+1 − yk∥2 = � k≥0 τkλ−1 k λ − 2p p−1 k+1 < +∞, where τ0 = 0 and τk := �k−1 i=0 λi for every k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' We are in position to apply Lemma 2 with (a, b, c) := � 1, 1, 2p p−1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' We get τk+1 ≥ C3k2− 1 p ∀k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (81) Therefore � k≥0 λk = limk→+∞ τk = +∞, and we can apply Theorem 8 to obtain the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' ⊓⊔ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='3 Adaptive stepsize resulting from the discretization of the gradient based system Now let us specialize the algorithm PEAS to the case where G(yk, yk−1) = ∥∇f(yk)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Algorithm 3: Proximal algorithm with adaptive step size defined via gradient Input: y0 ∈ H 1 for k = 0, 1, · · · do 2 if ∇f (yk) = 0 then 3 stop 4 else 5 λk := ∥∇f (yk)∥− p−1 p 6 yk+1 := proxλkf (yk) 7 end 8 end Theorem 10 Let (yk)k≥0 be the sequence generated by Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Then it holds f (yk+1) − inf H f = o � 1 k2− 1 p � as k → +∞ and the sequence of iterates (yk)k≥0 converges weakly as k → +∞, and its limit belongs to S = argminH f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Proof In this case we have from Theorem 8 (ii) � k≥0 τkλk ∥∇f (yk+1)∥2 = � k≥0 τkλkλ − 2p p−1 k+1 < +∞, (82) where τ0 = 0 and τk := �k−1 i=0 λi for every k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Let us establish an inequality of the type ∥∇f (yk)∥ ≤ Ck ∥∇f (yk+1)∥ , for some sequence Ck > 0 which is to be precised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' We have for all k ≥ 0 ∥∇f (yk)∥2 = ∥∇f (yk+1)∥2 − 2 ⟨∇f (yk+1) , ∇f (yk+1) − ∇f (yk)⟩ + ∥∇f (yk+1) − ∇f (yk)∥2 = ∥∇f (yk+1)∥2 + 2 λk ⟨yk+1 − yk, ∇f (yk+1) − ∇f (yk)⟩ + ∥∇f (yk+1) − ∇f (yk)∥2 ≤ ∥∇f (yk+1)∥2 + �2L λk + L2 � ∥yk+1 − yk∥2 = (1 + Lλk)2 ∥∇f (yk+1)∥2 ≤ (1 + Lλk+1)2 ∥∇f (yk+1)∥2 (83) where L > 0 denotes the Lipschitz constant of ∇f on a bounded set containing the sequence (yk)k≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Combining (82) and (83), we get � k≥0 τkλk 1 (1 + Lλk+1)2 ∥∇f (yk)∥2 = � k≥0 τkλk 1 (1 + Lλk+1)2 λ − 2p p−1 k < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (84) 21 Let us now show that limk→+∞ λk = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' According to the decreasing property of the sequence (f (yk) − infH f)k≥0, by summing inequalities (76) we get � k≥0 λk ∥∇f (yk+1)∥2 < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (85) From the closed-loop rule we deduce that � k≥0 λkλ − 2p p−1 k+1 < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (86) Therefore lim k→+∞ λkλ − 2p p−1 k+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Since (λk)k≥0 is increasing, let us denote by l > 0 its limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' If l is finite then, by passing to the limit on the above inequality we get l1− 2p p−1 = 0, a clear contradiction with l > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Therefore lim k→+∞ λk = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' In this case 1 (1+Lλk+1)2 ∼ (Lλk+1)−2, which gives � k≥0 τkλ 1− 2p p−1 k λk+1 −2 < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (87) We are in position to apply Lemma 2 with (a, b, c) := � 1, 2p p−1 − 1,2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' We get τk+1 ≥ C3k2− 1 p ∀k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' We have � k≥0 λk = limk→+∞ τk = +∞, and we can apply Theorem 8 to obtain, as k → +∞ f (yk+1) − inf H f = o � 1 k2− 1 p � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' ⊓⊔ Remark 3 Note that the closed-loop control of the velocity and the closed-loop control of the gradient give the same convergence rate of the values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Clearly, we have obtained a faster convergence result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' 5 Inertial proximal algorithms obtained by closed-loop damping Let us now consider the convergence properties of the sequences (xk)k≥0 which are obtained by applying the averaging process to the sequences generated by Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Indeed, we limit our investigation to the closed loop control of the velocity, the case of the closed loop control of the gradient is very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Let us discretize the continuous averaging relation ˙x(t) + ˙τ(t) τ(t) (x(t) − y(t)) = 0 as follows (recall that, because of the choice q = 1, we have ˙τ(t) = λ(t)) xk+1 − xk + λk τk+1 (xk − yk+1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Equivalently xk+1 = � 1 − λk τk+1 � xk + λk τk+1 yk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' This gives the following proximal inertial algorithm: Theorem 11 Let (xk)k≥0 be the sequence generated by Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Then it holds f (xk) − inf H f = O � 1 k2− 1 p � as k → +∞ and the sequence of iterates (xk)k≥0 converges weakly as k → +∞, and its limit belongs to S = argminH f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' 22 Algorithm 4: Proximal inertial algorithm with adaptive step size defined via velocity Input: τ0 := 0 and x0, y0 ̸= y−1 ∈ H 1 for k = 0, 1, · · · do 2 if ∇f (yk) = 0 then 3 stop 4 else 5 λk := ∥yk − yk−1∥− p−1 p 6 yk+1 := proxλkf (yk) 7 τk+1 := τk + λk 8 xk+1 := � 1 − λk τk+1 � xk + λk τk+1 yk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' 9 end 10 end Proof Let k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' By definition of xk+1 we have τk+1xk+1 = (τk+1 − λk)xk + λkyk+1 which gives (recall that τk+1 = �k i=0 λi) τk+1xk+1 − τkxk = (τk+1 − λk)xk + λkyk+1 − τkxk = (τk+1 − τk − λk)xk + λkyk+1 = λkyk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Therefore, by telescoping arguments we obtain xk+1 = �k i=0 λiyi+1 τk+1 ∀k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (88) By convexity of f we infer f (xk+1) − inf H f = � f − inf H f � (xk+1) = � f − inf H f � ��k i=0 λiyi+1 τk+1 � ≤ 1 τk+1 k � i=0 λi � f − inf H f � (yi+1) = 1 τk+1 k � i=0 λi � f (yi+1) − inf H f � , By Theorem 8 (i), we have � k≥0 λk (f (yk+1) − infH f) < +∞, and by (81) we have τk+1 ≥ k2− 1 p , which gives the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' The weak convergence of (xk)k≥0 to an element in S = argminH f follows from the weak convergence of (yk)k≥0 and the Stolz-Ces´aro Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' ⊓⊔ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='1 Geometric interpretation of PEAS First note that (PEAS) can be equivalently written as follows xk+1 = � 1 − λk τk+1 � xk + λk τk+1 proxλkf � xk−1 + τk λk−1 (xk − xk−1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (89) Since τk λk−1 > 1, the algorithm first involves an extrapolation step (this is the inertial aspect), then a proximal step, and finally a relaxation step which balances the inertia effect and dampens the oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' This is shown in the figure below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' We set θk = λk τk+1 ∈]0,1[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Despite some analogies, (PEAS) is different from the relaxed inertial proximal algorithm (RIPA) con- sidered by Attouch and Cabot in [7], and which writes � yk = xk + αk(xk − xk−1) xk+1 = (1 − ρk)yk + ρk proxλkf(yk) (90) As main difference, in (PEAS) relaxation is taken between xk and proxλkf(yk), while in (RIPA) it is taken between yk and proxλkf(yk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Consequently (PEAS) involves a Hessian damping effect which is not present in (RIPA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Note in (PEAS) the balance between the extrapolation (inertial, acceleration) effect and the relaxation effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Moreover, our construction provides coefficients which are generated automatically in closed loop way, whereas in (RIPA) they require subtle adjustment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' The importance of the relaxation technique when combined with inertia has been put to the fore in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' According to (88), xk+1 can be 23 yk = xk−1 + 1 θk−1 (xk − xk−1) xk xk−1 xk+1 = (1 − θk) xk + θk proxλkf (yk) proxλkf(yk) S � � �� � � � � � � ❛❛❛❛❛❛❛❛❛❛❛❛❛❛ ✚ ✚ ✚ ❂ ✄ ✄ ✄✄ ✪ ✪ ✪ ✪ Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='1 A geometrical illustration of algorithm PEAS interpreted as an average of the {yn : 0 ≤ n ≤ k + 1}, which makes our approach somewhat analogous to the nonlinear averaging technique developed in [27], where it is assumed that there is a unique minimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Averaging techniques have also been used in [26] in the context of hybrid systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Indeed, adjusting the damping in a closed-loop ad hoc manner bears some analogy to restarting methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' 6 Conclusion, perspective Our study proposes new fast adaptive optimization methods for convex optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' We have shown that the time scaling and averaging technique, previously developed by the authors in the context of non-autonomous systems, can be developed by taking closed-loop time parameterization, giving rise to au- tonomous dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' The method turns out to be flexible, because it is based on elementary mathematical tools, namely the dynamics of the steepest descent, and the operations of temporal parameterization and averaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' It is therefore not necessary to redo a Lyapunov analysis, one relies on the classic results for the steepest descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' The results obtained for the continuous dynamics pass quite naturally to the correspond- ing proximal algorithms, where the iterates are expressed in a direct way according to the proximal terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' This study is one of the very first to develop an algorithmic framework based on autonomous dynamics and which, when specialized, provides the convergence rates of the dynamical surrogate of the Nesterov acceler- ation gradient method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Another important aspect of our analysis is that it exhibits Hessian-driven damping, which plays a key role in damping oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Our work opens up many perspectives, our method natu- rally extending to gradient algorithms, proximal-gradient algorithms for composite optimization, cocercive monotone operators, and the study of the stochastic version, to name only a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' 7 Appendix 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='1 Classical facts concerning the continuous steepest descent Consider the classical continuous steepest descent (SD) ˙z(t) + ∇f(z(t)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (91) Under the standing assumption (A) on f, we know that, for any z0 ∈ H there exists a unique classical global solution z ∈ C1([t0, +∞[: H) of (SD) satisfying z(t0) = z0, see [5, Theorem 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' We fix t0 as the origin of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Recall classical facts concerning the continuous steepest descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Theorem 12 Suppose that f : H → R satisfies (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Let z : [t0, +∞[ → H be a solution trajectory of ˙z(t) + ∇f(z(t)) = g(t) (92) where g: [t0, +∞[ → H is such that � +∞ t0 ∥g (t)∥ dt < +∞ and � +∞ t0 t ∥g (t)∥2 dt < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (93) Then the following statements are satisfied: 24 (i) (convergence of gradients towards zero) ∥∇f (z (t))∥ = o � 1 √ t � as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (ii) (integral estimate of the velocities) � +∞ t0 t ∥ ˙z (t)∥2 dt < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (iii) (integral estimate of the gradients) � +∞ t0 t ∥∇f (z (t))∥2 dt < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (iv) (convergence of values) f (z (t)) − infH f = o �1 t � as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (v) (improved convergence rates of gradients) if g(t) ≡ 0, then ∥∇f (z (t))∥ = o �1 t � as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (vi) The solution trajectory z(t) converges weakly as t → +∞, and its limit belongs to S = argminf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' If g(t) ≡ 0 we have that t �→ ∥∇f (z (t))∥ is nonincreasing since in this case d dt ∥∇f (z (t))∥2 = 2 � ∇f (z (t)) , d dt∇f (z (t)) � = −2 � ˙z (t) , d dt∇f (z (t)) � ≤ 0 ∀t ≥ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Therefore, from the integral estimate of the gradients we deduce that ∥∇f (z (t))∥ = o � 1 t �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content='2 Auxiliary result Opial’s Lemma is a basic ingredient of the convergence analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Lemma 3 (Opial) Let S be a nonempty subset of H and let (xk)k≥0 be a sequence in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Assume that (i) for every z ∈ S, limk→+∞ ∥xk − z∥ exists;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' (ii) every weak sequential limit point of (xk)k≥0, as k → +∞, belongs to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Then (xk)k≥0 converges weakly as k → +∞, and its limit belongs to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' Funding The research of RIB and DKN has been supported by FWF (Austrian Science Fund), projects W 1260 and P 34922-N, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQfzfkv/content/2301.00701v1.pdf'} +page_content=' References 1.' 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0000000000000000000000000000000000000000..ff1dd984cbee4ad0afe0c2ce32fdf1a1e72505a9 --- /dev/null +++ b/wNFLT4oBgHgl3EQfki_9/content/tmp_files/2301.12116v1.pdf.txt @@ -0,0 +1,971 @@ +arXiv:2301.12116v1 [cond-mat.stat-mech] 28 Jan 2023 +Non-quasistatic response coefficients and dissipated availability for macroscopic thermodynamic +systems +Yuki Izumida +Department of Complexity Science and Engineering, Graduate School +of Frontier Sciences, The University of Tokyo, Kashiwa 277-8561, Japan∗ +The characterization of finite-time thermodynamic processes is of crucial importance for extending equilib- +rium thermodynamics to nonequilibrium thermodynamics. The central issue is to quantify responses of ther- +modynamic variables and irreversible dissipation associated with non-quasistatic changes of thermodynamic +forces applied to the system. In this study, we derive a simple formula that incorporates the non-quasistatic +response coefficients with Onsager’s kinetic coefficients, where the Onsager coefficients characterize the relax- +ation dynamics of fluctuation of extensive thermodynamic variables of semi-macroscopic systems. Moreover, +the dissipated availability that quantifies the efficiency of the irreversible thermodynamic process is formulated +in terms of the derived non-quasistatic response coefficients. The present results are demonstrated by using an +ideal gas model. The present results are, in principle, verifiable through experiments and are thus expected to +provide a guiding principle for the nonequilibrium control of macroscopic thermodynamic systems. +I. +INTRODUCTION +A quasistatic thermodynamic process is an important build- +ing block of thermodynamics. Consider a response of thermo- +dynamic variables X upon quasistatically added small pertur- +bation F: +x = χF, +(1) +where x is the deviation of X from the equilibrium value, and +χ denotes the quasistatic response coefficient to the perturba- +tion F. According to equilibrium statistical mechanics, the +quasistatic response coefficient χ can be calculated based on +the equilibrium correlation function of fluctuations of thermo- +dynamic variables in the absence of perturbation, which is re- +ferred to as the fluctuation-response relation [1]. +When the small perturbation is added slowly in time but +non-quasistatically, Eq. (1) may be generalized as +x = χF − R˙F, +(2) +where the dot denotes the time derivative, and we call R the +non-quasistatic response coefficients. +Using the linear re- +sponse theory [2, 3], in the same manner as the quasistatic +response coefficient χ, R can be expressed in terms of tem- +poral equilibrium correlation functions of fluctuations [4, 5]. +A similar quantity has been studied extensively [4–12], some- +times referred to as the friction tensor [4] or the generalized +friction coefficient [5], which has been used to formulate the +dissipated availability. The dissipated availability quantifies +the efficiency of irreversible thermodynamic processes carried +out by controlling parameters of a system in finite time [6]. It +was originally proposed for macroscopic endoreversible sys- +tems using extensive variables as control parameters [6], while +recent applications mainly focus on stochastic systems [4, 8]. +In particular, recent applications include those to stochastic +heat engine cycles [13–22], where both extensive and inten- +sive variables, such as volume and temperature, are used as +control parameters. +∗ izumida@k.u-tokyo.ac.jp +In this paper, we derive the non-quasistatic response co- +efficients of extensive thermodynamic variables for macro- +scopic thermodynamic systems against external thermody- +namic forces which are intensive. Through the application +of a singular perturbation theory to the dynamics of fluctua- +tions of extensive thermodynamic variables in the presence of +external thermodynamic forces, the non-quasistatic response +coefficients in terms of Onsager’s kinetic coefficients are de- +rived. The Onsager’s kinetic coefficients govern the relax- +ation dynamics of fluctuations of extensive variables of semi- +macroscopic systems in the vicinity of the equilibrium state +without the external thermodynamic forces. Moreover, the +derived non-quasistatic response coefficients are used for the +formulation of the dissipated availability. Our results provide +a guiding principle for the nonequilibrium control of macro- +scopic thermodynamic systems and contribute to the funda- +mental understanding of nonequilibrium thermodynamics. +The rest of the paper is organized as follows. In Sec. II, +we describe the setup of the theory. In Sec. III, we derive the +non-quasistatic response coefficients and the dissipated avail- +ability as our main results. In Sec. IV, we demonstrate the +main results by using an ideal gas model. Finally, in Sec. V, +we summarize the paper with discussion. +II. +SETUP +Consider a semi-macroscopic thermodynamic system. Let +ˆX = ( ˆX1, · · · , ˆXn)T (n ≥ 2) be independent extensive thermo- +dynamic variables of the system, where ˆX1 = ˆU and ˆX2 = ˆV +are the internal energy and the volume by taking an entropy +representation [23]. Here, the quantities with a hat denote +random variables. The system is in contact with a large bath +with externally controllable intensive parameters, such as a +heat bath, a pressure bath, and a particle bath, or the system is +a small partial system of such a bath. The system exchanges +its extensive variables with the bath. The bath is assumed to be +sufficiently large so that its intensive parameters do not change +upon exchanges of the extensive variables with the system. +The system may also be controlled by external forces, such as + +2 +a mechanical force through a piston. +Let ˆx = δ ˆX ≡ ˆX − Xeq be the fluctuation of ˆX, where Xeq is +the equilibrium value of ˆX. At equilibrium, the fluctuation ˆx +obeys the celebrated Einstein’s fluctuation formula [1, 24]: +Peq(ˆx) ∝ eδ2S (ˆx)/kB, +(3) +where Peq(ˆx) is the equilibrium probability distribution of ˆx, +kB is the Boltzmann constant, and δ2S (ˆx) is the second-order +entropy variation of the system from the equilibrium value +serving as a potential function of the fluctuation ˆx: +δ2S (ˆx) = −1 +2 ˆxTSˆx, +(4) +where S is a positive definite symmetric matrix and T denotes +the transpose. In general, in the vicinity of the equilibrium +state, the dynamics of the fluctuation ˆx obeys the following +Langevin equation [1, 24]: +dˆx +dt = L∂δ2S (ˆx) +∂ˆx ++ ξ, +(5) +where L is Onsager’s kinetic coefficients. Onsager’s kinetic +coefficients L are symmetric as L = LT, assuming that ˆx is +the time-reversely symmetric quantities under time-reversely +symmetric microscopic dynamics. +Moreover, L is a posi- +tive definite matrix, as shown below. ξ is Gaussian white +noise satisfying ⟨ξ(t)⟩ = 0 and +� +ξ(t)ξ(t′)T� += 2LkBδ(t − t′), +which assures that the stationary probability distribution of +ˆx agrees with the Einstein’s fluctuation formula Eq. (3) (the +fluctuation-dissipation theorem). +By defining the thermodynamic forces ˆy as restoring forces +to the equilibrium as +ˆy ≡ −∂δ2S (ˆx) +∂ˆx += Sˆx, +(6) +Eq. (5) may be expressed as the linear flux-force relation +dˆx/dt = −Lˆy + ξ. By defining an ensemble average x ≡ ⟨ˆx⟩ +and y ≡ ⟨ˆy⟩, the positive definiteness of L is concluded from +the positivity of dδ2S (x)/dt during relaxation to the equilib- +rium [24]: dδ2S (x)/dt = −(Sx)T˙x = yTLy ≥ 0, where we +used ˙x = −Ly and the equality holds for y = 0. +By defining a relaxation matrix A as +A ≡ LS, +(7) +which is a positive definite matrix reflecting the stability of +the equilibrium state, we can write Eq. (5) as +dˆx +dt = −Aˆx + ξ. +(8) +We add small external thermodynamic forces F(t/TF) chang- +ing slowly with time from t = 0 to t = TF (0 ≤ t ≤ TF), which +are intensive quantities as are constituted with variations of +intensive thermodynamic variables of the bath or external me- +chanical forces. Then, the dynamics Eq. (5) may be altered +as [24] +dˆx +dt = L ∂ +∂ˆx +� +δ2S (ˆx) + ˆxTF(t/TF) +� ++ ξ, +(9) +or, equivalently, +dˆx +dt = −L(ˆy − F(t/TF)) + ξ, +(10) +using ˆy, with ˆy − F being considered as the effective thermo- +dynamic forces. Using Eq (7), we can rewrite Eq. (9) as +dˆx +dt = −Aˆx + LF(t/TF) + ξ. +(11) +By taking an ensemble average of both sides, we obtain the +following: +dx +dt = −Ax + LF(t/TF). +(12) +Or, equivalently, by taking an ensemble average of both sides +of Eq. (10), we obtain the following: +dx +dt = −L(y − F(t/TF)). +(13) +III. +MAIN RESULTS +A. +Non-quasistatic response coefficients +Equation (12) can be solved perturbatively using a two- +timing method [25]. We introduce a small dimensionless pa- +rameter ǫ ≡ ts/TF ≪ 1 defined as the ratio of the typical time +scale characterizing the relaxation of the system ts to the dura- +tion TF required for a process changing F(t/TF) (0 ≤ t ≤ TF). +By introducing the dimensionless time ˜t ≡ t/ts (0 ≤ ˜t ≤ 1/ǫ), +we expand x(˜t) in terms of the fast and slow time scales τ ≡ ˜t +and T +≡ ǫ˜t as x(˜t, ǫ) = x(0)(τ, T ) + ǫx(1)(τ, T ) + O(ǫ2). +The time-differential operator thus becomes dx/d˜t = ∂x/∂τ + +ǫ∂x/∂T . By putting this into Eq. (12), the following equation +for each order of ǫ is obtained: +O(1) : ∂x(0) +∂τ += −Ax(0) + LF(T ), +(14) +O(ǫ) : ∂x(1) +∂τ += −Ax(1) − ∂x(0) +∂T . +(15) +For each order, we consider a stationary solution with respect +to the fast time, ∂τx(0) +s += ∂τx(1) +s += 0, under the assumption of +time-scale separation: +x(0) +s += A−1LF(T ) = S−1F(T ), +(16) +and +x(1) +s += −A−1 ∂x(0) +∂T , +(17) +respectively. Note that stationary x(0) +s +and x(1) +s +are dependent +only on the slow time scale T . Consequently, this yields +xs = χF − R˙F = χF − ǫRF′(T ), +(18) + +3 +where the prime denotes the derivative with respect to T , and +the quasistatic response coefficients χ and the non-quasistatic +response coefficients R are given as +χ = S−1, +(19) +R = A−1S−1 = S−1L−1S−1, +(20) +respectively. It is remarkable that the non-quasistatic response +coefficient R is directly related to Onsager’s kinetic coeffi- +cients L that govern the relaxation dynamics of fluctuations +of thermodynamic variables. +Several remarks are in order with respect to the results +Eqs. (18)–(20). +From Eq. (20), it can be concluded that R is symmetric as +R = RT, using the symmetric Onsager’s kinetic coefficients +L and the symmetric S. Moreover, R is a positive definite +matrix. This is confirmed by noticing that R in Eq. (20) can be +written as R = (S−1)TL−1S−1, where we used S−1 = (S−1)T. +This shows that L−1 and R are congruent, and because L−1 is +positive definite, R is also positive definite. +Equations (18)–(20) are consistent with the linear response +theory. We can show +� +ˆxˆxT� +eq = kBS−1, which implies that χ +is calculated using the equilibrium correlation function of ˆx as +χ = +� +ˆxˆxT� +eq /kB, where ⟨·⟩eq is taken with respect to Peq(ˆx) in +Eq. (3). Moreover, in the linear response theory, R can also be +calculated using the time integral of the temporal equilibrium +correlation function: +R = 1 +kB +� ∞ +0 +� +ˆx(t)ˆx(0)T� +dt. +(21) +This can be easily shown by substituting the explicit solution +of Eq. (11) +ˆx(t) = e−Atˆx(0) + +� t +0 +e−A(t−t′)ξ(t′)dt′ +(22) +into Eq. (21). Thus, Eq. (20) shows the detailed constituents +of R for the fluctuations of thermodynamic variables governed +by Eq. (5). +B. +Dissipated availability +The dissipated availability Adiss, which quantifies the ef- +ficiency of irreversible thermodynamic processes [6], is for- +mulated by using the entropy variation δ2S (x) serving as the +potential function of x. For a periodic thermodynamic force +F(0) = F(1) with period TF, we can show (see Appendix A +for the derivation) +Adiss ≡ +� TF +0 +dδ2S (x) +dt +dt ≃ +� TF +0 +˙FTR˙Fdt ≥ 0, +(23) +where the inequality holds by using the positive semi- +definiteness of R shown above. +Moreover, by using the +Cauchy-Schwartz inequality, we can obtain the tighter bound +for Adiss than Eq. (23): +Adiss ≥ L2 +TF +≡ A∗ +diss, +(24) +where +L ≡ +� +γF +√ +dFTRdF = +� 1 +0 +� +F′TRF′dT +(25) +is the thermodynamic length for the closed path γF on the +control space of thermodynamics forces F with R the metric +tensor defined on it [6, 13]. The equality of Eq. (24), the min- +imum dissipated availability A∗ +diss, is achieved for a geodesic +path that yields the constant dissipation such that the integrand +of Eq. (25) becomes constant [6]. As R is the constant matrix +evaluated at Xeq, this equality is expected to be realized for, +e.g., periodic F with linear profile symmetric in time. +It is also noteworthy that Adiss in Eq. (23) can also be ex- +pressed in terms of ˙xs instead of ˙F: +Adiss = +� TF +0 +˙xT +s L−1˙xsdt ≡ +� TF +0 +Φ (˙xs) dt ≥ 0, +(26) +where we used ˙F = S˙xs derived from the time derivative of +Eq. (16) and Eq. (20). The integrand Φ (˙xs) of Eq. (26) is es- +sentially the same quantity as the dissipation function [7]. The +inequality in Eq. (26) is assured by the positive definiteness of +L−1. +Moreover, when expressed in terms of the effective thermo- +dynamic force y − F, we obtain +Adiss = +� TF +0 +(ys − F)TL(ys − F)dt ≡ +� TF +0 +σ (ys) dt ≥ 0, (27) +where we used Eq. (13) in the first equality. The integrand +σ (ys) of Eq. (27) is the “familiar” total entropy production +rate. The inequality in Eq. (27) is assured by the positive def- +initeness of L. These apparently different but the equivalent +expressions Eqs. (23), (26), and (27) are informative, which +will be used in Sec. IV B for numerical demonstration of the +present theory. We should note that as what is controlling +here is the intensive thermodynamic force F, the expression +Eq. (23) using ˙F serves as the basic form. This may contrast +to both the original formulation of the dissipated availability +using the extensive variables [6] as control parameters and the +recent formulation of thermodynamic geometry using exten- +sive and intensive parameters [13] as control parameters. +IV. +EXAMPLE +A. +Ideal gas model +As a demonstration of our theory, we identify the non- +quasistatic response coefficients of a macroscopic phe- +nomenological model of a thermodynamic system. Our model +is an ideal gas confined in a cylinder with a movable pis- +ton subject to an external mechanical force f(t/TF) in con- +tact with a heat bath at changeable temperature TR(t/TF) by +a heater (Fig. 1). +Let X = (U, V) (n = 2) be the exten- +sive thermodynamic variables of the ideal gas, where U = +CVT = fdNkBT/2 is the internal energy with the constant- +volume heat capacity CV, the temperature of the gas T, and + +4 +FIG. 1. Schematic illustration of a thermodynamic system under con- +sideration: The ideal gas is confined in the cylinder enclosed by the +insulated walls with a movable piston on one side to which the ex- +ternal mechanical force f is applied. Further, it is in thermal contact +with a heat bath at changeable temperature TR by a heater. The sys- +tem and the heat bath exchange their extensive quantities, internal +energy and volume. +the internal degrees of freedom fd. Further, V is the volume +of the ideal gas. Under an assumption of spatial uniformity, +macroscopic dynamics for X is given by +dU +dt = κ(TR(t/TR) − T) − f(t/TF) +σ +dV +dt , +(28) +dV +dt = σ2 +Γ +�NkBT +V +− f(t/TF) +σ +� +. +(29) +Equation (28) is the first law of thermodynamics (the energy +conservation law) per unit time, where the first term on the +right-hand side denotes the Fourier’s law for the heat flow with +κ the thermal conductance and the second term does the work +flow done by the gas against the mechanical force f(t/TF) +acting on the piston in one direction, where σ is the section +area of the piston. Equation (29) is the equation of motion for +an overdamped piston whose inertia can be neglected, where +Γ is the friction coefficient of the piston. The piston moves +back and forth subject to the net force due to the internal gas +pressure NkBT/V and the mechanical pressure f(t/TF)/σ. +For the fixed temperature and mechanical force TR(t/TF) = +Teq and f(t/TF) = feq, respectively, the stationary state Xeq = +(Ueq, Veq) = (CVTeq, σNkBTeq/feq) satisfying dX/dt = 0 cor- +responds to the equilibrium state. +Equations (28) and (29) around the equilibrium state Xeq +can be rewritten into the form Eq. (12). First, consider the case +of fixed temperature and mechanical force TR(t/TF) = Teq +and f(t/TF) = feq. +By expanding Eqs. (28) and (29) in +terms of x around Xeq, dx/dt = −Ax is obtained, neglect- +ing the higher order terms of x, where A11 = +κ +CV + feq σ +Γ +NkB +CVVeq , +A12 = −feq σ +Γ +NkBTeq +V2eq , A21 = − σ2 +Γ +NkB +CVVeq , and A22 = σ2 +Γ +NkBTeq +V2eq . The +Einstein’s fluctuation formula for the corresponding fluctua- +tion ˆx = δ ˆX = (δ ˆU, δ ˆV) is given as [24] +δ2S (ˆx) = −1 +2 + +1 +CVT 2eq +δ ˆU2 + NkB +V2eq +δ ˆV2 + , +(30) +from which S11 = +1 +CVT 2eq , S12 = S21 = 0, and S22 = NkB +V2eq are +identified. Then, we obtain the quasistatic response coefficient +χ = S−1 as +χ = + +CVT 2 +eq +0 +0 +V2 +eq +NkB + . +(31) +We can also obtain the Onsager’s kinetic coefficients L = +AS−1 as +L = + κT 2 +eq + +f 2 +eq +Γ Teq −feq σ +Γ Teq +−feq σ +Γ Teq +σ2 +Γ Teq + . +(32) +Next, we consider the effect of time-dependent temperature +of the heat bath TR(t/TF) = Teq + δTR(t/TF) and mechanical +force f(t/TF) = feq + δf(t/TF), where |δTR(t/TF)| ≪ Teq +and |δf(t/TF)| ≪ feq. By expanding Eqs. (28) and (29) up to +O(x, δTR, δf), we obtain +dx +dt = −Ax + +� +κδTR(t/TF) + feq +Γ δf(t/TF) +− σ +Γ δf(t/TF) +� +. +(33) +Through comparisons of Eq. (33) and Eq. (12), the external +thermodynamic force F(t/TF) acting on x is identified as +F(t/TF) = + +δTR(t/TF) +T 2eq +δpR(t/TF) +Teq +− δ f(t/TF)/σ +Teq + , +(34) +where δpR(t/TF) ≡ NkBδTR(t/TF)/Veq. Here, F2(t/TF) de- +notes the net force increment acting on the piston that arises +owing to the combined effect of the change of the gas pressure +upon the temperature change of the heat bath and the change +of the external mechanical force. Using Eqs. (20), (31), and +(32), the non-quasistatic response coefficient R is derived as +R = + +C2 +VT 2 +eq +κ +CVTeqVeq +κ +CVTeqVeq +κ +V2 +eq +κ + +f 2 +d ΓV4 +eq +4σ2C2 +VTeq + . +(35) +Through explicit writing, the following is obtained as +Eq. (18): +δUs(T ) = χ11F1(T ) − ǫR11F′ +1(T ) − ǫR12F′ +2(T ), +(36) +δVs(T ) = χ22F2(T ) − ǫR21F′ +1(T ) − ǫR22F′ +2(T ). +(37) +Note that while the non-diagonal components of χ in Eq. (31) +identically vanish, the non-diagonal components of R in +Eq. (35) are non-zero. Therefore, the cross effect between δU +and δV appears as a consequence of a nonequilibrium control +of the external thermodynamic forces. +B. +Numerical demonstration +The calculations in Sec. IV A can be confirmed numeri- +cally. For this purpose, Eqs. (28) and (29) are made nondi- +mensional as follows: +d ˜U +d˜t = ˜TR(ǫ˜t) − ˜U − ˜f(ǫ˜t)d ˜V +d˜t , +(38) +d ˜V +d˜t = 1 +˜Γ +� 2 ˜U +fd ˜V − ˜f (ǫ˜t) +� +, +(39) + +5 +TABLE I. Summary for nondimensionalized quantities +Time t +˜t = t/ts with ts = CV/κ +Internal energy U +˜U = U/Ueq = U/(CVTeq) +Volume V +˜V = V/Veq +Temperature T, TR +˜T = T/Teq, ˜TR = TR/Teq +Friction coefficient Γ +˜Γ = Γ/(σ2C2 +VTeq/κV2 +eq) +Mechanical force f +˜f = f /(σCVTeq/Veq) +TABLE II. Summary for nondimensionalized F, χ, and R +F1(T ) +˜F1(T ) = TeqF1(T ) = δ ˜TR(T ) +F2(T ) +˜F2(T ) = +Veq +CV F2(T ) = 2δ ˜TR(T )/fd − +δ ˜f(T ) +χ11 +˜χ11 = 1 +χ22 +˜χ22 = fd/2 +R11 +˜R11 = 1 +R12 +˜R12 = 1 +R21 +˜R21 = 1 +R22 +˜R22 = 1 + f 2 +d ˜Γ/4 +respectively, where we put T = U/CV on the right-hand sides +of Eqs. (28) and (29) before the nondimensionalization and +the nondimensionalized quantities are summarized in Table I. +The quantities with a tilde denote nondimensionalized quan- +tities. Then, Eqs. (36) and (37) are also nondimensionalized +as +δ ˜Us(T ) = ˜χ11 ˜F1(T ) − ǫ ˜R11 ˜F′ +1(T ) − ǫ ˜R12 ˜F′ +2(T ), +(40) +δ ˜Vs(T ) = ˜χ22 ˜F2(T ) − ǫ ˜R21 ˜F′ +1(T ) − ǫ ˜R22 ˜F′ +2(T ), +(41) +respectively, where the nondimensionalized F, χ, and R are +summarized in Table II. +In Fig. 2, the theoretical results Eqs. (40) and (41) for T = 1 +are compared with the numerical results obtained by solving +Eqs. (38) and (39) numerically with the following linear pro- +file as δ ˜TR(T ) and δ ˜f (T ): +δ ˜TR(T ) = ∆ ˜TT (0 ≤ T ≤ 1), +(42) +δ ˜f(T ) = ∆ ˜f T (0 ≤ T ≤ 1), +(43) +where ∆ ˜T ≪ 1 and ∆ ˜f ≪ 1 are small increments. As evident, +for sufficiently small ǫ, the numerical results are consistent +with the theoretical results as expected, with the discrepan- +cies observed with an increase in ǫ where the nonlinear effects +begin to appear. +Finally, we demonstrate the dissipated availability Adiss in +Eq. (23). It is nondimensionalized as +˜A∗ +diss = A∗ +diss +CV += ǫ +�� 1 +0 +� +˜F′T ˜R˜F′dT +�2 += ǫ ˜L2. +(44) +As we know from Eq. (27) that the dissipated availability is +equivalent to the total entropy production for small ǫ, we in- +troduce the entropy production of the total system (system and + 0.0007 + 0.0008 + 0.0009 + 0.001 + 0 + 0.1 + 0.2 + 0.3 +Numerical +Theory + 0 + 0.0001 + 0.0002 + 0.0003 + 0 + 0.1 + 0.2 + 0.3 +Numerical +Theory +-0.0003 +-0.0002 +-0.0001 + 0 + 0 + 0.1 + 0.2 + 0.3 +Numerical +Theory +-0.0015 +-0.0012 +-0.0009 +-0.0006 +-0.0003 + 0 + 0.1 + 0.2 + 0.3 +Numerical +Theory +(a) +(b) +(c) +(d) +FIG. 2. (a) and (b): ǫ dependence of δ ˜Us(1) for (a) ˜F′ +2(1) = 0 and (b) +˜F′ +1(1) = 0. The theory denotes Eq. (40) with T = 1. (c) and (d): ǫ +dependence of δ ˜Vs(1) for (c) ˜F′ +2(1) = 0 and (d) ˜F′ +1(1) = 0. The theory +denotes Eq. (41) with T = 1. As parameter values, fd = 3, ˜Γ = 1, +∆ ˜T = 10−3, and ∆ ˜f = 10−3 were used. The fourth Runge–Kutta +method with time step 10−3 was used for obtaining the numerical +results. + 0 + 2x10-7 + 4x10-7 + 6x10-7 + 0 + 0.05 + 0.1 + 0.15 +Numerical (linear) +Numerical (quadratic) +Theory +FIG. 3. ǫ dependence of the total entropy production over one cycle +∆ ˜S tot in Eq. (46) calculated for Eqs. (47) and (48) (bold solid line) +and for Eqs. (49) and (50) (thin solid line). The theory denotes the +theoretical minimum dissipated availability ˜A∗ +diss in Eq. (51). The +same parameter values and numerical scheme as those in Fig. 2 were +used. +bath) over one cycle for comparison with Adiss: +∆S tot ≡ − +� TF +0 +κ(TR(t/TF) − T(t)) +TR(t/TF) +dt, += −κ +� TF +0 +� +1 − +T(t) +TR(t/TF) +� +dt, +(45) +which coincides with the entropy change of the heat bath as +the system’s state returns to the initial state after one cycle. Its +nondimensionalized form reads +∆ ˜S tot = ∆S tot +CV += − +� 1/ǫ +0 +� +1 − +˜T(˜t) +˜TR(ǫ˜t) +� +d˜t. +(46) +For the demonstration, we use F with the following periodic +linear profile as δ ˜TR(T ) and δ ˜f(T ) as the geodesic path on the + +6 +control space: +δ ˜TR(T ) = + +2∆ ˜TT (0 ≤ T ≤ 1/2), +2∆ ˜T(1 − T ) (1/2 < T ≤ 1), +(47) +δ ˜f (T ) = + +2∆ ˜fT (0 ≤ T ≤ 1/2), +2∆ ˜f(1 − T ) (1/2 < T ≤ 1). +(48) +For comparison, we also use F with the following periodic +quadratic profile as the non-geodesic path: +δ ˜TR(T ) = + +4∆ ˜TT 2 (0 ≤ T ≤ 1/2), +4∆ ˜T(1 − T )2 (1/2 < T ≤ 1), +(49) +δ ˜f(T ) = + +4∆ ˜f T 2 (0 ≤ T ≤ 1/2), +4∆ ˜f (1 − T )2 (1/2 < T ≤ 1). +(50) +As we noted in Sec. III B, the minimum dissipated availabil- +ity ˜A∗ +diss is expected to be achieved for periodic linear pro- +files such as Eqs. (47) and (48). Explicitly, ˜A∗ +diss calculated for +Eqs. (47) and (48) is given as +˜A∗ +diss = ǫ +� +4∆ ˜T 2 + 4∆ ˜T +�4∆ ˜T +fd +− 2∆ ˜f +� ++ +1 + f 2 +d ˜Γ +4 + +�4∆ ˜T +fd +− 2∆ ˜f +�2� +. +(51) +In Fig. 3, we show the total entropy production ∆ ˜S tot nu- +merically calculated for the periodic linear profile Eqs. (47) +and (48) and for the periodic quadratic profile Eqs. (49) and +(50), with a comparison with the minimum dissipated avail- +ability ˜A∗ +diss in Eq. (51). We can confirm that ∆ ˜S tot for the +periodic linear profile as the geodesic path agrees with ˜A∗ +diss in +the small ǫ region as expected. Meanwhile, we can also con- +firm that ∆ ˜S tot for the periodic quadratic profile as the non- +geodesic path is larger than ˜A∗ +diss in the small ǫ region, which +is consistent with the theoretical prediction. +V. +SUMMARY AND DISCUSSION +In summary, we derived a simple formula for the non- +quasistatic response coefficients for macroscopic thermody- +namic systems. The formula revealed the general relation- +ship between the non-quasistatic response coefficients and +Onsager’s kinetic coefficients that govern the relaxation dy- +namics of fluctuations of semi-macroscopic thermodynamic +variables. Similarly to the quasistatic response coefficients +whose symmetry has been already established in equilibrium +thermodynamics, the non-quasistatic response coefficients are +also symmetric as R = RT, which is related to the symmetry +of Onsager’s kinetic coefficients. Moreover, we also formu- +lated the dissipated availability Adiss for the present system +that quantifies the efficiency of irreversible thermodynamic +processes. It is expressed in terms of the time derivative of +the entropy variation, and the equivalent expressions using the +dissipation function or the total entropy production were pro- +vided. Our theory was demonstrated by using the ideal gas +model. +It is expected to measure the predicted symmetry of R and +the dissipated availability Adiss in realistic nonequilibrium ex- +periments such as those illustrated in Fig. 1 where Eqs. (28) +and (29) serve as a good approximation. Real systems may be +accompanied by complicated fluid motion as well as the non- +uniform heat conduction inside the system associated with a +piston motion. However, from recent experiments on real heat +engines using a gas [26, 27], it can be concluded that simple +models similar to those presented in Eqs. (28) and (29) explain +the experimental results to a sufficiently good approximation. +Therefore, experimental verification will be an important di- +rection for future investigations. +ACKNOWLEDGMENTS +This work was supported by JSPS KAKENHI Grant Num- +bers 19K03651 and 22K03450. +Appendix A: Derivation of Eq. (23) +First, we evaluate the time derivative of the entropy varia- +tion dδ2S (x)/dt: +dδ2S (x) +dt += −(Sx)T˙x += −yT˙x += yTL(y − F), +(A1) +where we used Eq. (13) in the third equality. By applying S +to both sides of Eq. (18), we can approximate y up to O(ǫ) as +y ≃ ys ≡ Sxs = F − ǫSRF′. +(A2) +By substituting Eq. (A2) into Eq. (A1), we obtain +dδ2S (x) +dt +≃ −ǫFTS−1F′ + ǫ2F′TRF′ += −FTS−1 ˙F + ˙FTR˙F += − d +dt +�1 +2FTS−1F +� ++ ˙FTR˙F. +(A3) +The first term vanishes by the time-integration of Eq. (A3) +from t = 0 to t = TF and using F(0) = F(1) for the periodic +thermodynamic force, and we derive Eq. (23). +[1] Y. Oono, Perspectives on Statistical Thermodynamics, (Cam- +bridge University Press, Cambridge, 2017). +[2] R. Kubo, M. Toda, and N. Hashitsume, Statistical Physics +II: Nonequilibrium Statistical Mechanics, 2nd ed. (Springer, + +7 +Berlin, 1991). +[3] U. M. B. Marconi, A. Puglisi, L. Rondoni, and A. 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E 103, +L050101 (2021). +[19] Y. Izumida, Irreversible efficiency and Carnot theorem for heat +engines operating with multiple heat baths in linear response +regime, Phys. Rev. Research 4, 023217 (2022). +[20] G. Watanabe and Y. Minami, Finite-time thermodynamics of +fluctuations in microscopic heat engines, Phys. Rev. Research +4, L012008 (2022). +[21] P. Abiuso and M. Perarnau-Llobet, Optimal Cycles for Low- +Dissipation Heat Engines, Phys. Rev. Lett. 124, 110606 (2020). +[22] H. J. D. Miller, M. H. Mohammady, M. Perarnau-Llobet, and +G. Guarnieri, Thermodynamic Uncertainty Relation in Slowly +Driven Quantum Heat Engines, Phys. Rev. Lett. 126, 210603 +(2021). +[23] H. B. Callen, Thermodynamics and an Introduction to Thermo- +statistics, 2nd ed. (Wiley, New York, 1985). +[24] L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, +Vol. 5. Statistical Physics, 3rd ed. (Elsevier, 1980), Part 1. +[25] S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applica- +tions to Physics, Biology, Chemistry, and Engineering (West- +view Press, 2001). +[26] Y.-H. Ma, R.-X. Zhai, J. Chen, C. P. Sun, and H. Dong, Exper- +imental Test of the 1/τ-Scaling Entropy Generation in Finite- +Time Thermodynamics, Phys. Rev. Lett. 125, 210601 (2020). +[27] S. Toyabe and Y. Izumida, Experimental characterization of +autonomous heat engine based on minimal dynamical-system +model, Phys. Rev. Research, 2, 033146 (2020). + diff --git a/wNFLT4oBgHgl3EQfki_9/content/tmp_files/load_file.txt b/wNFLT4oBgHgl3EQfki_9/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..75b90b8d7c496b9aff10a8c2ac399bb706b5578f --- /dev/null +++ b/wNFLT4oBgHgl3EQfki_9/content/tmp_files/load_file.txt @@ -0,0 +1,448 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf,len=447 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content='12116v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content='stat-mech] 28 Jan 2023 Non-quasistatic response coefficients and dissipated availability for macroscopic thermodynamic systems Yuki Izumida Department of Complexity Science and Engineering, Graduate School of Frontier Sciences, The University of Tokyo, Kashiwa 277-8561, Japan∗ The characterization of finite-time thermodynamic processes is of crucial importance for extending equilib- rium thermodynamics to nonequilibrium thermodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' The central issue is to quantify responses of ther- modynamic variables and irreversible dissipation associated with non-quasistatic changes of thermodynamic forces applied to the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' In this study, we derive a simple formula that incorporates the non-quasistatic response coefficients with Onsager’s kinetic coefficients, where the Onsager coefficients characterize the relax- ation dynamics of fluctuation of extensive thermodynamic variables of semi-macroscopic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Moreover, the dissipated availability that quantifies the efficiency of the irreversible thermodynamic process is formulated in terms of the derived non-quasistatic response coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' The present results are demonstrated by using an ideal gas model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' The present results are, in principle, verifiable through experiments and are thus expected to provide a guiding principle for the nonequilibrium control of macroscopic thermodynamic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' INTRODUCTION A quasistatic thermodynamic process is an important build- ing block of thermodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Consider a response of thermo- dynamic variables X upon quasistatically added small pertur- bation F: x = χF, (1) where x is the deviation of X from the equilibrium value, and χ denotes the quasistatic response coefficient to the perturba- tion F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' According to equilibrium statistical mechanics, the quasistatic response coefficient χ can be calculated based on the equilibrium correlation function of fluctuations of thermo- dynamic variables in the absence of perturbation, which is re- ferred to as the fluctuation-response relation [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' When the small perturbation is added slowly in time but non-quasistatically, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (1) may be generalized as x = χF − R˙F, (2) where the dot denotes the time derivative, and we call R the non-quasistatic response coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Using the linear re- sponse theory [2, 3], in the same manner as the quasistatic response coefficient χ, R can be expressed in terms of tem- poral equilibrium correlation functions of fluctuations [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' A similar quantity has been studied extensively [4–12], some- times referred to as the friction tensor [4] or the generalized friction coefficient [5], which has been used to formulate the dissipated availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' The dissipated availability quantifies the efficiency of irreversible thermodynamic processes carried out by controlling parameters of a system in finite time [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' It was originally proposed for macroscopic endoreversible sys- tems using extensive variables as control parameters [6], while recent applications mainly focus on stochastic systems [4, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' In particular, recent applications include those to stochastic heat engine cycles [13–22], where both extensive and inten- sive variables, such as volume and temperature, are used as control parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' ∗ izumida@k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content='u-tokyo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content='jp In this paper, we derive the non-quasistatic response co- efficients of extensive thermodynamic variables for macro- scopic thermodynamic systems against external thermody- namic forces which are intensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Through the application of a singular perturbation theory to the dynamics of fluctua- tions of extensive thermodynamic variables in the presence of external thermodynamic forces, the non-quasistatic response coefficients in terms of Onsager’s kinetic coefficients are de- rived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' The Onsager’s kinetic coefficients govern the relax- ation dynamics of fluctuations of extensive variables of semi- macroscopic systems in the vicinity of the equilibrium state without the external thermodynamic forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Moreover, the derived non-quasistatic response coefficients are used for the formulation of the dissipated availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Our results provide a guiding principle for the nonequilibrium control of macro- scopic thermodynamic systems and contribute to the funda- mental understanding of nonequilibrium thermodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' II, we describe the setup of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' III, we derive the non-quasistatic response coefficients and the dissipated avail- ability as our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' IV, we demonstrate the main results by using an ideal gas model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Finally, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' V, we summarize the paper with discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' SETUP Consider a semi-macroscopic thermodynamic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Let ˆX = ( ˆX1, · · · , ˆXn)T (n ≥ 2) be independent extensive thermo- dynamic variables of the system, where ˆX1 = ˆU and ˆX2 = ˆV are the internal energy and the volume by taking an entropy representation [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Here, the quantities with a hat denote random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' The system is in contact with a large bath with externally controllable intensive parameters, such as a heat bath, a pressure bath, and a particle bath, or the system is a small partial system of such a bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' The system exchanges its extensive variables with the bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' The bath is assumed to be sufficiently large so that its intensive parameters do not change upon exchanges of the extensive variables with the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' The system may also be controlled by external forces, such as 2 a mechanical force through a piston.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Let ˆx = δ ˆX ≡ ˆX − Xeq be the fluctuation of ˆX, where Xeq is the equilibrium value of ˆX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' At equilibrium, the fluctuation ˆx obeys the celebrated Einstein’s fluctuation formula [1, 24]: Peq(ˆx) ∝ eδ2S (ˆx)/kB, (3) where Peq(ˆx) is the equilibrium probability distribution of ˆx, kB is the Boltzmann constant, and δ2S (ˆx) is the second-order entropy variation of the system from the equilibrium value serving as a potential function of the fluctuation ˆx: δ2S (ˆx) = −1 2 ˆxTSˆx, (4) where S is a positive definite symmetric matrix and T denotes the transpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' In general, in the vicinity of the equilibrium state, the dynamics of the fluctuation ˆx obeys the following Langevin equation [1, 24]: dˆx dt = L∂δ2S (ˆx) ∂ˆx + ξ, (5) where L is Onsager’s kinetic coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Onsager’s kinetic coefficients L are symmetric as L = LT, assuming that ˆx is the time-reversely symmetric quantities under time-reversely symmetric microscopic dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Moreover, L is a posi- tive definite matrix, as shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' ξ is Gaussian white noise satisfying ⟨ξ(t)⟩ = 0 and � ξ(t)ξ(t′)T� = 2LkBδ(t − t′), which assures that the stationary probability distribution of ˆx agrees with the Einstein’s fluctuation formula Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (3) (the fluctuation-dissipation theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' By defining the thermodynamic forces ˆy as restoring forces to the equilibrium as ˆy ≡ −∂δ2S (ˆx) ∂ˆx = Sˆx, (6) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (5) may be expressed as the linear flux-force relation dˆx/dt = −Lˆy + ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' By defining an ensemble average x ≡ ⟨ˆx⟩ and y ≡ ⟨ˆy⟩, the positive definiteness of L is concluded from the positivity of dδ2S (x)/dt during relaxation to the equilib- rium [24]: dδ2S (x)/dt = −(Sx)T˙x = yTLy ≥ 0, where we used ˙x = −Ly and the equality holds for y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' By defining a relaxation matrix A as A ≡ LS, (7) which is a positive definite matrix reflecting the stability of the equilibrium state, we can write Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (5) as dˆx dt = −Aˆx + ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (8) We add small external thermodynamic forces F(t/TF) chang- ing slowly with time from t = 0 to t = TF (0 ≤ t ≤ TF), which are intensive quantities as are constituted with variations of intensive thermodynamic variables of the bath or external me- chanical forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Then, the dynamics Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (5) may be altered as [24] dˆx dt = L ∂ ∂ˆx � δ2S (ˆx) + ˆxTF(t/TF) � + ξ, (9) or, equivalently, dˆx dt = −L(ˆy − F(t/TF)) + ξ, (10) using ˆy, with ˆy − F being considered as the effective thermo- dynamic forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Using Eq (7), we can rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (9) as dˆx dt = −Aˆx + LF(t/TF) + ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (11) By taking an ensemble average of both sides, we obtain the following: dx dt = −Ax + LF(t/TF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (12) Or, equivalently, by taking an ensemble average of both sides of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (10), we obtain the following: dx dt = −L(y − F(t/TF)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (13) III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' MAIN RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Non-quasistatic response coefficients Equation (12) can be solved perturbatively using a two- timing method [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' We introduce a small dimensionless pa- rameter ǫ ≡ ts/TF ≪ 1 defined as the ratio of the typical time scale characterizing the relaxation of the system ts to the dura- tion TF required for a process changing F(t/TF) (0 ≤ t ≤ TF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' By introducing the dimensionless time ˜t ≡ t/ts (0 ≤ ˜t ≤ 1/ǫ), we expand x(˜t) in terms of the fast and slow time scales τ ≡ ˜t and T ≡ ǫ˜t as x(˜t, ǫ) = x(0)(τ, T ) + ǫx(1)(τ, T ) + O(ǫ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' The time-differential operator thus becomes dx/d˜t = ∂x/∂τ + ǫ∂x/∂T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' By putting this into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (12), the following equation for each order of ǫ is obtained: O(1) : ∂x(0) ∂τ = −Ax(0) + LF(T ), (14) O(ǫ) : ∂x(1) ∂τ = −Ax(1) − ∂x(0) ∂T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (15) For each order, we consider a stationary solution with respect to the fast time, ∂τx(0) s = ∂τx(1) s = 0, under the assumption of time-scale separation: x(0) s = A−1LF(T ) = S−1F(T ), (16) and x(1) s = −A−1 ∂x(0) ∂T , (17) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Note that stationary x(0) s and x(1) s are dependent only on the slow time scale T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Consequently, this yields xs = χF − R˙F = χF − ǫRF′(T ), (18) 3 where the prime denotes the derivative with respect to T , and the quasistatic response coefficients χ and the non-quasistatic response coefficients R are given as χ = S−1, (19) R = A−1S−1 = S−1L−1S−1, (20) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' It is remarkable that the non-quasistatic response coefficient R is directly related to Onsager’s kinetic coeffi- cients L that govern the relaxation dynamics of fluctuations of thermodynamic variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Several remarks are in order with respect to the results Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (18)–(20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (20), it can be concluded that R is symmetric as R = RT, using the symmetric Onsager’s kinetic coefficients L and the symmetric S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Moreover, R is a positive definite matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' This is confirmed by noticing that R in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (20) can be written as R = (S−1)TL−1S−1, where we used S−1 = (S−1)T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' This shows that L−1 and R are congruent, and because L−1 is positive definite, R is also positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Equations (18)–(20) are consistent with the linear response theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' We can show � ˆxˆxT� eq = kBS−1, which implies that χ is calculated using the equilibrium correlation function of ˆx as χ = � ˆxˆxT� eq /kB, where ⟨·⟩eq is taken with respect to Peq(ˆx) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Moreover, in the linear response theory, R can also be calculated using the time integral of the temporal equilibrium correlation function: R = 1 kB � ∞ 0 � ˆx(t)ˆx(0)T� dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (21) This can be easily shown by substituting the explicit solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (11) ˆx(t) = e−Atˆx(0) + � t 0 e−A(t−t′)ξ(t′)dt′ (22) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Thus, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (20) shows the detailed constituents of R for the fluctuations of thermodynamic variables governed by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Dissipated availability The dissipated availability Adiss, which quantifies the ef- ficiency of irreversible thermodynamic processes [6], is for- mulated by using the entropy variation δ2S (x) serving as the potential function of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' For a periodic thermodynamic force F(0) = F(1) with period TF, we can show (see Appendix A for the derivation) Adiss ≡ � TF 0 dδ2S (x) dt dt ≃ � TF 0 ˙FTR˙Fdt ≥ 0, (23) where the inequality holds by using the positive semi- definiteness of R shown above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Moreover, by using the Cauchy-Schwartz inequality, we can obtain the tighter bound for Adiss than Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (23): Adiss ≥ L2 TF ≡ A∗ diss, (24) where L ≡ � γF √ dFTRdF = � 1 0 � F′TRF′dT (25) is the thermodynamic length for the closed path γF on the control space of thermodynamics forces F with R the metric tensor defined on it [6, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' The equality of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (24), the min- imum dissipated availability A∗ diss, is achieved for a geodesic path that yields the constant dissipation such that the integrand of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (25) becomes constant [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' As R is the constant matrix evaluated at Xeq, this equality is expected to be realized for, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=', periodic F with linear profile symmetric in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' It is also noteworthy that Adiss in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (23) can also be ex- pressed in terms of ˙xs instead of ˙F: Adiss = � TF 0 ˙xT s L−1˙xsdt ≡ � TF 0 Φ (˙xs) dt ≥ 0, (26) where we used ˙F = S˙xs derived from the time derivative of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (16) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' The integrand Φ (˙xs) of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (26) is es- sentially the same quantity as the dissipation function [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' The inequality in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (26) is assured by the positive definiteness of L−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Moreover, when expressed in terms of the effective thermo- dynamic force y − F, we obtain Adiss = � TF 0 (ys − F)TL(ys − F)dt ≡ � TF 0 σ (ys) dt ≥ 0, (27) where we used Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (13) in the first equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' The integrand σ (ys) of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (27) is the “familiar” total entropy production rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' The inequality in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (27) is assured by the positive def- initeness of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' These apparently different but the equivalent expressions Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (23), (26), and (27) are informative, which will be used in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' IV B for numerical demonstration of the present theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' We should note that as what is controlling here is the intensive thermodynamic force F, the expression Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (23) using ˙F serves as the basic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' This may contrast to both the original formulation of the dissipated availability using the extensive variables [6] as control parameters and the recent formulation of thermodynamic geometry using exten- sive and intensive parameters [13] as control parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' EXAMPLE A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Ideal gas model As a demonstration of our theory, we identify the non- quasistatic response coefficients of a macroscopic phe- nomenological model of a thermodynamic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Our model is an ideal gas confined in a cylinder with a movable pis- ton subject to an external mechanical force f(t/TF) in con- tact with a heat bath at changeable temperature TR(t/TF) by a heater (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Let X = (U, V) (n = 2) be the exten- sive thermodynamic variables of the ideal gas, where U = CVT = fdNkBT/2 is the internal energy with the constant- volume heat capacity CV, the temperature of the gas T, and 4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Schematic illustration of a thermodynamic system under con- sideration: The ideal gas is confined in the cylinder enclosed by the insulated walls with a movable piston on one side to which the ex- ternal mechanical force f is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Further, it is in thermal contact with a heat bath at changeable temperature TR by a heater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' The sys- tem and the heat bath exchange their extensive quantities, internal energy and volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' the internal degrees of freedom fd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Further, V is the volume of the ideal gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Under an assumption of spatial uniformity, macroscopic dynamics for X is given by dU dt = κ(TR(t/TR) − T) − f(t/TF) σ dV dt , (28) dV dt = σ2 Γ �NkBT V − f(t/TF) σ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (29) Equation (28) is the first law of thermodynamics (the energy conservation law) per unit time, where the first term on the right-hand side denotes the Fourier’s law for the heat flow with κ the thermal conductance and the second term does the work flow done by the gas against the mechanical force f(t/TF) acting on the piston in one direction, where σ is the section area of the piston.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Equation (29) is the equation of motion for an overdamped piston whose inertia can be neglected, where Γ is the friction coefficient of the piston.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' The piston moves back and forth subject to the net force due to the internal gas pressure NkBT/V and the mechanical pressure f(t/TF)/σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' For the fixed temperature and mechanical force TR(t/TF) = Teq and f(t/TF) = feq, respectively, the stationary state Xeq = (Ueq, Veq) = (CVTeq, σNkBTeq/feq) satisfying dX/dt = 0 cor- responds to the equilibrium state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Equations (28) and (29) around the equilibrium state Xeq can be rewritten into the form Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' First, consider the case of fixed temperature and mechanical force TR(t/TF) = Teq and f(t/TF) = feq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' By expanding Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (28) and (29) in terms of x around Xeq, dx/dt = −Ax is obtained, neglect- ing the higher order terms of x, where A11 = κ CV + feq σ Γ NkB CVVeq , A12 = −feq σ Γ NkBTeq V2eq , A21 = − σ2 Γ NkB CVVeq , and A22 = σ2 Γ NkBTeq V2eq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' The Einstein’s fluctuation formula for the corresponding fluctua- tion ˆx = δ ˆX = (δ ˆU, δ ˆV) is given as [24] δ2S (ˆx) = −1 2 \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 1 CVT 2eq δ ˆU2 + NkB V2eq δ ˆV2 \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb , (30) from which S11 = 1 CVT 2eq , S12 = S21 = 0, and S22 = NkB V2eq are identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Then, we obtain the quasistatic response coefficient χ = S−1 as χ = \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed CVT 2 eq 0 0 V2 eq NkB \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (31) We can also obtain the Onsager’s kinetic coefficients L = AS−1 as L = \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed κT 2 eq + f 2 eq Γ Teq −feq σ Γ Teq −feq σ Γ Teq σ2 Γ Teq \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (32) Next, we consider the effect of time-dependent temperature of the heat bath TR(t/TF) = Teq + δTR(t/TF) and mechanical force f(t/TF) = feq + δf(t/TF), where |δTR(t/TF)| ≪ Teq and |δf(t/TF)| ≪ feq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' By expanding Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (28) and (29) up to O(x, δTR, δf), we obtain dx dt = −Ax + � κδTR(t/TF) + feq Γ δf(t/TF) − σ Γ δf(t/TF) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (33) Through comparisons of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (33) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (12), the external thermodynamic force F(t/TF) acting on x is identified as F(t/TF) = \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed δTR(t/TF) T 2eq δpR(t/TF) Teq − δ f(t/TF)/σ Teq \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 , (34) where δpR(t/TF) ≡ NkBδTR(t/TF)/Veq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Here, F2(t/TF) de- notes the net force increment acting on the piston that arises owing to the combined effect of the change of the gas pressure upon the temperature change of the heat bath and the change of the external mechanical force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (20), (31), and (32), the non-quasistatic response coefficient R is derived as R = \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed C2 VT 2 eq κ CVTeqVeq κ CVTeqVeq κ V2 eq κ + f 2 d ΓV4 eq 4σ2C2 VTeq \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (35) Through explicit writing, the following is obtained as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (18): δUs(T ) = χ11F1(T ) − ǫR11F′ 1(T ) − ǫR12F′ 2(T ), (36) δVs(T ) = χ22F2(T ) − ǫR21F′ 1(T ) − ǫR22F′ 2(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (37) Note that while the non-diagonal components of χ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (31) identically vanish, the non-diagonal components of R in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (35) are non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Therefore, the cross effect between δU and δV appears as a consequence of a nonequilibrium control of the external thermodynamic forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Numerical demonstration The calculations in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' IV A can be confirmed numeri- cally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' For this purpose, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (28) and (29) are made nondi- mensional as follows: d ˜U d˜t = ˜TR(ǫ˜t) − ˜U − ˜f(ǫ˜t)d ˜V d˜t , (38) d ˜V d˜t = 1 ˜Γ � 2 ˜U fd ˜V − ˜f (ǫ˜t) � , (39) 5 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Summary for nondimensionalized quantities Time t ˜t = t/ts with ts = CV/κ Internal energy U ˜U = U/Ueq = U/(CVTeq) Volume V ˜V = V/Veq Temperature T, TR ˜T = T/Teq, ˜TR = TR/Teq Friction coefficient Γ ˜Γ = Γ/(σ2C2 VTeq/κV2 eq) Mechanical force f ˜f = f /(σCVTeq/Veq) TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Summary for nondimensionalized F, χ, and R F1(T ) ˜F1(T ) = TeqF1(T ) = δ ˜TR(T ) F2(T ) ˜F2(T ) = Veq CV F2(T ) = 2δ ˜TR(T )/fd − δ ˜f(T ) χ11 ˜χ11 = 1 χ22 ˜χ22 = fd/2 R11 ˜R11 = 1 R12 ˜R12 = 1 R21 ˜R21 = 1 R22 ˜R22 = 1 + f 2 d ˜Γ/4 respectively, where we put T = U/CV on the right-hand sides of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (28) and (29) before the nondimensionalization and the nondimensionalized quantities are summarized in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' The quantities with a tilde denote nondimensionalized quan- tities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Then, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (36) and (37) are also nondimensionalized as δ ˜Us(T ) = ˜χ11 ˜F1(T ) − ǫ ˜R11 ˜F′ 1(T ) − ǫ ˜R12 ˜F′ 2(T ), (40) δ ˜Vs(T ) = ˜χ22 ˜F2(T ) − ǫ ˜R21 ˜F′ 1(T ) − ǫ ˜R22 ˜F′ 2(T ), (41) respectively, where the nondimensionalized F, χ, and R are summarized in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' 2, the theoretical results Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (40) and (41) for T = 1 are compared with the numerical results obtained by solving Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (38) and (39) numerically with the following linear pro- file as δ ˜TR(T ) and δ ˜f (T ): δ ˜TR(T ) = ∆ ˜TT (0 ≤ T ≤ 1), (42) δ ˜f(T ) = ∆ ˜f T (0 ≤ T ≤ 1), (43) where ∆ ˜T ≪ 1 and ∆ ˜f ≪ 1 are small increments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' As evident, for sufficiently small ǫ, the numerical results are consistent with the theoretical results as expected, with the discrepan- cies observed with an increase in ǫ where the nonlinear effects begin to appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Finally, we demonstrate the dissipated availability Adiss in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' It is nondimensionalized as ˜A∗ diss = A∗ diss CV = ǫ �� 1 0 � ˜F′T ˜R˜F′dT �2 = ǫ ˜L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (44) As we know from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (27) that the dissipated availability is equivalent to the total entropy production for small ǫ, we in- troduce the entropy production of the total system (system and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content='0007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content='0008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content='0009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content='001 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content='3 Numerical Theory 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content='0002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content='0003 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content='3 Numerical Theory 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content='0003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content='0002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content='0001 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content='3 Numerical Theory 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content='0015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content='0012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content='0009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content='0006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content='0003 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content='3 Numerical Theory (a) (b) (c) (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (a) and (b): ǫ dependence of δ ˜Us(1) for (a) ˜F′ 2(1) = 0 and (b) ˜F′ 1(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' The theory denotes Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (40) with T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (c) and (d): ǫ dependence of δ ˜Vs(1) for (c) ˜F′ 2(1) = 0 and (d) ˜F′ 1(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' The theory denotes Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (41) with T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' As parameter values, fd = 3, ˜Γ = 1, ∆ ˜T = 10−3, and ∆ ˜f = 10−3 were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' The fourth Runge–Kutta method with time step 10−3 was used for obtaining the numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' 0 2x10-7 4x10-7 6x10-7 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content='15 Numerical (linear) Numerical (quadratic) Theory FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' ǫ dependence of the total entropy production over one cycle ∆ ˜S tot in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (46) calculated for Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (47) and (48) (bold solid line) and for Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (49) and (50) (thin solid line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' The theory denotes the theoretical minimum dissipated availability ˜A∗ diss in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' The same parameter values and numerical scheme as those in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' 2 were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' bath) over one cycle for comparison with Adiss: ∆S tot ≡ − � TF 0 κ(TR(t/TF) − T(t)) TR(t/TF) dt, = −κ � TF 0 � 1 − T(t) TR(t/TF) � dt, (45) which coincides with the entropy change of the heat bath as the system’s state returns to the initial state after one cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Its nondimensionalized form reads ∆ ˜S tot = ∆S tot CV = − � 1/ǫ 0 � 1 − ˜T(˜t) ˜TR(ǫ˜t) � d˜t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (46) For the demonstration, we use F with the following periodic linear profile as δ ˜TR(T ) and δ ˜f(T ) as the geodesic path on the 6 control space: δ ˜TR(T ) = \uf8f1\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f3 2∆ ˜TT (0 ≤ T ≤ 1/2), 2∆ ˜T(1 − T ) (1/2 < T ≤ 1), (47) δ ˜f (T ) = \uf8f1\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f3 2∆ ˜fT (0 ≤ T ≤ 1/2), 2∆ ˜f(1 − T ) (1/2 < T ≤ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (48) For comparison, we also use F with the following periodic quadratic profile as the non-geodesic path: δ ˜TR(T ) = \uf8f1\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f3 4∆ ˜TT 2 (0 ≤ T ≤ 1/2), 4∆ ˜T(1 − T )2 (1/2 < T ≤ 1), (49) δ ˜f(T ) = \uf8f1\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f3 4∆ ˜f T 2 (0 ≤ T ≤ 1/2), 4∆ ˜f (1 − T )2 (1/2 < T ≤ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (50) As we noted in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' III B, the minimum dissipated availabil- ity ˜A∗ diss is expected to be achieved for periodic linear pro- files such as Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (47) and (48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Explicitly, ˜A∗ diss calculated for Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (47) and (48) is given as ˜A∗ diss = ǫ � 4∆ ˜T 2 + 4∆ ˜T �4∆ ˜T fd − 2∆ ˜f � + \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed1 + f 2 d ˜Γ 4 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 �4∆ ˜T fd − 2∆ ˜f �2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (51) In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' 3, we show the total entropy production ∆ ˜S tot nu- merically calculated for the periodic linear profile Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (47) and (48) and for the periodic quadratic profile Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (49) and (50), with a comparison with the minimum dissipated avail- ability ˜A∗ diss in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' We can confirm that ∆ ˜S tot for the periodic linear profile as the geodesic path agrees with ˜A∗ diss in the small ǫ region as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Meanwhile, we can also con- firm that ∆ ˜S tot for the periodic quadratic profile as the non- geodesic path is larger than ˜A∗ diss in the small ǫ region, which is consistent with the theoretical prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' SUMMARY AND DISCUSSION In summary, we derived a simple formula for the non- quasistatic response coefficients for macroscopic thermody- namic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' The formula revealed the general relation- ship between the non-quasistatic response coefficients and Onsager’s kinetic coefficients that govern the relaxation dy- namics of fluctuations of semi-macroscopic thermodynamic variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Similarly to the quasistatic response coefficients whose symmetry has been already established in equilibrium thermodynamics, the non-quasistatic response coefficients are also symmetric as R = RT, which is related to the symmetry of Onsager’s kinetic coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Moreover, we also formu- lated the dissipated availability Adiss for the present system that quantifies the efficiency of irreversible thermodynamic processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' It is expressed in terms of the time derivative of the entropy variation, and the equivalent expressions using the dissipation function or the total entropy production were pro- vided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Our theory was demonstrated by using the ideal gas model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' It is expected to measure the predicted symmetry of R and the dissipated availability Adiss in realistic nonequilibrium ex- periments such as those illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' 1 where Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (28) and (29) serve as a good approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Real systems may be accompanied by complicated fluid motion as well as the non- uniform heat conduction inside the system associated with a piston motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' However, from recent experiments on real heat engines using a gas [26, 27], it can be concluded that simple models similar to those presented in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (28) and (29) explain the experimental results to a sufficiently good approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Therefore, experimental verification will be an important di- rection for future investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work was supported by JSPS KAKENHI Grant Num- bers 19K03651 and 22K03450.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' Appendix A: Derivation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (23) First, we evaluate the time derivative of the entropy varia- tion dδ2S (x)/dt: dδ2S (x) dt = −(Sx)T˙x = −yT˙x = yTL(y − F), (A1) where we used Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (13) in the third equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' By applying S to both sides of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (18), we can approximate y up to O(ǫ) as y ≃ ys ≡ Sxs = F − ǫSRF′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (A2) By substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (A2) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (A1), we obtain dδ2S (x) dt ≃ −ǫFTS−1F′ + ǫ2F′TRF′ = −FTS−1 ˙F + ˙FTR˙F = − d dt �1 2FTS−1F � + ˙FTR˙F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (A3) The first term vanishes by the time-integration of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFLT4oBgHgl3EQfki_9/content/2301.12116v1.pdf'} +page_content=' (A3) from t = 0 to t = TF and using F(0) = F(1) for the periodic 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